A Visual, Intuitive Guide to Imaginary Numbers

Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories:

  • It’s a mathematical abstraction, and the equations work out. Deal with it.
  • It’s used in advanced physics, trust us. Just wait until college.

Gee, what a great way to encourage math in kids! Today we’ll assault this topic with our favorite tools:

And our secret weapon: learning by analogy. We’ll approach imaginary numbers by observing its ancestor, the negatives. Here’s your guidebook:

imaginary number properties

It doesn’t make sense yet, but hang in there. By the end we’ll hunt down i and put it in a headlock, instead of the reverse.

Video Walkthrough:

Really Understanding Negative Numbers

Negative numbers aren’t easy. Imagine you’re a European mathematician in the 1700s. You have 3 and 4, and know you can write 4 – 3 = 1. Simple.

But what about 3-4? What, exactly, does that mean? How can you take 4 cows from 3? How could you have less than nothing?

Negatives were considered absurd, something that “darkened the very whole doctrines of the equations” (Francis Maseres, 1759). Yet today, it’d be absurd to think negatives aren’t logical or useful. Try asking your teacher whether negatives corrupt the very foundations of math.

What happened? We invented a theoretical number that had useful properties. Negatives aren’t something we can touch or hold, but they describe certain relationships well (like debt). It was a useful fiction.

Rather than saying “I owe you 30” and reading words to see if I’m up or down, I can write “-30” and know it means I’m in the hole. If I earn money and pay my debts (-30 + 100 = 70), I can record the transaction easily. I have +70 afterwards, which means I’m in the clear.

The positive and negative signs automatically keep track of the direction — you don’t need a sentence to describe the impact of each transaction. Math became easier, more elegant. It didn’t matter if negatives were “tangible” — they had useful properties, and we used them until they became everyday items. Today you’d call someone obscene names if they didn’t “get” negatives.

But let’s not be smug about the struggle: negative numbers were a huge mental shift. Even Euler, the genius who discovered e and much more, didn’t understand negatives as we do today. They were considered “meaningless” results (he later made up for this in style).

It’s a testament to our mental potential that today’s children are expected to understand ideas that once confounded ancient mathematicians.

Enter Imaginary Numbers

Imaginary numbers have a similar story. We can solve equations like this all day long:

\displaystyle{x^2 = 9}

The answers are 3 and -3. But suppose some wiseguy puts in a teensy, tiny minus sign:

\displaystyle{x^2 = -9}

Uh oh. This question makes most people cringe the first time they see it. You want the square root of a number less than zero? That’s absurd! (Historically, there were real questions to answer, but I like to imagine a wiseguy.)

It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must have seemed crazy at first. There’s no “real” meaning to this question, right?

Wrong. So-called “imaginary numbers” are as normal as every other number (or just as fake): they’re a tool to describe the world. In the same spirit of assuming -1, .3, and 0 “exist”, let’s assume some number i exists where:

\displaystyle{i^2 = -1}

That is, you multiply i by itself to get -1. What happens now?

Well, first we get a headache. But playing the “Let’s pretend i exists” game actually makes math easier and more elegant. New relationships emerge that we can describe with ease.

You may not believe in i, just like those fuddy old mathematicians didn’t believe in -1. New, brain-twisting concepts are hard and they don’t make sense immediately, even for Euler. But as the negatives showed us, strange concepts can still be useful.

I dislike the term “imaginary number” — it was considered an insult, a slur, designed to hurt i‘s feelings. The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it.

Visual Understanding of Negative and Complex Numbers

As we saw last time, the equation x^2 = 9 really means:

\displaystyle{1 \cdot x^2 = 9}


\displaystyle{1 \cdot x \cdot x = 9}

What transformation x, when applied twice, turns 1 to 9?

The two answers are “x = 3” and “x = -3”: That is, you can “scale by” 3 or “scale by 3 and flip” (flipping or taking the opposite is one interpretation of multiplying by a negative).

Now let’s think about x^2 = -1, which is really

\displaystyle{1 \cdot x \cdot x = -1}

What transformation x, when applied twice, turns 1 into -1? Hrm.

  • We can’t multiply by a positive twice, because the result stays positive
  • We can’t multiply by a negative twice, because the result will flip back to positive on the second multiplication

But what about… a rotation! It sounds crazy, but if we imagine x being a “rotation of 90 degrees”, then applying x twice will be a 180 degree rotation, or a flip from 1 to -1!

Imaginary Number Rotation

Yowza! And if we think about it more, we could rotate twice in the other direction (clockwise) to turn 1 into -1. This is “negative” rotation or a multiplication by -i:

Negative Rotation

If we multiply by -i twice, the first multiplication would turn 1 into -i, and the second turns -i into -1. So there’s really two square roots of -1: i and -i.

This is pretty cool. We have some sort of answer, but what does it mean?

  • i is a “new imaginary dimension” to measure a number
  • i (or -i) is what numbers “become” when rotated
  • Multiplying i is a rotation by 90 degrees counter-clockwise
  • Multiplying by -i is a rotation of 90 degrees clockwise
  • Two rotations in either direction is -1: it brings us back into the “regular” dimensions of positive and negative numbers.

Numbers are 2-dimensional. Yes, it’s mind bending, just like decimals or long division would be mind-bending to an ancient Roman. (What do you mean there’s a number between 1 and 2?). It’s a strange, new way to think about math.

We asked “How do we turn 1 into -1 in two steps?” and found an answer: rotate it 90 degrees. It’s a strange, new way to think about math. But it’s useful. (By the way, this geometric interpretation of complex numbers didn’t arrive until decades after i was discovered).

Also, keep in mind that having counter-clockwise be positive is a human convention — it easily could have been the other way.

Finding Patterns

Let’s dive into the details a bit. When multiplying negative numbers (like -1), you get a pattern:

  • 1, -1, 1, -1, 1, -1, 1, -1

Since -1 doesn’t change the size of a number, just the sign, you flip back and forth. For some number “x”, you’d get:

  • x, -x, x, -x, x, -x…

This idea is useful. The number “x” can represent a good or bad hair week. Suppose weeks alternate between good and bad; this is a good week; what will it be like in 47 weeks?

\displaystyle{x \cdot (-1)^{47} = x \cdot -1 = -x}

So -x means a bad hair week. Notice how negative numbers “keep track of the sign” — we can throw (-1)^47 into a calculator without having to count (”Week 1 is good, week 2 is bad… week 3 is good…“). Things that flip back and forth can be modeled well with negative numbers.

Ok. Now what happens if we keep multiplying by i?

\displaystyle{1, i, i^2, i^3, i^4, i^5}

Very funny. Let’s reduce this a bit:

  • \displaystyle{1 = 1} (No questions here)
  • \displaystyle{i = i} (Can’t do much)
  • \displaystyle{i^2 = -1} (That’s what i is all about)
  • \displaystyle{i^3 = (i \cdot i) \cdot i = -1 \cdot i = -i} (Ah, 3 rotations counter-clockwise = 1 rotation clockwise. Neat.)
  • \displaystyle{i^4 = (i \cdot i) \cdot (i \cdot i) = -1 \cdot -1 = 1} (4 rotations bring us “full circle”)
  • \displaystyle{i^5 = i^4 \cdot i = 1 \cdot i = i} (Here we go again…)

Represented visually:

imaginary number cycle

We cycle every 4th rotation. This makes sense, right? Any kid can tell you that 4 left turns is the same as no turns at all. Now rather than focusing on imaginary numbers (i, i^2), look at the general pattern:

  • X, Y, -X, -Y, X, Y, -X, -Y…

Like negative numbers modeling flipping, imaginary numbers can model anything that rotates between two dimensions “X” and “Y”. Or anything with a cyclic, circular relationship — have anything in mind?

‘Cos it’d be a sin if you didn’t. There’ll de Moivre be more in future articles. [Editor’s note: Kalid is in electroshock therapy to treat his pun addiction.]

Understanding Complex Numbers

There’s another detail to cover: can a number be both “real” and “imaginary”?

You bet. Who says we have to rotate the entire 90 degrees? If we keep 1 foot in the “real” dimension and another in the imaginary one, it looks like this:

imaginary number i plus i

We’re at a 45 degree angle, with equal parts in the real and imaginary (1 + i). It’s like a hotdog with both mustard and ketchup — who says you need to choose?

In fact, we can pick any combination of real and imaginary numbers and make a triangle. The angle becomes the “angle of rotation”. A complex number is the fancy name for numbers with both real and imaginary parts. They’re written a + bi, where

  • a is the real part
  • b is the imaginary part

imaginary number a plus bi

Not too bad. But there’s one last question: how “big” is a complex number? We can’t measure the real part or imaginary parts in isolation, because that would miss the big picture.

Let’s step back. The size of a negative number is not whether you can count it — it’s the distance from zero. In the case of negatives this is:

\displaystyle{\text{Size of } \ -x = \sqrt{(-x)^2} = |x|}

Which is another way to find the absolute value. But for complex numbers, how do we measure two components at 90 degree angles?

It’s a bird… it’s a plane… it’s Pythagoras!

Geez, his theorem shows up everywhere, even in numbers invented 2000 years after his time. Yes, we are making a triangle of sorts, and the hypotenuse is the distance from zero:

\displaystyle{\text{Size of } \ a + bi = \sqrt{a^2 + b^2}}

Neat. While measuring the size isn’t as easy as “dropping the negative sign”, complex numbers do have their uses. Let’s take a look.

A Real Example: Rotations

We’re not going to wait until college physics to use imaginary numbers. Let’s try them out today. There’s much more to say about complex multiplication, but keep this in mind:

Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for every 4 units North. I want to change my heading 45 degrees counter-clockwise. What’s the new heading?

imaginary number example

Some hotshot will say “That’s simple! Just take the sine, cosine, gobbledegook by the tangent… fluxsom the foobar… and…“. Crack. Sorry, did I break your calculator? Care to answer that question again?

Let’s try a simpler approach: we’re on a heading of 3 + 4i (whatever that angle is; we don’t really care), and want to rotate by 45 degrees. Well, 45 degrees is 1 + i (perfect diagonal), so we can multiply by that amount!

imaginary number example

Here’s the idea:

  • Original heading: 3 units East, 4 units North = 3 + 4i
  • Rotate counter-clockwise by 45 degrees = multiply by 1 + i

If we multiply them together we get:

(3 + 4i) \cdot (1 + i) &= 3 + 3i + 4i + 4i^2 \\ &= 3 + 7i \hspace{8mm} + 4(-1) \\ &= -1 + 7i

So our new orientation is 1 unit West (-1 East), and 7 units North, which you could draw out and follow.

But yowza! We found that out in 10 seconds, without touching sine or cosine. There were no vectors, matrices, or keeping track what quadrant we are in. It was just arithmetic with a touch of algebra to cross-multiply. Imaginary numbers have the rotation rules baked in: it just works.

Even better, the result is useful. We have a heading (-1, 7) instead of an angle (atan(7/-1) = 98.13, keeping in mind we’re in quadrant 2). How, exactly, were you planning on drawing and following that angle? With the protractor you keep around?

No, you’d convert it into cosine and sine (-.14 and .99), find a reasonable ratio between them (about 1 to 7), and sketch out the triangle. Complex numbers beat you to it, instantly, accurately, and without a calculator.

If you’re like me, you’ll find this use mind-blowing. And if you don’t, well, I’m afraid math doesn’t toot your horn. Sorry.

Trigonometry is great, but complex numbers can make ugly calculations simple (like calculating cosine(a+b) ). This is just a preview; later articles will give you the full meal.

Aside: Some people think “Hey, it’s not useful to have North/East headings instead of a degree angle to follow!”

Really? Ok, look at your right hand. What’s the angle from the bottom of your pinky to the top of your index finger? Good luck figuring that out on your own.

With a heading, you can at least say “Oh, it’s X inches across and Y inches up” and have some chance of working with that bearing.

Complex Numbers Aren’t

That was a whirlwind tour of my basic insights. Take a look at the first chart — it should make sense now.

There’s so much more to these beautiful, zany numbers, but my brain is tired. My goals were simple:

  • Convince you that complex numbers were considered “crazy” but can be useful (just like negative numbers were)
  • Show how complex numbers can make certain problems easier, like rotations

If I seem hot and bothered about this topic, there’s a reason. Imaginary numbers have been a bee in my bonnet for years — the lack of an intuitive insight frustrated me.

Now that I’ve finally had insights, I’m bursting to share them. But it frustrates me that you’re reading this on the blog of a wild-eyed lunatic, and not in a classroom. We suffocate our questions and “chug through” — because we don’t search for and share clean, intuitive insights. Egad.

But better to light a candle than curse the darkness: here’s my thoughts, and one of you will shine a spotlight. Thinking we’ve “figured out” a topic like numbers is what keeps us in Roman Numeral land.

There’s much more complex numbers: check out the details of complex arithmetic. Happy math.

Epilogue: But they’re still strange!

I know, they’re still strange to me too. I try to put myself in the mind of the first person to discover zero.

Zero is such a weird idea, having “something” represent “nothing”, and it eluded the Romans. Complex numbers are similar — it’s a new way of thinking. But both zero and complex numbers make math much easier. If we never adopted strange, new number systems, we’d still be counting on our fingers.

I repeat this analogy because it’s so easy to start thinking that complex numbers aren’t “normal”. Let’s keep our mind open: in the future they’ll chuckle that complex numbers were once distrusted, even until the 2000’s.

If you want more nitty-gritty, check out wikipedia, the Dr. Math discussion, or another argument on why imaginary numbers exist.

Other Posts In This Series

  1. A Visual, Intuitive Guide to Imaginary Numbers
  2. Intuitive Guide to Angles, Degrees and Radians
  3. Intuitive Arithmetic With Complex Numbers
  4. Understanding Why Complex Multiplication Works
  5. Intuitive Understanding Of Euler's Formula
  6. An Interactive Guide To The Fourier Transform
  7. Intuitive Understanding of Sine Waves
  8. An Intuitive Guide to Linear Algebra

Questions & Contributions


  1. I enjoy reading your intuitive approach to math and hadn’t really considered “i” normal until recently. (My favorite math formula contains all sorts of “non-existent” numbers — e^(pi*i)=-1.)

    Anyway, need to point out a simple error in your article. (-1)^48 is 1, not -1. It’s a small issue, but didn’t want others to be confused.

    Happy Holidays,
    . John

  2. I must thank you for this wonderful site. It has opened up my eyes to many things that I knew how to use, but never truly understood. This article in particular made me say, “HOLY CRAP! That’s freaking awesome!” Thank you very much for your work, and please keep it up.

  3. Nice article, but I always found the “best” way to understand math is by its history, especially how mathematical idea came into being. No one actually wanted to solve

    x^2 = -9

    , nor want to “take the square root of nothing”. But in the 1500s, Bombelli wanted to use one of Cardano’s formula to solve

    x^3 = 15x + 4

    , and get

    x = cuberoot(2 + sqrt(-121)) + cuberoot(2 – sqrt(–121))

    After figuring that
    cuberoot(2 + sqrt(–121)) = 2 + sqrt(–1)
    cuberoot(2 – sqrt(–121)) = 2 – sqrt(–1)
    , he found the real solution
    x = 4
    The idea was that this number sqrt(-1) was actually useful!
    And yeah, everyone should also see the (simple) proof of Euler’s formula. It is Euler’s formula that links trigonometry to arithmetic (and allows for a geometric interpretation of complex numbers as a result).

  4. @John: Thanks for the catch, I just fixed it. I’m a big fan of the e^i*pi = -1 formula as well.

    @JB: Thanks! Yes, I had a similar “wow” moment and just wanted to share it. There are so many things we think we “know” (because we learned them a decade ago), but never bother to revisit with a fresh set of eyes. I’ll keep the articles coming.

    @Bryan: I agree — I needed to see the diagram before it clicked. I don’t know why it’s not taught visually either — it makes students think imaginaries are entirely made up and unintuitive.

    @Chick: Thanks for the background info! I had to plug in the numbers myself to see myself:

    (2 + i)^3 = 2 + 11i = 2 + sqrt(-121)

    There’s more details here as well:


  5. Actually, your rotation calculation is wrong, depends on what you really mean by “heading”. If you only want to rotate by 45 and not to scale, you have to multiply by a complex value with length 1. 1+i has length sqrt(2) so the final answer is thus -1/sqrt(2)+7i/sqrt(2).

  6. Yeah, I wanted to leave out the discussion of scaling until the next article. The meaning of heading was just the “angle”, so the scale shouldn’t matter in this case. Also, a triangle of sides 1/sqrt(2) + 7i/sqrt(2) is hard to draw :)

  7. Thanks Chaz! Yeah, if negatives are “mirror images”, then complex numbers are “rotations”. I wish I had been taught that analogy first, instead of some arcane symbols which *later* get shown to have a geometric interpretation. We’re visual creatures! :)

    Thanks for the link, I’ll have to check it out.

  8. A very nice explanation, I’ve never thought of it that way before. So how would you describe an x,y plot where both x and y are complex? (I’m not trying to be a smart a, I’m sincerely curious). Or maybe the question should be if you add rotation to an x,y co-ordinates you then get something else (quaternions?).


  9. @Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space. They are not quaternion, however, although both are 4 dimensional. Quaternion have three imaginary axes i,j, and k; with non-commutative multiplication. But they are actually used in your favorite FPS games: Halo, Doom, etc., in, 3D rotation. Surprise!

    And then here’s the octonions with seven imaginary axes and non-associative multiplication…

  10. George,

    I loved your book but you never answered the question posed by the title.

    As far as analogies are concerned, thinking about imaginary numbers as rotation is a good start but I think periodicity goes deeper. Your book kind of touched on that in the e^i*pi = -1 section.

  11. @George: Actually, I haven’t read that book — all analogies and mistakes came from my brain :). I’m a fan of using analogies to understand difficult topics, and they’ll continue appearing in my articles.

    @Chick: Thanks for the details, I’m not familiar with quaternion but am looking forward to learning.

    @bayareaguy: Yep, the rotation analogies go much deeper with Euler’s formula. But all that would be too much for one sitting :). It’ll be in a future article.

  12. hi,
    suppose x^2=a
    then x can have two values sqrt(a) or – sqrt(a)
    will this same rule not apply to imaginary no.s??

    ie. i^2 can be equal to + or – 1

    i= sqrt(-1)

    i^2 = sqrt(-1)* sqrt(-1)
    = sqrt( -1 * -1 )
    = sqrt(1)
    = 1

  13. I never had imaginary numbers in school, but I think I can deal with them now, thanks to your explanation :).

    By the way, there’s a word missing here:

    what confounded ancient mathematicians DIDN’T.

  14. @abc: Actually, it’s the other way around: if you have x^2 = a, then sqrt(a) is either +x or -x. For example, sqrt(9) is either +3 or -3.

    So, there are 2 values of sqrt(-1): +i and -i. There’s only one value of i^2, which is -1. (Just like there is only one value of -3^2, which is -9).

    @Robin: Thanks, glad you found it useful! Yes, imaginary numbers are weird at first but I’m getting a handle on them also. Btw, I also fixed up the sentence to be more clear.

  15. @Kalid: Sorry, the sentence was right. I just didn’t know the verb “to confound”, so I thought “confounded” was an adjective meaning something like “wise” :).

  16. I prefer seeing Euler’s equation as

    e^(i*pi) + 1 = 0

    because then it brings together FIVE (5) really special numbers in one equation.

    Good post. I learned about i as a rotational operator, and I’m surprised that it isn’t taught that way (in addition to “follow the math” ways).

  17. Thanks Burton, glad you liked it. I like that representation of Euler’s formula also — I’m gearing up to cover it in an upcoming post (first we need a bit more on e and imaginary numbers :) ).

    I too am shocked that the “rotation” analogy wasn’t shown when I originally learned about i (in high school). For a long time I thought “i” was just an artificial abstraction used to fill in a gap in our number system (“Well, we need *something* to be the square root of -1, so let’s just stick i in there.”).

  18. This is a nice presentation — thanks for creating it!

    I’ve successfully used this analogy to explain complex numbers to my children, and a few other elementary schoolers. I’m not sure where I first saw the seeds of it, but I recommend Hestenes’ Oersted lecture on geometric algebra for how to extend this idea and where to find lots more like this.

    One thing that comes up is that a sensible answer to “What transformation x, when applied twice, turns 1 into -1?” is to subtract one. If you point out that this doesn’t work on 2, the child may reply that the answer is to subtract x, whatever it is, twice. I’ve had to clarify at this point that a transformation can only look at its input, which is a single number. The problem with “subtract again” is that it doesn’t know whether the zero that it gets after one transformation came from 1 or from 2, and it’s not allowed to remember where it started from. The problem is to find an instruction that two different people could do (in series), without sharing any information except for the intermediate number.

    It’s also helpful, when explaining this in person and giving the student a chance to come up with the answer, to rotate a pencil *out* of (perpendicular to) the page or to rotate your arm out of the blackboard, and then back into the plane in the negative direction. This doesn’t give the answer away as much as showing a rotation within the plane, but it’s a nice intermediate clue that “primes the pump” for the explicit explanation, and also adds a somatic modality.

  19. Mind blown here, genius way to describe imaginary numbers visually and to actually use it in real life situation without using fancy methods like sine and cosine. Thanks

  20. @Oliver: Thanks for the insightful comment! I really like that way of looking at it: you need to do *something* twice, and you can’t tell different types of “zero” apart (1-1 or 2-2). Giving hints like rotating the pencil out of the paper is a nice trick as well. I think kids would be able to pick up on these ideas (better than adults even!) and it’s cool you are introducing it to your children.

    @Darius: You’re welcome, I’m glad you found it useful. There are “everyday” uses of imaginary numbers, but nobody seems to talk about them!

  21. Well, just an idea to discuss on: now we could think about a+bi+cj numbers :) Or we could think about four-dimension numbers too: a+bi+cj+dk
    And so on…

  22. Well Done! I share share your frustration at the fact that most high school mathematics courses do not explain complex numbers adequately.

  23. @Alessio: Thanks for the suggestion. Yes, I want to learn more about quaternions, imaginary numbers extended to more dimensions :).

    @Ivan: Thanks, glad you liked the article.

  24. I’m glad to hear you’ll be covering Euler’s equation again in an upcoming post (I haven’t checked back until now). On the same topic, I thought you might want to check out some installation art i did on the topic a couple of years ago.

  25. Hi Burton, thanks for dropping by — I like the message on that art :). Yeah, I want to cover Euler’s equation, but would like to lay a bit of groundwork (more about e & pi) to help it really sink in.

    Also, I like what you said about math being a language that is self-describing to some extent; you can communicate with others *and* discover new ideas by using it.

  26. Hi Kalid,

    Yes, I agree with the others: nice job on this page!

    One comment just for fun: Did you know that engineers (at least electrical engineers) use “j” instead of “i” to denote sqrt(-1)? We need to reserve “i” for electrical current (very important!). BTW, electrical engineering makes very *heavy* use of complex math. So “our” version of Euler’s equation is e^(j*pi)+1=0. It’s only a difference in the use of a symbol, but I think it’s a rather interesting “cultural” difference to know about.


  27. Thanks Peyton, glad you liked it! Yes, those “cultural differences” (I like that phrase) are quite interesting. Another way to set off a cultural war is to ask what base “log” refers to (e, 10, or 2).

  28. Nice post. As a future maths teacher I found it very interesting.

    One minor point, though. When you say “complex numbers aren’t”, it’s not technically true. Complex refers to something made from more than one part (in this case the real and imaginary parts)Think of a complex of buildings.What you mean is that they aren’t complicated.

    Alessio: It may interest you to know that the 3 dimensional system you suggest, a+bi+cj, has been proven not to work. The 4 dimensional system, a+bi+cj+dk, only works if you remove the insistence on associativity,ie in the quaternions a*b=b*a doesn’t hold in the general case.

  29. Great explanation. Byt it begs the next question.

    Negative numbers complete the “real” numbers in a one-dimensional number line. Imaginary nu,mbers open that out into a two dimensional complex number space. So what is in the three, four and higher dimesnional number spaces?

  30. Hi Alec, great question. There are ways to consider i, j and k to handle more degrees of rotation (called quaternions, I don’t have much experience with them). At this point, it’s probably easier to use linear algebra (matrices) to keep track of multi-dimensional data. Any set of x, y and z coordinates can be represented in a matrix, and other matrices can represent transformations like rotation and scaling.

  31. But why were imaginary numbers first used? I understand they have many uses today, but what were they used for in 1572 when they were first discovered?

  32. Actually, what makes to think of an imaginary axis. Where can be this imaginary number stuff be applied?

    I’m not getting the right image on how you were looking at the number.

  33. @maheshexp: Take a look at the example in the article — imaginary numbers help deal with rotations, without having to use trigonometry.

    In general, imaginary numbers are good for things that move in cycles (since i can be seen as rotations about a center point). In Physics and Electrical Engineering, imaginary numbers are used to describe electric current and other things that can have cyclical patterns. It can often make the math much easier.

  34. Khalid, now I could make a pretty difference between ‘Complex Number’ & Trigs.

    Trig -> Given your position or distance (eg: 4N,3E), what angle should you move from the current point.

    Complex Number -> Given an angle, solves what would be the new position

  35. Hi maheshexp, that’s an interesting observation. Yes, trig mostly deals with the raw angles, while complex numbers have you think about distances.

  36. i never understood math in school (geometry being the exception). but just recently it clicked, and now i think math is beautiful. i spend more time reading these sorts of pages than on actual schoolwork, and i don’t even have any math classes this semester!

  37. Thanks, Kalid (is that “Khalid”?) for making maths so easy. I’m 38, very interested in Science, Tech, Electronics, Computers et al but maths has always been my “Achilles’ heel”. I had an almost-aha moment when I was studying remedial maths at the South Tyneside College back in ’93. That was calculus.

    Sorry for intruding between all the other ‘Math’ (why not Maths?) wiz kids(?) – I’m still just learnin

  38. I always had I hard time understanding eulers formula. After all we are used to think expotential functions as something that grows.

    But then I thought about the multiplication rule, and that e^i is really just a point on the circle with angle 1.

    And when you do e^ix = (e^i)^x with x being an integer, you are really just multiplying this point by itself, and thereby adding angle 1 each time.

    Just like you could use the formula i^x=cos(pi/2*x)+i*sin(pi/2*x) as the complex number (0+1i) lies on this circle 😀

    I’m very happy with your article.

  39. @Anonymous: Thanks, glad you enjoyed it! That’s a great insight about moving around on the unit circle, I like that interpretation too :).

  40. Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?


  41. Hi Peter, great question. Yep, there’s something called quaternions which are complex numbers extended to 4 dimensions, and are used in graphics programming for 3d rotations (like complex numbers can be used for 2d rotations).

    If you want even more dimensions, you can use linear algebra (vectors & matrices) to represent data with n dimensions. For example, Google uses matrices to represent the multi-dimensional relationships between web pages in its ranking algorithms, when a single number (or quaternion) won’t do it justice :).

    Hope this helps,


  42. Thank you for this great explanation. I have been reading about vectors for the first time recently, and your article makes me wonder whether the “ship’s heading” example could be solved similarly with vectors. More generally, can imaginary numbers be seen as a specific case of wider vector-based concepts?

    Thanks for your thoughts!

  43. Hi Weston, happy you enjoyed it! Yep, imaginary numbers can be considered a type of vector, with certain rules for how to multiply & add them.

    You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra). They’re actually a neat way to introduce the topic of keeping more than 1 piece of information in a single “number” (or vector :) ).


  44. Outstanding! I never thought of the complex plane in terms of rotation before. I’m teaching myself 3d graphics, and I’m not satisfied with “put the model in the engine and don’t worry your pretty little head over how it works”, and one area that is blowing up my head is in the area of quaternions. This might help crack that area open for me.

    … But on the off chance that I or others still can’t get any traction on the topic, you think you could cover quaternions in a future article? :)

  45. @vaibhav: Glad you enjoyed it.

    @chuck: Thanks! Yes, I’m never satisfied with “plug and chug” either. Sure, I’ll add quaternions to the topic list — no estimate on the ETA though :).

  46. Hey, I’ve got a few questions/comments myself.

    1. If you have a TI Calc, put the following into a calculator.

    ANS ^ ( 1 / i )
    ENTER/SUBMIT/EQUALS (whatever it is) Three Times

    You get a seemingly weird result. You want to explain that one geometrically?

    Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above. I showed it to the Pre-Cal teacher (as I was in a programming class she taught) and she got confused by it. This was two years ago, in a different school, when I was High School Freshman. What took me a few hours of actual thought on it (counting time sleeping, about 15), I managed to recreate something that took most mathematicians a few years to grasp. Sadly, I was rejected when showing the idea…but you’ve turned my thoughts back around to proving me true.

    A point on saying that numbers have two dimensions. I’m guessing you didn’t take the next step for simplicity, and just say that numbers have an infinite amount of dimensions.

  47. hi,
    thanks a lot for these explainations,
    i start to understand it better,
    i must say these imaginary numbers have been a full stop to my scientific studies about twenty years ago,
    turned towards languages sports & phylosophy for my graduation,
    really messed my studies after that, gave it all up
    to professional life,
    been musician,
    till the day i found out all that not satisfying
    as some light missing,
    really loved science as kid so turned back to those teachings,
    & making an aim of understanding those imaginary numbers,
    intuitivly thinking they where the key i needed for that door,
    the candle to light so i can take the next step

    thanks a lot


  48. Thanks for doing this, I totally support what you’re doing. I don’t understand why understanding is so seldom taught, if not consistently assaulted. Several cynical possibilities spring to mind.

    I think so many people are cut off from the beauty of the mathematics as a result. For me it’s like having been deprived of a sense for most of my life. We would not think it fair to stop someone being able to smell or taste, nor is it fair to deprive them of those beautiful moments of silent, wordless understanding, the “a-ha!”s. Our joy consists in coming to better know a piece of nature.

    I’ve always been interested in physics, but I can’t go any further without more mathematics; something I’ve always found difficult. What a chore, I thought. Then I had the revelation that maths was beautiful if one tried to undertand it like physics. It’s almost physics in reverse, unpacking the maths into intuitions, rather than packing intuitions into maths.

    Wikipedia can be disheartening because it often comes across as preaching to the choir – showing off to people who already know the subject and wouldn’t need to look it up anyway. What good is the sum of all human knowledge if you can’t understand it? What you’re doing here is really valuable.

  49. @Psy: Thanks for the comment! I agree, I don’t know why understanding (not memorization) is the focus — I suspect it’s because memorization is easier to test.

    I like that point about the beauty of math and physics — many people scoff that notion, but the beauty really is there! I don’t know how to explain it either, but the way such simple rules can create something so complex and powerful is mind-boggling.

    Also agree on Wikipedia — it’s a great reference, but not a good learning aid as it’s often written at the most detailed level (by experts, for experts). Ideas die unless they can be understood and learned by later generations. Thanks again for the comment.

  50. hi kalid,
    thank you very much. i have not found any better explanation than this one on complex numbers.

    — abu ihsan,
    Kuala Lumpur, Malaysia.

  51. OMG!!! this made so much sense, and was very very easy to follow. it wasnt very mindblowing, and it actually made sense!!!! i like having answers to why i is i and all that jazzz….
    thanks a ton!!!

  52. In high school my teacher could not get me to believe that there could be any such thing as the square root of a negative number. Your site has made me a believer!

  53. @Ti: Glad you found it helpful!

    @Daniel: Thanks, I didn’t “believe” in i for a long time either :). Only when I started accepting that numbers could have more than 1 dimension (why not?) did it click.

  54. Hi Kalid, great article – I wish I had something like this when I was learning it in school. Believe it or not, I only understood this when I was actually working on complex power as an electrical engineer. Yep that’s right, I was being paid and didn’t even realise that a complex number was just a number with two dimensions. I could do all the algebra, but didn’t understand the fundamentals. Scary thing is, most engineers I know don’t really understand this either… it’s just not taught properly.

    One thing you could add is something brief on complex exponentials, which is another thing I never understood properly at uni. Euler’s law is usually just taught in passing on the way to Fourier transforms and series. But it is seldom derived and I thought it was magic for a long time! I personally found the derivation based on the Taylor expansion of e to be the most intuitive.

  55. @Jules: Thanks for the comment! I know what you mean — it took me a long, long time to see complex numbers beyond what I was initially taught. I think people are a bit wary to raise the question in school, and go on using a memorized definition.

    I’m really looking forward to covering complex exponentials, thanks for the suggestion. For me, I want to find an explanation that goes beyond symbol manipulation (which is what most of the proofs I’ve seen are) and dives into what transformation it represents. Thanks for the suggestion!

    @Jarek: Interesting paper! I like the effort to distinguish a number from the operation, it’s a theme I agree with.

  56. Wow! I have “easy maths” in school, but I’ve always been very curious about so-called imaginary numbers. This was a very fun experience, you really do learn something new every day!
    Thank you,

  57. Seems pretty formal, maybe this should be pushed to be learn’t in math C classes even further since im in one this isn’t explained in so much detail, thank you.

  58. @Brian: Interesting, I’ll have to check it out.

    @Stary: You’re welcome — yes, it’d be nice if students got to see the geometric viewpoint along with the pure algebra approach.

  59. thanks so much for all your great tutorials on this site. I can actually grok imaginary numbers now. All it took was to consider them as another dimension, geometric or otherwise. wonderful stuff you’re doing here. And I hope you enjoyed electroshock therapy!

  60. @Deryk: Really glad it helped! Yeah, I don’t think the therapy stuck :).

    @Jeff: Thanks, I’ll try to keep putting them out!

  61. Thank you Kalid.

    I direct my students here. You’re providing a wonderful resource.

    Please keep on writing, you’re single-handedly making a huge difference to the world.


  62. I remember learning about ‘i’ in highschool and was basically given the first explanation. I did the equations but never really felt satisfied, it was after I read your article on finding pi that I got more interested in math.

  63. Although I’m in my 60’s, I still like to learn new stuff and imaginary numbers always puzzled me. No more however, thanks to your excellent tutorial. I wish you had taught me math back in the dark old days when it was “get it, or get a smack across the head”. (I have a suspicion that half our teachers didn’t know subjects themselves well enough to really explain things. What a wonderful (obvious) concept: “Better explained”. I love it!

  64. thanks a lot.

    this complx numbers are like vectors.real number represents x-direction and imaginary number represents y-direction.if we can’t add two quantities directly(i.e is x-direction’s magnitude and y-direction’s magnitude), then we use cmplx numbers in which two quantities are dealed seperately and in the end result ,the real and imaginary parts are used to find the magnitude using pythegorean theorem.

  65. In the boat example given above we are interested only in direction.what if we want to know the magnitude?. consider two forces say a,b (one in x-direction and other one in y-direction) acting on a body.Then the resultant force is a + ib.if we want to rotate this resultant force 45 degree in counter clockwise direcion, then multiplying the number a+ib with 1+i will change the direction of the resultant force to 45 degree in counter clockwise direction.but what about the magnitude of the force?.it will not be same as sqrt(a^2 + b^2).but it will be 1.414*sqrt(a^2 + b^2).because the magnitude of 1+i is not equal to 1 .it is equal to sqrt(1^2 + 1^2)=1.414.multiplying a+ib with 1+i actually increasing the original magnitude of the force along with rotating it 45 degree
    so to rotate the force vector we have to multiply it by unit vector.unit vector can be obtained just by dividing any vector with its own magnitude.In our case it is (1+i)/1.414
    so our answer will be (a+ib)*(1+i)/1.414 which is same as a+ib offset 45 degree from current position with same magnitude.

  66. Thank you very much , i keep thinking trought you explanation about real dimension of the number have 2 dimensions this focus is related to solve euler rotations problem with quaternions but i thik probably is the same problem of fermat’s last theorem one number can’t hold more than squareroot of x^2 + y^2 or probably not but thanks anyway !!!

  67. Great Imagination in your explanation, Kalid :)

    If “i” can be thought of as a rotation of 90 degrees, why isn’t it “2i” for 180, as opposed to i^2?

  68. @free: Thanks!

    @giles: Thanks for the kind words! Yes, I feel that most education is of the smack-you-across-the-head variety, and the teachers themselves may not deeply know what they are explaining. I’m really glad it helped!

    @Eduard: Thanks!

    @John: Great question, I had to think about it for a minute.

    Regular multiplication (times 2) increases the magnitude (size) of a number. All of the “two-ness” is going towards increasing the number.

    When we multiply by i, we are dealing with rotation. All of the “i-ness” goes into rotating the number.

    Multiplying by 2i is the same as multiplying by 2, and then by i. The “two-ness” increases the size, but only the “i-ness” rotates the number. So if we want to keep rotating it, we need to use i * i (rotate by 90 degrees, and rotate by 90 degrees again) or multiply by i^2 = -1.

    I hope this helps!

  69. When we write x = 1 we can interpret it as a length, it is a direct value which we call positive. We can have the opposite of it, which we call a negative number y=-1. x+y=0 , that opposite cancels that direct value. We have that horizontal number line from negative and positive numbers.

    x^2=1 is basically a length multiplied by another length (area), a square with direct value 1 of length x and height x.

    However what is the opposite of that square? one which has an opposite value area, we want x^2+y^2=0. We are looking for that length y that gives y^2=-1, that number y is square root of -1 but that doesn’t mean anything since square root is only defined for positive numbers so we need a new symbol for that unit. We have the symbol “1” for a positive unit, we have the symbol “-1” for the negative unit but now we need a symbol for the lateral unit (the “imaginary” unit) which is i and it’s opposite “-i” which basically is laterally moving away from the horizontal number line into another dimension.

    That’s the way I think of it, I think it’s correct.

  70. Great articles! I had always hated math classes because of dreadful approaches they take in teaching.
    Reading your articles gives deep insights into maths (fills the gaps and links things together).
    After all, math is not a super-natural thing as it seemed so to me before!
    I think what is taught in years could be actually taught in much less time if this smart intuitive approach is implemented!
    I am compelled to tell my friends about this website. They will LOVE me for that!
    Thanks Kalid!

  71. Thanks, thanks, thanks! Kalid, you are providing an amazing service. I hope Better Explained never goes offline, and you continue to add to it. It’s one of the best resources I know of.

    The more math I learn, the more I’m frustrated that my teachers never took the time to show me and my classmates the intuitive aspects of what we were learning. When my kids are learning math, I want them to be taught by someone like you. Of course, you’re obviously overqualified to teach grade school math – you’ve got far better opportunities. So I wonder what it would take to get people like you to teach, even if it’s just for a couple years. I would love to know what you think.

  72. @Nasser: Thanks for the note, really glad it helped! Yes, I really wish schools would take 5 minutes to explain the intuition of a concept before diving in; it makes a huge difference in understanding.

    @Gary: Wow, I really appreciate the kind words!

    I was similarly frustrated by the lack of insight in my math education and started writing notes to myself, which eventually turned into this site. I don’t know why the present approach persists, likely because “some” students manage to figure it out and others resign themselves to think “I’m no good at math.”. Yet for reading and writing, we simply do not accept “I’m no good at writing” — yes, there are differing levels of ability, but everyone is _expected_ to have the competency to at least read a newspaper and write basic prose. Math doesn’t have that expectation, so it’s “ok” if you don’t get algebra.

    That’s a great question about teaching. In college, as I realized I liked education more and more, I started looking into what I could do. Unfortunately, what I learned about the school system turned me away. I don’t like the idea of unions (or rewards based on seniority, not merit), and the over-testing of kids. The latter leads to teaching to the test, which could lead to even worse results as people focus on short-term “get through this class” vs. long-term insights which improve their entire academic perspective.

    And unfortunately, there’s the issue of compensation as well. I love contributing, but compared to industry it’s a very difficult pill to swallow, esp. with loans, etc. to pay. I think it’s ridiculous that one of the ‘most important jobs’ could be rewarded the same as a janitor or toll-booth worker.

    Writing here is my effort to help share what I know, and I want to help teachers help students. I think that might be the most practical approach actually — give teachers better analogies and examples to help improve education. I think the focus on “computerization” and technical improvements are vastly overrated — the best results, from what I’ve seen, are from enthusiastic teachers or better ways of looking at problems.

    A bit of a rant — I think teaching for a few years in Teach for America could be an option. To be honest though, I feel like such an example is like joining the Peace Corps. Yes, I really love teaching and sharing knowledge, but coming out of college I wasn’t sure I wanted to dedicate my life to it in such a way. The more I think about it now, the more sure I am that I want it to be a large and growing part of my life. I’d love for one day to make something along the lines of Wikipedia in terms of breadth and impact: to give really useful, insightful ways of understanding problems (not just the facts about them).

    Phew — a bit rambling, but hope this helps give my mindset!

  73. Thanks for sharing your thoughts, Kalid!. I agree that superb teaching doesn’t seem to be the goal around which public education is organized, and incentives are a big part of that. But private schools don’t seem to be orders of magnitude more attractive in terms of compensation, so I suppose it’s possible that many people aren’t willing to pay what high quality teaching would cost.

    The internet resources you’re creating are a great way to make the cost of great insights more palatable (downright delicious, actually) and make them more widely available at the same time. Keep up the great work – you’re helping a lot of people!

  74. @Gary: You’re more than welcome, I’m always happy for the discussion!

    Yes, private schools don’t seem vastly different from public ones in the compensation regard. I suspect that because the direct impact of teaching is so nebulous and long-term, it’s difficult to ‘measure’ the value of a good teacher. Like anything else, the important things that are far off can seem less pressing than present-day trivialities :).

    I’m really happy you’re able to enjoy the articles, I’ll definitely keep at them :).

  75. Kalid, I just ran across http://www.khanacademy.org/ and thought of you. You two share a gift for explaining ideas. He focuses a lot on giving the intuition, as you do. Check it out! Maybe a chance to team up? Kalid and Sal together would be unstoppable :)

    Also, I posted a link to Khan Academy on the BetterExplained links page, but it seems that page may be getting overrun by spammers. Just a heads up.

  76. @Gary: Thanks for the note! I ran across Khan academy a while back, I appreciate the reminder — I think I’ll reach out and see if there’s a way to work together :)

    Thanks for adding it to the link site — the spammers started up recently (no idea why, it was very quiet before) so I’ve turned off anonymous submissions. That site is powered by a 3rd-party service so I hope they have other anti-spam measures I can use. Thanks again for the note!

  77. Excellent post from Kalid.

    Another very good explaination I get about complex number is from Prof. David Joyce http://www.clarku.edu/~djoyce/complex/

    A real number is a point on the number line, a complex number represents a vector on the two dimension plane.

    David gave detailed geometrical interpretations on all complex number arithmatic. It is very enlightening.

  78. I am still trying to wrap my brain around the concept of “i” so my thinking may be off. If so please correct me. One of the statments you made was “Also, keep in mind that having counter-clockwise be positive is a human convention — it easily could have been the other way.”
    Are you sure this is a correct statement?

    It would seem to me that possitive “i” has to be counter-clockwise according to the way you explain it with the graphs. Here is my reasoning.

    -i*i=1 or -(i^i) would be -(-1)=1

    If we start with “-i” and multiply by “i” we would go one rotation counterclockwise which is 1.

    If we start with “i” and multiply by “-i” we would go one rotation clockwise which is still 1 so everything makes sense so far.

    However if we made possitive “i” a clockwise rotation we would end up with “-i*i=-1 since we are multipling by possitive “i” which we are now rotating clockwise it would be -1.

    So doesn’t postive “i” have to be a counterclockwise rotation to get the right answer or did I just thoroughly confuse myself and everyone else?

  79. @Daini: Thanks, I’ll check it out!

    @Mike: Great question! For your last statement you wrote:

    However if we made possitive “i” a clockwise rotation we would end up with “-i*i=-1 since we are multipling by possitive “i” which we are now rotating clockwise it would be -1.

    If “i” was clockwise, then “-i * i” would be “start at the North position (-i, since -i must be counter clockwise!) and then rotate clockwise by i (putting you at East).” This would give you 1, just as before — remember, if i means clockwise rotation, then “regular i” would point South, and -i would point North.

    Hope this helps!

  80. Ok, now it makes sense! On a side note I just wanted to thank you for this wonderful site. I think I can finally wrap my brain around the concept “i” and your section on “e” and natural logs is wonderful too. I wish it would have been explained to me that way in school. It is nice to have an explanation on why they work instead of the usually “Here is the rule. Trust me it just works!” Thanks!

  81. @Mike: Awesome, glad it worked for you! You’re more than welcome, there’s so many concepts that I’ve struggled to understand, I just want to share them and help spare some pain :).

  82. It was really really really really really coooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooool!!!!!!!!!

  83. Absolutely fantastic…. I regret havin learnt maths without undersatndin this analogy all these years….

  84. Nice guide, I was just studying the (inverse) euler formulas and didn’t really understand complex numbers.. way easier now!

  85. An excellent and very clear explanation of imaginary and complex numbers. Thank you so much!

    A request: Imaginary numbers: Part Deux including just what the heck it means to raise a real number to a complex power. I have never understood that and even using the above rotational analogy I don’t get it.


  86. @Craig: Great question! I have a post on Euler’s Formula which is drafted up, and that may help explain it.

    But for a quick preview: I consider an exponent being “decide the rate and direction of growth for the base”. I.e, 2^3 really means 1 * 2^3 which means take 1, grow it by 200% for a duration of 3 units.

    When you have a complex rate of growth (2^i), instead of growing linearly (in the same dimension), your rate of growth is rotated. With a rotated rate of growth, you grow tangentially and follow a circle instead of growing ‘straight ahead’ on a line. The upcoming post will make this more clear… but that’s my intuition :). [And if you can guess, (2^i)^i would be a “twice-rotated” rate of growth which should SHRINK you (growing straight to growing tangentially to growing AGAINST your initial direction), and indeed it does, as (2^i)^i = 2^(i*i) = 2^-1].

  87. Man, thinking of complex numbers as rotations…that just opened up a whole new world for me as did thinking of basic arithmetic as transformations.

    I guess the fact that I got through high school math and am currently dealing with college math just means I eventually accepted the definitions I was taught with no understanding.

    Yeah this intuitive understanding of complex numbers is very wonderful. It’s gets me excited that I’m actually getting this stuff.

  88. Is there a way to rotate 60 degrees?
    Is there a way to cubeRoot?
    What is cubeRoot(-8)?
    Does 2*cubeRoot(-8)=sqrt(-4)?
    Can you hypercubeRoot?
    What is hypercubeRoot(-16)?
    What does 2*hypercubeRoot(-16)=cubeRoot(-8)?
    How would one visualize hypercube (AKA 4D) space?
    Can calculus be adapted for 4D visualization?

  89. I guess I should clarify on the 60 degrees.
    Is there a way to rotate a non-complex non-zero number 60 degrees without a non-zero real result?

  90. Khalid, Thanks a lot for all your insightful, intuitive topics on Math. I have one question here-

    If you consider i as a 90 degree rotation, and 1+i as a 45 degree rotation, (1+i)^4 is a 180 degree rotation? but it also scales 1 by 400% where as a 2*90 degree rotation doesn’t.

    Also in the heading analogy, the magnitude of the vector isn’t the same when rotated by 45 degrees. Shouldn’t the magnitude remain the same before and after rotation?

  91. I’m coming back to imaginary numbers as I need to understand them better before I learn about quaternions. This is the best explanation for these I’ve seen or heard – and I thought I’d already understood them. Thanks.

  92. @Sesh: Great question! Multiplications can do two things: scale (change the size) or rotate (change the orientation).

    When we multiply by 1+i, we rotate by 45 degrees, but we also scale by the “size” of 1+i. (For example, when we multiply by -2, we flip the number, but also make it twice as large: -2 * 3 = -6).

    In this case we need the pythagorean theorm to find the “size” of 1+i, which is the distance from 0: sqrt(1^2 + 1^2) = sqrt(1 + 1) = sqrt(2). That means each time we multiply by 1+i, we’re rotating by 45 degrees and multiplying by sqrt(2) also.

    So you’re right — after 4 multiplications, we make a 180 degree rotation and a sqrt(2)*sqrt(2)*sqrt(2)*sqrt(2) = 4 times increase in size. There’s more in the follow-up article:


    @Steve: Awesome, glad it helped! Good luck with quaternions, I need to learn them too :).

  93. You rock!! I am in 4th year engineering, I use complex numbers every bloody day, never really understood it. Now a lot of things make sense to me.

  94. What a buncha bullcrap !!! These mathematician people him-haw around and make all kinds of complexicated gobbledygook that new students can’t understand, when all complex numbers are is square roots of negative numbers, and they NEVER REALLY SAY THAT! It took 2 days for me to figure this out, when all someone would have had to do is just start from that point and I would have had no trouble piecing the rest of it together. People just get their heads all up in the air and spew out all kinds of confusing formulas without explaining the basic idea of it.

  95. wait how did you get (3=4i)*(1+i)= 3+4i+3i+4i^2= 3-4+7i =-1+7i

    more specifically how did you get he 3-4

  96. @Anonymous: I used FOIL (first, outer, inner, last) like this:

    (3 + 4i) * (1 + i)
    = (3 * 1) + (3 * i) + (4i * 1) + (4i * i)
    = 3 + 3i + 4i + 4i*i
    = 3 + 3i + 4i + 4i^2
    = 3 + 7i + 4(-1)
    = -1 + 7i

    Hope that helps! I wrote it in a slightly different order in the article, but the 3 comes from 3 * 1 (“first”) and the -4 comes from 4i * i (“last”).

  97. my teacher is making me and my friends do a debate on imaginary numbers. i’m on the “against” side. i’m against having to learn them in high school. Any one got any good reasons why they shouldn’t be?

  98. @Anonymous: Hrm — imaginary numbers are a name we’ve given to a relationship. It’s like debating whether “green” exists — things are out there in nature, and we classify certain ones as “green”. Similarly, we classify certain operations as “imaginary” (which might otherwise be called rotations). At the end of the day, it’s a human convention, but a useful one.

    @Esteban: I agree, they rock.

  99. I have to be honest, I remember learning about imaginary numbers in middle school. I remember working with them in calculus and differential equations. I took an upper-division course in complex number analysis. I took 3 courses in electrical signal theory (fourier analysis which is all complex number theory). I’ve taken and taught circuit theory which uses phasors (Euler’s equation). I’ve probably used Euler’s equation as an electrical engineer more than all other mathematical formulae combined. And this is the first time in 15 years of using imaginary numbers that I feel like I understand what imaginary numbers represent. Bravo on a good, intuitive explanation.

  100. @George: Wow, thank you for the heartfelt comment — it means a lot to me! I was the same way, having used imaginary numbers so many times in school and college, but never really having them click. Really glad it helped.

  101. luv ya! I understood it perfectly! When I first encountered them I could only think of them as a some sort of magical tool – when you build a brick house you use instruments but in the end your house consists only of bricks. Allegory’s still true but I now know exactly what this tool does. Not a witchcraft anymoar (I really believed it was) 😀 Well I shall move on to polynomials. Wish me luck 😀

  102. @reiiya: All I can say is… good luck! I think you’re getting it though, that’s exactly the feeling you want, where it leaves the realm of magic and becomes something you can just use like an everyday concept. It takes time, but it seems like you are halfway there already :).

  103. You have written as follows:-
    Let’s try a simpler approach: we’re on a heading of 3 + 4i (whatever that angle is; we don’t really care), and want to rotate by 45 degrees. Well, 45 degrees is 1 + i, so we can multiply by that amount!
    How do we know intuitively that 45 degrees is 1+i ?. What is 74 degrees then?

  104. Oh no! If I just found this article a little bit earlier, I could’ve appreciated my math subjects even more.

    Putting jokes aside, thank you very much for this wonderful article.

  105. This project is a bright idea,

    As a 5 yr engineering student, the best times are going back to the basics and realizing you can take away more.

  106. As an engineering student, sometimes I get so wrapped up charging through material and equations that I forget to step back and take a deeper look. Thanks for encouraging me to do so.

  107. Hey Kalid,well done, your style of teaching reminds me of an old electrical engineering lecturer who presented imaginary numbers in pretty much the same way you have. He used to joke “Gentlemen, if you’re unfortunate enough to receive a few hundred amps of current across the heart that is 90 deg out of phase, be sure to let us all know how imaginary that felt to you, assuming you’re still alive that is” (In electrical engineering, ‘J’ is used to to represent the imaginary component because ‘i’ is reserved for current)

  108. ~ a very informative and followable explanation , thankyou !
    so numbers can be 2 dimensinal
    what happens if we propose now a 3 dimensional number ? We have 3 D space , and it seems a fair logical progression to assume 3D numbers ..
    for example
    x = (a + ib + mc )
    lets call m a magical number say ,
    we have already defined i as square root of (-1)
    does it now make sense to define m as the square root of -(a+ib) ?
    oh dear , i see from previous posts that multi-dimensional numbers have already been mentioned … and i thought i was on to something new … still im fascinated … could they be used to generate 3D fractal geometry for example ?
    I’d love to research further ..

  109. The url included here:


    is my attempt to prove the pythagorean theorem by multiplying conjugates (a+bi) and (a-bi) and looking the geometric result. The diagram seems to suggest that c/a = (a^2+b^2)/c and I am curious why the extra a appears? Is my diagram correct? I tried to follow the ideas you wrote on this page… Please email me with any ideas on why this “variant” of the pythagorean theorem occurs..

  110. So, what’s the difference between a complex number and a 2D vector? I remember back in the time when i used to explore math with my HP 48G that it has two datatypes: complex numbers and vectors (which may be 2D, 3D, etc.) and they seemed too similar to be different things…

    “””Have a question? Know an explanation that caused your own a-ha moment? Write about it here.”””

    This was my a-ha moment:
    a-ha moment

  111. @Anonymous: Thanks — and yes, “imaginary” is a word we use to describe a relationship! There’s nothing fictional about that current :).

  112. I am totally overjoyed. When I read your article, my heart was pumping much faster than usual. The culprit was that awesome simplicity with which you have presented the material. Thank you SO much. You are my hero:)

  113. @Olenka: Wow, thank you for the wonderful comment — it made my day! I love running across the simple underpinnings of these complex ideas, so much is just in getting the right approach. Thanks so much!

  114. “What transformation x, when applied twice, turns 1 into -1? Hrm.”

    That right there was my aha moment. Okay actually it was two seconds later when you answered the question because, lets be honest, I didn’t think of rotation on my own. But once I read your answer that question really drove home how imaginary numbers relate to the rest of the number system.

    Fantastic explanation presented with a good sense of humor (“Crack. Did I break your calculator?” …hehe). Can you puh-lease find an agent and write a textbook or two asap? thanks.


  115. Well in Pre -Calculus, we were given an extra credit assignment to type a 500 word essay on the applications of imaginary numbers in the real world. I must say this helped me to further understand imaginary numbers and that they aren’t imaginary, they have feelings too :). This article made me laugh and learn :O, and as a math/science nerd I’m excited to read more of your posts. ^_^

  116. @Karina: Ah, thanks for sharing the exact moment of your aha! That’s really interesting for me, as I love discovering what makes an idea really click (I think education would be so much easier if we could share and focus on those crucial moments, instead of getting lost in the details we can read on our own anyway).

    On publishing, I’d love to! I’m getting some experience collecting my thoughts into ebooks, and I’d love to flesh it out further :).

    @Kisame: Awesome, glad it was entertaining for you :).

  117. Hi Khalid,
    When I was half way through reading this article it was like, I have found a treasure…I guess I would not have been so much excited even I had found a treasure full of gold instead of this article. :)
    >>>‘Cos it’d be a sin if you didn’t
    Just for a second after reading this line, I was like what ??? and then immediately, lol lol 😀
    I havent seen your complete profile,I am so excited I want leave a comment before I see your other articles…

  118. Nice tutorial, I now understand imaginary numbers!

    One thing though. The new heading isn’t -1,7… You still have to multiply both numbers by sin(a) to get the accurate results of -0.71,4.95 where a is the angle you changed (in your case 45°).

    Anyway, very well explained, I had to read it twice to fully understand it, but that’s just because I had to adjust my brain 😉 Thanks!


  119. Wow! How did you come up with this stuff? It’s really well explained. You’ve convinced me that imaginary numbers exist!

  120. @Anonymous: It’s funny — this interpretation of imaginary numbers didn’t come about until decades after they were invented! Until then, people weren’t very sure how they were useful. Glad to bring them into the real world :).

  121. what do u mean by i times.like if something to power of i.considering what you said about exponents how can we state i amount of time..

  122. Im not in school or anything, just always wondered about imaginary numbers. The finding patterns bit is what did it for me, when I was able to see the continuous reapplication of i on itself producing 90 degree rotations NOW i see how there can be sqrt’s of negative numbers, simply mind blowing, I mean..there’s no words for this.

  123. Your simplified view of rotations isn’t so simple after all. It’s obvious that the complex number 1+i is at an angle of 45°. But what about the angle 23°? Let’s say I want to rotate the point (3,4) 23°. I’d first have to find a complex number with that angle. How do I do this? Trigonometry. You need to take the sin and cos of 23°, which isn’t so simple or trivial, and isn’t too pretty.
    Alternatively, you can just use the following two identities, which are easily proved graphically and analytically using only “nice” properties of trigonometry:
    x’= xcosa -ysina
    y’= xsina +ycosa
    Where (x,y) is your old coordinate, (x’,y’) is your new coordinate and a is your angle.

    Also, that formula truly rotates something; it doesn’t just generate your heading, which your method does (since your dilate your vector).

    In fact, matrices are used to do most rotations. They’re quite simple, and extend easily and intuitively to higher dimensions. Also, they go well together with any other arbitrary transformation.

  124. This is the best explainer on imaginary numbers I’ve ever read. Unfortunately, it took me through grad school to find it :/

  125. @Dennis: Thanks! Unfortunately a lot of “textbook” definitions are just about memorizing the formulas and moving on. Glad it helped.

  126. @Ethan: Great question re: other angles. The goal is to show an application of what you can do without calculators / tables of sines, cosines, tangents. The assumption is that if you aren’t using calculators, etc. then you would never really ask “Let’s rotate 23 degrees”. Such a precise answer is probably the result wof working backwards (I see an object in the distance, I measure the triangle it makes with my trajectory, take the inverse tangent and get 23 degrees).

    In this calculator-less world, presumably you see the object in the distance and just want to rotate face that. You get the triangle representing the desired trajectory, and instead of taking the tangent to get an angle that you then convert again using sine/cosine, you just make a rotation off that initial triangle. In a world where you could know an angle is exactly 23 degrees, sure, you’d use sine/cosine, but the intent is to show how you can do calculations by hand without a calculator / tables of sine/cosine/tan. Hope this helps clarify.

    Expanding to matrices is definitely a nice addition to the toolset.

  127. Was a great article.but according to this explaination what do you mean by i times.you know like 2 to the power of i.meaning scaling 2 i times.how can we explain that?

  128. if you have nodes A, B, C, and know the distances between them dAB, dAC, dBC, then you can generate coordinates for A,B, C, etc. using law of cosines and pythagorean theorem. Imaginary numbers come up whenever distances violate the triangle rule such as dAB=3,dAC=4, dBC=9 if node Axy is 0,0; Bxy is 3,0, then Cxy is (x=-9.33, y=8.43i).

  129. Hey man, thanks for shedding some light on i. I am getting ready for Network Theory 2 (AC Circuits) and this “strange” number showed up. Good write though really. I wish I learned this in a classroom too! :)

  130. Excellent explanation. I really like your explanation of our number system being two dimensional. My understanding of complex number is now much clearer. Thanks very much for your efforts.

    Regards from Australia

  131. I just want to say that, as an MIT ee&cs grad who has taken 5 math courses and 2 physics courses, I have *just now* understood an intuition behind complex numbers. Thank you.

  132. You absolute legend! Now the only question is why do they wait until college to teach something so simple while torturing kids with the sin-tan-quadrant-hoo-ha? Oh well!

  133. ‘So-called “imaginary numbers” are as normal as every other number (or just as fake): they’re a tool to describe the world.’

    Well said! That’s another aspect that is not taught to kids (or really outside of philosophy classes). Numbers are in our minds, not “out there” in the physical universe.

    Excellent article!

  134. This all made sense until the putting-ketchup-on-a-hot-dog part. I mean, that’s just insane.

  135. Hi you say “Multiplying by a complex number rotates by its angle” which definitely seems to be true, but why is that? Thanks :)

  136. @steve: Greetings! :) Glad it was helpful.

    @fastfinner: Thanks, I was the same way — I only felt comfortable long after the courses were over!

    @fedora: Thanks!

    @Mark: Yes, that’s exactly it! We’re just trying to understand models we’ve came up with on our own.

    @Jesse: Math makes me think some crazy thoughts.

    @Killian: Great question, I was thinking about that this weekend! I’ll be writing a post on it soon :)

  137. Just taught this to a bunch of year 14 years olds, and it helped my understanding to read it to! Thanks!

  138. Negative numbers allow you to flip in one dimension. Imaginary numbers allow you to enter into two dimensions. What comes after that for 3, 4, etc. dimensions? In a sense we can do any number of dimensions using x,y,z,etc. coordinates, but that’s not exactly what we’re talking about in this article.

    If there isn’t anything like that then why is that? If the answer is that matrices and/or x,y,z coordinates are good enough for additional dimensions then why is i necessary for rotations as described in this article?

  139. @Geoff: As you say, negatives are 1-dimensional — and only invented in the 1700s! They opened our mind to what’s possible. Imaginary numbers can later (2d), and I think people realized you could have any number of dimensions in linear algebra / matrixes (and didn’t have to visualize them).

    For reasons I don’t understand, I don’t think you can have 3 dimensions easily (due to symmetry) but there are 4-dimensional numbers (1, i, j, k) called Quaternions (http://en.wikipedia.org/wiki/Quaternion) that are actually used in video game programming to model rotations! There are even 8-dimensional numbers too… though at some point, it becomes easier to just use a matrix.

    I think the neat thing about “i are rotations” is that it expanded our mathematical perspective to “accept” that numbers can be 2d. It would be hard to really believe that numbers could have two perpendicular components without an analogy like that. After i broke the ice, we can accept that numbers can have any number of perpendicular components :).

    That’s my 2c on it anyway!

  140. oh, shit, how come I did not know this 20 years ago? It was very very frustrating, which even made me frustrated with math from high school to uni, even today, but now i am released.

    unlimited thanks to your explanation for ever. If given any chance, I will share your comments with as many others as possible.

  141. absolutely fantastic explanation. I am a college lecturer and always thought that i was the absolute best at explaining things( I teach finance and accounting and taxation). I am impressed to say the least. I was pretty good at maths in school. but dint really understand a lot of things. this is the first time i have understood the concept of complex numbers. I used your explanation to explain complex numbers to my 13yr old daughter and she also thought that the explanation was awesome. Indebted to you for life. thanks

  142. Thanks a lot for the great article simplifying the abstract idea.

    About the last note you wrote, I’m also fascinated by the invention of number 0 and I think it might be a cultural concept in India before it became mathematics. The ancient india believe in the Ether element that is void but upholds the physicality of the other elements. The representation of that is probably 0 , and when big numbers are represent as 100 the zeros are not actually there but only upholds the realness of the concept of hundred.

  143. Wow!!! This was a very interesting and amazing explanation of imaginary numbers. I’m 38 years old and always love to study science, technology and math. However, it’s been a while since I don’t get into the topic of imaginary numbers and this helps me a lot to “break the ice”. Congratulations!

  144. I don’t have the words to express my gratitude to you the author. You relieved me from a deep hatred of complex numbers since school (I’m 50 now). And you truly enlightened me.
    Thank you so much for this gift.

  145. Absolutely incredible.

    What an innovative and inspiring site you have here Kalid, I have just come back from writing an exam (I am actually a student taking maths as a major at Varsity) and literally thought I could not take in any information for the day but your articles have lead me from one to the other endlessly, and I am more in awe and get more excited after each one!

    Keep up the good work, idea’s and creations like your’s motivate me to do my own bit for changing the way people view maths and physics and basically any of the sciences.

    I look forward to hearing and reading more, especially as I plan to continue studying maths for the next couple of years!

  146. @Brad: Thanks, I’m happy the article was able to stay palatable even after finishing the test! I think one of the keys is when we approach math with a curiosity to learn, it starts becoming invigorating/energizing on its own, and we just want to keep learning more. I really appreciate the support — definitely try your hand at helping people understand math better :).

  147. Thanks so much for finally clearing up for me what exactly complex numbers (and i) really mean! I really appreciate it!

  148. This is beautiful! I presented it to one of my Algebra 2 classes today, and I’ll show it to two more tomorrow. Eyes bugging out!!! Thank you for giving i meaning!!!

  149. @Bala: Awesome, glad it helped!

    @Diana: That’s so cool!!! I love knowing when it’s coming handy in the real world :).

  150. That’s amazing!!! I just sat back in wonder when i read your explanation of how complex numbers were rotations…and then again when you explained how to use them instead of finding angles. I really like the way you explain math, it’s all very visual, and even the parts that don’t have pictures really conjure up understandable mental images in the reader.
    I only wish high schools taught math like this. Most teachers don’t even introduce the unit circle when teaching trig (which is horrible!) and I know that my fellow ib nerds would love to be taught this way.

  151. @Manasa: Wow, really glad it clicked for you! I’m glad the analogies/visual approach was helpful, thanks for letting me know.

    I agree with you, I wish people in high school / college focused on the intuition of truly understanding the concepts.

  152. Homework: I understand that (i) is similar to 0 and that it’s a weird way of making math easier. I understand WHY it was created but I’m having a littel difficulties with understanding exactly what it represents. I know that the sqaure root of negative 64 is 8i but I’m confused as to what the (i) stands for. This helped me understand though !

  153. @Jyeisha: Glad it was helpful. To me, when I see a number like -1, the negative sign encompasses the “opposite-ness” of the number. -1 is an “opposite 1”.

    When you see “i”, you can consider it encompassing the “rotation-ness” of a number. i (really, 1*i) is a “rotated 1”. The key trick is allowing numbers to exist in 2 dimensions (and why not? Numbers are ideas used to keep track of things).

  154. i like to read what people like you, like you me writes about novel things and find (y)our satisfaction. i didn’t get c/x numbers but no worries, my brain will keep on working until it makes some sense out of it.

    in other words (maybe, probably not but anyways)


    sorry for my engrish mate!

  155. My teacher wants me to make a power point on “The real life application of imaginary numbers” hurrray. Now i get to research. Thanks for the article and helping explaint to me what imaginary and complex numbers are all about.

  156. @sqlguy: Great question! Let’s think about what it means: What transformation, when applied twice, turns 1 into -i?

    Well, we want to get from 1 (“due East”) to -i (“due South”) in 2 steps. Each step should be a negative 45 degree turn, or a trajectory of (1, -1).

    But… we need to account for the scaling effect, so need the 45 degree turn to be normalized to the unit circle (length sqrt(2)), so the trajectory is (1/sqrt(2), -1/sqrt(2)).

    So ONE square root of -i is

    1/sqrt(2) – i/sqrt(2)

    The other square root can be found by getting to “due South” using two positive rotations :).

  157. Would that be:

    Sqrt(-i) = Sqrt(-1 x i) = Sqrt (-1) x Sqrt ( i) = i x Sqrt( i) = i^(3/2).

    Check: ( i^(3/2) ) ^2 = i^3 = i^2 x i = –1 x i = –i. :)

  158. (actually, just realised this is the same answer as above (225):

    1/sqrt(2) – i/sqrt(2) = (1-i)/sqrt(2) = ((1-i)^2) / 2

    = (1-2i+i^2) / 2 = -2i/2 = – i !!

    (you wouldn’t have thought it!)

  159. @kalid, @Rich
    Thanks guys! I guess I thought that since the square root of the negative unit in the real dimension was in a new dimension orthogonal to the real, that the square root of the negative of THAT unit might be in a new dimension mutually orthogonal to the complex plane. But actually raising i to any power has rotational effects in the complex plane, is that right?

  160. But I guess I’m still confused. If raising i to a whole exponent gets you either a completely real or completely imaginary number, does raising i to a fractional exponent give you a complex number?

  161. @sqlguy: Yep, you got it :). Raising i to a fractional exponent (like the square root, 0.5) will give a complex number halfway between real and imaginary, i.e. at a 45 degree angle. Basically, you need to get to i (90 degrees) in 2 steps, each of 45 degrees.

  162. So I’ve been tutoring a student in precalculus and noticed that he’s starting complex numbers. I’ve never really engaged any of their applications, and this kid always needs to know how math relates to the real world.

    It left me a bit puzzled, as I knew that just rattling off rules would shut him down and leave him hating complex numbers (as I did, too, for a long time). Luckily, I stumbled onto your blog and I’m shocked about how cool they are!

    An intuitive understanding behind complex numbers may not be taught in classrooms, but I assure you that after today—thanks to your blog—at least one inquisitive high school student will have a better intuition for these fascinating numbers.

  163. I spent an hour mucking about the internet trying to find some sort of decent explanation and example of practical applications of complex and imaginary numbers, and not one of the other sites I visited made the least amount of sense. Especially in my sleep-deprived state. This on the other hand, this was simple and elegant. Thanks!

  164. @sometimeszero: Wow, that is so great to hear! One of my goals for this site was arming teachers with the analogies that really helped things click for me, hoping they can modify and incorporate them into their own routines (just providing ammo in the fight against boring, ill-understood math). Thanks so much for the comment!

  165. @Anonymous: Awesome, glad it helped! My goal isto write in a way that would have helped me when I was a sleep deprived student too :)

  166. @nuur leina:
    You simply carry out the multiplication on the right side (as you would do in multiplying two polynomials: term by term) and then equate real and imaginary parts of both sides.
    (3+i)(2-3i) = 6 – 9i + 2i – 3(i^2)
    = 6 – 7i + 3
    = 9 – 7i
    Then x + yi = 9 – 7i.
    Then x = 9 and y = -7.

  167. Thank you very much. Very clear explanation of imaginary numbers! Helpful to understanding imaginary time.

  168. @Cheryl: Sorry to hear that — please feel free to leave any questions. The main insight is that numbers can be 2-dimensional. In most circumstances, 1 dimensional numbers are fine (money in my bank account is positive or negative), but sometimes we want to consider things in multiple dimensions (such as the trajectory of a ship).

    @Mr r: Thanks!

  169. Hello Kalid, thanks for taking the time to share your explanations of mathematics. I have two questions based on the teachings and comments above.

    1. This one his hard for me to articulate but here is my attempt. Under the section “Understanding Complex Numbers” you wrote, “We’re at a 45 degree angle, with equal parts in the real and imaginary (1 + i).” But, there is no clear explanation as to why (1+i) necessarily equals 45 degrees. I could see this if the magnitude of i equals the magnitude of 1 (because you could show a 90 degree triangle with two sides equal to 1, that is an isosceles right triangle). But, saying that the magnitude of i equals the magnitude of 1 seems inconsistent with the rest of the teachings. So what am I missing ? Maybe I am actually confused with i altogether – is it meant to represent the concept of rotation around an origin (similar to a negative sign representing the flip across a vertical axis) or is it a type of number (imaginary) which is said to be the square root of i^2, or is it meant to represent both with the use of only one symbol ?

    2. In one of your comments above you wrote, “Multiplications can do two things: scale (change the size) or rotate (change the orientation).” Given two different descriptions of the term “Multiplication”, I assume it is intended to say that these are two interpretations of the term Multiplication. So then, what do the two interpretations have in common that maps back to an abstract definition of Multiplication from which all interpretations adhere to ? In other words, one asks – “So then, what does Multiplication in its purest form mean ?”. Maybe it is me, but with something as exact as mathematics, I can not stand the idea of having one term with two different interpretations, yet nothing concrete that drives the concept behind both of them.


  170. @Alonzo: No problem, happy to help

    1. (1 + i) is a 45 degree angle because it’s equal parts real and imaginary, similar to how a NE (northeast) trajectory is at a 45-degree angle because it’s equal parts North and East.

    Think of “i” as an indication of direction — we’re used to most numbers being on a single dimension, but “i” is an indication that a number is in this new direction. Confusingly, i is often written by itself (instead of 1i) so it gets mixed in. The magnitudes being the same is similar to “1 mile East is the same distance from me as 1 mile North”, i.e., both are a single mile away. Similarly, 1 and 1i are both “1 unit” away from 0, but in separate directions. This is what the magnitude is meant to measure: how far are you from zero?

    Yep, it’s best to think of “i” as a type of symbol indicating direction [similar to a negative sign], and in many cases you’ll see numbers written as “a + bi” (i.e., the number is a in the East direction, and b in the North direction). Because we’re sloppy sometimes, we might just write 3i instead of (0 + 3i) to indicate 3 units only in the i direction (i.e. 3 units in the North direction).

    2. I should have clarified: when I wrote “Multiplications can do two things: scale (change the size) or rotate (change the orientation).” I meant that multiplication can do one, the other, or both :).

    * “times 3” means scale up by 3
    * “times negative 1” means rotate 180 degrees (if you were going forward, go backwards)
    * “times negative 3” means rotate 180 degrees AND scale up (if you were going forwards 20mph, go backwards at 60mph).

    Imaginary numbers give us a new type of rotation: we can rotate “partway” (i.e. 90 degrees, not the full 180), and we can still scale, so there is some complex number which is “double your speed and rotate 90 degrees counter-clockwise”. (In this case, rotating 90 degrees counter-clockwise is “times i” and doubling your speed is “times 2”, so “times 2i” would have both effects).

    You raise a very, very good question about multiplication. I think we’re still discovering what multiplication in it’s purest form is, just like we uncovered what “numbers” really were. We started out thinking numbers were things you could count (fingers, pebbles) and then realized “Hey, there are halfway numbers, like fractions!”. So we had 1/2, 1/3, etc. Then we realized there are certain numbers that are partway but are not fractions (like sqrt(2)) and we got the real numbers (some sequence of decimals). Then we realized that numbers could be negative (why can’t they be less than zero) or complex (why are they stuck in one dimension?).

    Each time we discovered new numbers, the meaning of multiplication expanded a bit. With pebbles, multiplication might mean repeated addition. With real numbers (like 13.45 hours * 54.12 miles per hour) we might think of “scaling” or similar (vs. repeated counting). With negatives and complex numbers, because they move in different directions, multiplication implies motion in that new direction to.

    To me, the essence of multiplication is “applying” one number to another. When you apply a number, you transfer its properties to the result. So multiplying by a negative number gives the “negativeness” to the other number. Multiplying by a complex / imaginary number gives the “rotation-ness” to the result. Multiplying by a large number gives the “largeness” to the result. Without getting too cute, the essence of multiplication is how to apply the “essence” of one number to another. Hope this helps!

  171. Hello Kalid, let me first start by thanking you for taking the time to provide a detailed response to my questions. Your insight not only helps me, but as a consequence it helps me when I try to explain things to younger folks in an attempt to keep keeps their interest and confidence with math (not a teacher nor a Math person, just someone who has seen one too many kids completely give up on math). Here are my follow-ups to your response.

    1. You wrote, “Similarly, 1 and 1i are both “1 unit” away from 0, but in separate directions.” So is this to say that i is the unit of measurement on the imaginary axis ? And with magnitude being a measurement of distance with respect to an origin, we get – in this case of ‘1i’ – the measurement of a 90 degrees rotation around an origin ? But, then is that to say that the units in the imaginary and the units on the real axises are equivalent – thus the units on the real axis are also based on ‘i’s unit notion of rotation? If this is so, then I think there may be hope for me. If not, then wouldn’t we have 2 axises with different units in a way similar to say.. feet on one axis and miles on the other? In these types of scenarios, don’t we have to convert one of the two units of measurement to match the other before we can do things such as calculate the magnitude of a vector/hypotenuse using the Pythagorean theorem (or at least account for the difference in units somewhere in the calculation)? So, if the units on the real and imaginary are (or can) be different, then how/when does this conversion occur with the use of complex numbers ? Perhaps the key to my confusion lies in a misunderstanding of what a “unit” is (but I am pretty sure I have the basic idea) and the rules invoked when calculating numbers that have different units. I can understand the scenario when you do something like (multiply 2 dogs by 7 cats = 14 dogs and cats), but if I had 20 miles on an x-axis and 20 feet on a y-axis, it seems like I would have to convert one of the two units, so that they are equivalent, at some point in the calculation of the magnitude.

    2. Thanks for giving me a non-symbolic definition for multiplication. Your answer is exactly the type of response I was interested in. I think one of the interesting aspects of your description is the use of the term “how” in the following, “how to apply the “essence” of one number to another?”. This question of “how” almost assumes that the essence of the two numbers can take on an interaction between each other. Do you believe your description can extend to concepts beyond numbers ? Speaking of extension, to extend this conversation even further, how would you then make a similar abstraction for Addition? What would you say are the distinguishing characteristics between the two abstractions such that they deserve their own term ? With the assumption that your abstraction of multiplication can be extended beyond numbers, and without yet having your abstraction of Addition, how would you categorize the interaction of 2 chemicals ? Would you say the essence of a mixture more so mimics Addition, Multiplication, or perhaps one followed by the other ? Assuming I interpreted you description of Multiplication properly, perhaps an example of Multiplication vs Addition could be found in nature: When reproduction occurs, Multiplication takes places (genetically) and the result is an Addition to the family. So in this sense, the interaction from the properties of the parent’s genes had a direct affect on the resulting offspring (Multiplication), and the Addition to the family (although grew the family) did not affect (at least genetically) any previous “Additions” that the family already had (i.e older sibling) …… Well, maybe you have a better analogy cause it doesn’t work so well against the concept of Negative.

    Thanks again for the response !

  172. Actually, the dogs and cats thing does not make sense to me. Why would you multiply 2 dogs by 7 cats. There seems to be no reason to do this (by the way, adding dogs and cats does make sense – thank goodness :-) ). Not sure where my head was at…perhaps a better example would be 4 rows times 5 people/row to get 20 people. But here the units cancel out and it makes sense to me. Oddly enough, the bizarreness of the cats and dogs scenario is equivalent to the bizarreness I find of the possibility that the real and imaginary units are not the same, nor are ever adjusted, for calculations such as magnitude.

  173. Hi Kalid, I wanted to let you know that, after further exploration, I am at now at peace with my level of understanding the imaginary number. Thanks again for the insight.

    However, when you have an opportunity, I am still interested in your thoughts on number 2 in my “Alonzo on March 13, 2012 at 1:11 pm” posting about multiplication vs addition.


  174. @Alonzo:

    No problem, you’re more than welcome. And it’s awesome that you’re helping encourage kids to keep going with math! :)

    1) Great question, let me try to clarify.

    * A “unit” is a measure of distance (like a mile)
    * An “axis” is a measure of direction (like North/South or East/West)

    Imagine a street map. There are 2 axes (North/South and East/West), and both are measured in units of a “mile” (let’s say). If we want to move somewhere, however, we might say “1 Mile North” or “3 miles East” — i.e., you need to specify the distance and direction.

    In the case of imaginary and real numbers, we have two axes (East/West = real (positive and negative), and North/South = imaginary (positive and negative)). Distance traveled on each axis is measured in the same “unit” (there’s no specific unit like miles, we just call them “units”).

    It’s true that you *could* have different measures of distance (i.e., the real axis uses “feet” and the imaginary axis uses “miles”) but in math, we assume the units are the same size on each dimension. (And if they weren’t, in some given application, you’d probably convert them first… i.e., a mile would be 5280 feet, and you’d write that down on the “feet” axis).

    So, the assumption is that when dealing with the real & imaginary axis, the distance measures are using the exact same units.

    2) Awesome questions.

    >> This question of “how” almost assumes that the essence of the two numbers can take on an interaction between each other. Do you believe your description can extend to concepts beyond numbers?

    Yep, I think the concept of “multiplication” can be used on lots of other ideas in math. In calculus, we “integrate” functions, which is a beefed-up form of multiplication (or alternatively, multiplication is a special-case of integration). The essence of “applying” one concept to another shows up in many places, but we don’t always use the term “multiply”.

    >> Speaking of extension, to extend this conversation even further, how would you then make a similar abstraction for Addition? What would you say are the distinguishing characteristics between the two abstractions such that they deserve their own term?

    You may like this article:


    To me, addition can mean “accumulate, slide or combine” depending on what is being added. In the case of chemistry, yep, I’d say addition corresponds to combining. (2 Hydrogen + 1 Oxygen = Water + Excess Energy).

    With nature, yes, a lot of it depends on the metaphors we’re using. Individual families may add new members, specials as a whole may multiply exponentially. I think part of it is stepping back and looking at what is happening, vs. what we named it :). We shouldn’t confuse the sign on the road for the road itself (very Zen!). Basically, we’ve given words some meaning (addition, multiplication, etc.) but we need to see what’s actually happening to the number, and whether a better word could apply (“Hrm, I’m saying add here, but do I mean combine? I’m saying multiply, but do I mean scale?”).

    For addition vs. multiplication, my intuitive meanings are basically:

    * Addition: accumulate, slide, combine
    * Multiplication: repeated counting, scale, apply

    Multiplication can look like “faster addition” if you’re talking about repeated counting vs. accumulating one at a time. But that’s only one use of the term, and other uses (like scaling something to make it larger or smaller) don’t have a direct analogue to addition.

  175. Hi Kalid, glad to hear that the units are assumed to be the same. This is consistent with my new understanding of what complex numbers are and how they are used. Nice !

    Ready for some more curious (but familiar) questions ?

    In your response to number 2, your wrote:

    “Basically, we’ve given words some meaning (addition, multiplication, etc.) but we need to see what’s actually happening to the number, and whether a better word could apply (“Hrm, I’m saying add here, but do I mean combine? I’m saying multiply, but do I mean scale?”).”

    So, what seems to happen is that depending on the area of math one is studying, you may be given a different definition and interpretation for operations such as Multiplication (I think even within the same area of Mathematics there will sometimes exist varying definitions). I find it odd that one can create a set of axioms for numbers and then say “Here is how we will define multiplication for this number system….”. And then, the next number system being stated will say: “And – here is how WE will define multiplication…” And of course, the two definitions do not always seem consistent as you try to interpret the implications/interpretations/intuitions. One intuition says scale, one says repeat, the other says rotate – all derived from the use of the same term. So, I have to ask…with all of these different definitions and interpretation of these operations that float around, do they all have to map back to some set of axioms or laws in order for a mathematical system to invoke its definition of multiplication ? (Sounds similar to my first question, but differs b/c now it is less about intuition and more about requirements)

    Or is it the case that the terms gets used in various systems of mathematics with no minimum criteria that must be met. That is, could I create a set of axioms and then use those axioms to define Multiplication as whatever I see fit – no relationship at all required with the previous use of term? Or is it that mathematicians use an agreed upon intuitive principle to define “how” Multiplication should look like in your system when trying to impose the properties of one mathematical object on another ?

    If you tell me, nope it is whatever you want to define it as (just depends on context, just like in the English language) and is just a tragedy (or perhaps a blessing in some unexpected way) in the field of mathematics – it would mean something to me. If you told me, Mathematicians use the high level principle which maps back to “how” you believe the essence should be applied b/w 2 numbers – it would mean something to me. If you say, Alonzo there are a set of axioms that are required to be true or properties that must hold before you can use the term multiplication and here they are – it would mean something to me. No matter what the answer, clearing it up would mean something to me. Although if you say I have no idea, that actually would mean something to me also because it tells me I am not the only one with this question.

    And maybe you have already said it in one of your previous posting to me and I missed it. For example, you wrote, “I think we’re still discovering what multiplication in it’s purest form”…so is this support of the idea that we don’t really know what it is or has become – so folks just define it in a way that is useful for their system? But then, wouldn’t this force the use of the term multiplication to become meaningless overtime. Why, why, why so many different definitions for the same term? How are we allowed to do this in such an precise field of study ? It seems like math is consistent once a term is defined for a particular system, but the field of study is not consistent with the use of its terms among the various areas. I am rambling now – so I’ll shut myself up.

  176. Hi Alonzo, glad if some things are coming together :). You’re raising very good, very deep math questions which are nudging up to the edge of my formal math knowledge!

    Intuitively, we’ve developed better and better understanding of what “numbers” are. We first thought numbers were for counting rocks (integers). But wait, we have fractions! (Rational numbers). And there are numbers between fractions! (Real numbers). And negatives (only discovered in the 1700s!). And complex numbers.

    You’re correct: each new set of numbers required us to redefine what “multiplication” meant (“Hrm, what does -1 x -3 mean? It can’t be repeated counting…”). It often happened that at the “lower” levels (integers), multiplication was a special case of what happened at the higher levels (repeated counting is a special case of scaling, when you’re dealing with whole units).

    Mathematically, the definitions of numbers and their operations is called an “algebra”. (“Elementary algebra” is what we think of as solving x^2 + 3x = 5; “Algebra” as a class is about the very definitions of math). In an algebra, you can define what a number is, what types of operations can be done (add / subtract / multiply / divide), whether there are any special elements (anything times 0 is 0), and so on.

    Wikipedia has an article (http://en.wikipedia.org/wiki/Algebra), but like many Wiki articles, you already need to be an expert to understand it. But there is a cool chart midway down that shows different types of numbers (Natural numbers, Integers, etc.) and the different operators and properties they have. Phew!

    Yes, for laymen, I think “multiplication” is changing over time, just as “number” is changing over time (500 years ago, numbers could only be positive; 2000 years ago, numbers could only be fractions… in 500 years, who knows what “number” will refer to?).

    In everyday discussions, I’d say “number” and “multiplication” refer to the most popular systems of the day, so “number” means real number (since people are comfortable with decimals, but not imaginary numbers) and “multiplication” means scaling (since they are comfortable with decimals, multiplication can stretch or shrink a number, but not rotate it).

    In more rigorous math discussions, however, you would need to mention what type of numbers you’re describing (one of the common types, like Integers or Complex numbers, or your own type). And if it’s your own type, you need to describe the behavior of the “x” symbol (which you may call Multiplication).

  177. Hello Kalid. Over the weekend, I gave some thought to what you wrote in your last post and I would like to see whether my understanding is on the right track. The following is a story I am trying to put together in my head by collection various pieces of information (I am sure there are gaps):

    It sounds like the Addition (along with Subtraction) of objects represents one of the most natural concepts to humans. Mathematics provided a formal definition for Addition (perhaps there were multiple definitions) to capture this natural observation and – from there- realized inherent properties and conclusions based on these definitions (and it was useful). But at some point, someone decided to define a more complex operation on objects, yet – it still had addition at its core. In the early days, the complex extension was in the form of repeated addition, thus we have the early definition of the term multiplication – and it was useful. However, overtime folks realized different forms/definitions of complex operations that also had addition at it’s core – and they were also useful. Because these operations seemed to share an enhanced complexity involving addition – they were also called Multiplication. Of course, based on the various definition, we would get various inherent properties and interpretations (though some were shared among the different definitions)

    So perhaps a pattern was established with these various forms of “complex addition” such that the operations result in a complex interactions between the objects involved. This complex interaction forces the very nature of their interaction to have a strong influence on what the result looks like. This leads to descriptions of Multiplication like you stated above, ” the essence of multiplication is how to apply the “essence” of one number to another”.

    So to generalize, Addition for your system asks, “how” do you want to define a simple union, collection, joining for the object in your system. Multiplication asks, “how” do you want to extend the use of the addition operation to define a more complex interaction among the objects in your system. Now your definitions will probably result in observed properties, and some of them may actually be useful. But, whether or not the properties or the system as a whole turns out to be useful – you get to define it. The questions of “how” associated with Addition and Multiplication asks you some common questions associated with defining a Mathematical system. Thus you gets statements like those from above: “And if it’s your own type, you need to describe the behavior of the “x” symbol (which you may call Multiplication)” – you get to define it.”

    I suppose you could generalize it even further by saying Addition is “how” you – the creator of a system – would like to define a simple interaction between your system’s objects and Multiplication is “how” you – the creator- would like to define a more complex interaction between your systems objects. But, at this point, it seems like the abstraction has gotten to a point where meaning and the ability to distinguish starts to be come lost.

    So Kalid, would it be possible for you to comment on this story I am trying to create for myself. Perhaps some things are factually incorrect. Other things are consistent with how things really work. In other words, as someone with much more knowledge on this subject that myself, would you critique the story ?


  178. Hi Alonzo, happy to comment. I think you’re on the right track: we started with very basic interactions (addition), came up with a concept for repeats of this interaction (multiplication as repeated addition), and realized this more complex interaction could have other properties of its own (scaling). I think the true mechanisms of multiplication probably dawned on people over time (like genetics; previously it was the “bloodline”, where tall people had tall children, and we later realized it wasn’t the blood so much as the DNA inside which determined hereditary traits… we got more specific and nuanced in our understanding).

    Looking back, we can now see that multiplication is a more advanced interaction which seems to transfer properties (like dimensions! inches x inches = square inches) whereas addition does not (inches + inches = inches). So, I think your story seems to make sense — humanity has had a gradual unfolding / clarification in the meaning of math.

  179. Kalid, many thanks for your patience and explanations to the crazy questions I have asked. Although I have expressed frustration in some of my previous threads with trying to grasp things – it has all been worth it! As I read back over the discussion, things really started to sink in. I truly have a perspective that was not available to me before. I simply don’t know why this perspective has comes so late in life. But, it will help me to help others with their frustrations. In fact, I plan on using our correspondence for the basis of conversations with folks that I think may be stuck on similar issues.

    But again, thanks for your time and insight. The education your providing people on this website probably has more of an affect than you know, so excellent job on making an impact !

    Until next time, take care.


  180. @Alonzo: You’re more than welcome, I really enjoy these types of conversations. There’s no time limit on when these aha! moments come, I’m coming back to topics I studied in school 10 years ago and still learning new things. I’m thrilled if you’re able to share your aha! moments with others, that’s what learning is truly about.

    Thanks again for your kind words — I love the interaction of sharing ideas, getting feedback on what works (or doesn’t) and polishing explanations into things which really click for us. Till next time!

  181. Wow. This is just phenomenal. This, coupled with the videos from Khan Academy really helped me truly understand this. I’m not completely comfortable with it, but its a huge improvement from where I was an hour ago. The chart comparing negative numbers and complex numbers, the idea of rotation and the example of the boat were amazing. I just realised my school did not even teach us any real life applications of this chapter, and that’s quite dissapointing. Its one of the reasons why some of my friends still struggle with this chapter. For me, If I don’t see any application, I usually show little interest in it, and this was a life-saver. Thanks!!

  182. I must be Thick! or have petrified thinking ( I’m 65) . If i can be any angle .
    How can i +1 be an angle of 45 degrees. Please don’t groan!! I know I’m missing the concept especially since Ethan notes that its obvious..

  183. @Dave: No worries, this is a tough concept :). “1 + i” is 45 degrees the same way “1 Mile East + 1 Mile North” is 45 degrees — you’ve moving the same amount in each direction, so are “diagonal” from your starting point. Hope this helps!

  184. Love the article; it made things so much clearer. The thing about turning 1 to -1 intrigued me, even though the concept seems so simple…. Are you saying that all multiplication sort of starts with 1? Likewise, does all arithmetic start with 0? I wouldn’t have thought this idea was all that important before reading this article, but now it does seem like it has some significance; the concept of getting from 1 to -1 wouldn’t make sense if 1 was not a starting point. This is because 1 is the multiplicative identity (if that’s the right term…), right?

  185. @Pandaroux: Great question! Yes, I’ve come to expand my understanding of basic arithmetic, and the idea that addition and multiplication are transformations to some base.

    “Plus 3” really means “slide something over”. And how much? Well the number 3, seen from an addition standpoint, is starting at 0 and sliding over 3 units (0 to 1, 1 to 2, 2 to 3).

    “Times 3” means “make it 3x larger” when using a multiplication standpoint. In this perspective, you start at 1 and scale to 3x the size.

    Breaking arithmetic down like this helps understand how funky numbers like i can work. There’s some more here:


    I didn’t understand i until I started thinking of it as “What transformation, when applied twice, could turn 1 to -1?”

  186. Thank you for this nice article and for making your ideas public with this website. Everything I always thought about learning math, but rarely found, I found here.
    I always thought that the kind of understand I wanted to have was a ideal thing, that only a few people would look for.
    And as I study medicine, I dont have time and energy to go through the process of learning on the formal way, and them get to “that” undestand I was looking for.
    Thats why its good to read your articles, you do the hard way for us , and bring it the way its easier to understand, something that all teachers shoud try to do.
    Its very good to know that there are more people, and very intelligent ones, and also graduated, that think like me.
    Really, congratulations for this site. And continue this work you are doing, because as it did to me, i am sure it helps the math learning, and contribute for creating more enthusiasm for knowledge for young and even older people who was educated to see math as something cold and static.
    Ok, here is my question. If the angle wasn’t 45º, but was like 58º, you would have to use pythagoras to discover the x and y of X + Yi, wouldn’t you? It wouldn’t be 1 + i
    Then even with imaginary, we still depends on pythagoras, am I right?
    sorry about my english, i am from Brazil

  187. @Douglas: Thanks for the comment! (Great English by the way). Really happy you’re enjoying the site, I like connecting with other people who want to find the intuition behind concepts. Math is only cold and static if we don’t really understand it :).

    Good question — if we have an angle like 58 degrees, we’d need to use sine/cosine/pythagorean theorem to figure out the x and y. In a world where we somehow know that the angle is exactly 58 degrees, we would also have calculators to compute sine and cosine.

    In a world where we just measured the angle (measuring the trajectory on a map, for example) we can just use the measurements directly. That’s the really neat thing about imaginary numbers!

  188. Hi Kalid,

    I already emailed you once thanking you for the very clear explanations that certainly helped me. It now strikes me that besides for finding roots of equations, complex math is used almost exclusively for wave-like phenomena, and the reason is because they occupy a closed circular space, where if you advance one full cycle you arrive back where you started, and yet it also has a magnitude dimension that multiplies the conventional way. A LOT of confusion could perhaps be avoided by re-naming the “imaginary” dimension to the “rotational” dimension! The weird thing about complex math is that it merges the linear and the rotational in a unique way, to create a linear/rotational hybrid with unique properties that mirror the amplitude and phase properties of wave phenomena.

  189. Hi Steven, thanks for dropping by — I must apologize, I believe I have an email response I owe you on some of your other work as well!

    Yes, exactly, a huge application for complex numbers is anything which cycles (since they seem to do so naturally, and are a great way to model it). They’re almost a way to use rectangular coordinates in a polar way (i.e., I want to rotate my x,y coords, so I multiply by i, or another complex number, instead of converting everything to polar).

    I’d be hesitant to rename to something as specific as “rotational” because there may be other (yet undiscovered) applications, but almost anything is better than “imaginary”! :) Perhaps “alternate” or “supplementary” numbers.

  190. @Jaycee: Awesome, so glad it helped! Getting imaginary numbers to click was one of my favorite moments in all of math.

  191. Brilliant explanation!!! This is by far the best and most helpfull math website, I have ever come across.
    However, I have a doubt. Earlier, in your rotation example, you rotated 3+4i, 45 degrees by multiplying with (1+i). while I understood this, I am thinking, “why dint he multiply by i/2″? if ” i ” rotates a heading by 90 degrees, the surely, “i/2” should rotate it 45 degress.

    i did calculate (3+4i) . i/2. Turns out, the new heading is 90 degree rotated. but the magnitude was halved. Still not getting why this is so.

  192. @prakash: Awesome, glad it’s clicking. Great question by the way.

    When you multiply by a complex number, you also scale by its size. i has size 1, and i/2 has size 1/2. So that’s why multiplication by i/2 halved the magnitude (if you like think of i/2 as i * 1/2, and do the multiplications separately).

    To keep the original distance the same, you want to multiply by (1 + i)/sqrt(2). (1 + i) has size sqrt(2) [by the Pythagorean theorem], so we divide by that amount to bring its size back down. There’s more in the follow-up article on complex numbers. Hope this helps.

  193. I’m a Biomedical Eng for a large research institution, my son just started 1st yr Engineering after taking a year off. When he asked me about Linear Algebra I kinda felt like I was having a flashback. My first yr of Engineering consisted of 48hr/wk ontop of a part time job, leaving me to have to make some sacrifices for time. Algebra was the victim (sorry), I only went to the first class, got the assignments and showed for the mid-term and final ( scored a B+ ).
    after reading this blog and watching the video it has made this subject crystal clear, if you aren’t a Phd by now you should be.

  194. Thanks Charles — really glad the article was able to click for you. Heh, I might like to go back to school one day, right now I’m having a lot of fun just studying math on my own :).

  195. gr88888888888888888888

    im in the last year of engineering and in quite a lot course i came across complex number and even before engineering. but bcz i never understood it, my interest was always very low. But now i know it would be different thanx to you.

    And yeah the way you take your time to respond to the long questions in the comments section is simply great.

    again thanx.

  196. Kalid, Awesome page! I’m a math professor and have taught about complex numbers from almost this exact same perspective for about a dozen years, ever since I had the same revelation: Numbers can be thought of as transformations, and complex numbers as transformations in two dimensions. You’ve done a great job getting this message out to so many people.

    One thing I’d like to mention for your readers: Lots of advanced math is about discovering transformation rules for more complicated kinds of data (like high-dimensional vectors and functions). For example, consider rotations in 3D – it matters what order you do them in! Try rotating a book about a vertical axis, then a horizontal one. Then try the other order!

  197. @Bob: Thanks for the comment, I love hearing from other teachers! Great point about the rotation — I had wondered why quaternions needed 4 items (not 3) and I think that’s unraveling it. I completely agree about math being about exploring transformations in general, I’m starting to see that as I look back. Another nice topic to write about :).

    @Ahmad: Awesome, glad it clicked. I really like having discussions with people, it helps cement understanding (my own as well — often I get questions that make me dive deeper for intuitive insights).

  198. great article Kalid – for the first time I feel I understand the damned things (which probably means I don’t 😉 – though I’d heard the rotation aspect before I hadn’t really understood it. Your style of explanation is excellent – I like the visual aspects of your methods, that helps a lot with me, and your conversational style keeps it light whilst not drifting off-topic.

    One thing that puzzled me was why adding 45* – whilst it was only talking about direction and not quantity – increased the quantity (the distance form zero) from 5 to just over 7 – I couldn’t get my head round why that should be, and a word or two of explanation might be helpful there (unless you consider it out of scope).

    “Today you’d call someone obscene names if they didn’t “get” negatives.” I suspect you will laugh yourself silly (or cry) at something that happened here in the UK a few years back: see http://menmedia.co.uk/manchestereveningnews/news/s/1022757_cool_cash_card_confusion !!!

    Thanks again for this site, a great resource and I completely agree that “it frustrates me that you’re reading this on the blog of a wild-eyed lunatic, and not in a classroom” – if only we’d had a lunatic like you teaching our maths class!

  199. Hi Richard, thanks for the note! Glad you’re enjoying the style, I try to write in the way I would want to be taught (in a casual, informal manner).

    Great question on the size. When multiplying, we give all “properties” to the result. When you multiply something by -2, not only does it become negative, it doubles as well.

    Imaginary / complex numbers will “give” their angle, but also their size. For simplicity, I made a 45-degree angle using a triangle of sides 1 and 1. But, the length of the diagonal is sqrt(2), or about 1.4 [by the Pythagorean theorem]. So, when multiplying, the final result was sqrt(2) or 1.4 times bigger than the original. In this case, we were only interested in the angle anyway, but for consistency we could divide by sqrt(2) to bring it back to the original size. I might need to make this more clear though, thanks.

    Ah, good old negatives, tricking us into the new millenium :). Although to be fair, it’s easier to think “lower” means smaller. Maybe a better wording would be “find a colder temperature” (not lower) and hopefully it’d be more clear.

    Glad you’re enjoying the site!

  200. Ah of course! That makes perfect sense now. Many thanks for taking the time to respond and make that clear to me.
    I’m looking forward to working through your other articles – all things that never made sense at school because of the way they were taught, unfortunately.
    I really applaud your work to explain concepts in a way that makes sense. You are in a very real sense empowering people – great work, please keep it up.

  201. Nice tutorials, Kalid, tyvm! I’m a math neophyte, but just enjoy numbers and came across your page and really enjoyed it. One thing I could not wrap my head around when you were discussing the heading problem was the fact that you MULTIPLIED the complex number (3+4i) by (1+i). I understood the multiplication process and even worked out the trig to make sure everything works, and it does. BUT it is not immediately evident to me WHY you multiply these two numbers rather than add them. Would you be good enough to point me in the right direction so I may understand, please? Thanks! -Tom

  202. Hi Tom, glad it’s clicking! Great question — basically, it comes down to the effect we want to have on the result.

    Multiplication gives the “properties” to the result. Starting with 3 and multiplying by -2 will infuse the size of -2 (which is 2) and the sign of -2 (negative) into the result. We’ll get -6.

    Addition is more of a “slide” — starting at 3, we slide along -2 units, to end up at 1. It wasn’t really a combination, more an adjustment.

    In the case of complex numbers, their angle is like the negative sign — it’s an intrinsic part of them. In order to “combine” the angles, we need to multiply the numbers together and infuse the result [this sounds a bit weird, but it’s just the analogy I use]. The actual reason this works is found here: http://betterexplained.com/articles/understanding-why-complex-multiplication-works/

    If we just wanted to slide the endpoints around [follow a trajectory of 3+4i, then follow another trajectory of 1+i afterwards] then we would just add them. That could be useful in another circumstance, i.e. we want our final position after taking two shorter legs.

  203. Thank you! The link was wonderful. Keep up the great work, it’s a lot of fun to read your pages. :) -Tom

  204. Thanks for this…imaginaries are now very clear for me. All I was ever taught in school is “i = -1” and the rules for manipulating powers of “i”. 😉

  205. Thanks Kalid for what you are doing, I know you know how much it is appreciated by many people.
    With this tutorial, however, I was with you all the way to this:
    “If we multiply them together we get:…” followed by the sums to get -1 + 7i (can’t copy n paste that part).
    The maths here has lost me. Where has 3i and 4i^2 squared come from??? :S
    Many thanks –

    A tired brain, UK

  206. Hi Ian, great question. When multiplying two complex numbers, we need to combine all the parts (more here: http://betterexplained.com/articles/how-to-understand-combinations-using-multiplication/). Imagine multiplying 13 x 14:

    (10 + 3)*(10 + 4) = 10*10 + 3*10 + 4*10 + 3*4 = 100 + 30 + 40 + 12 = 182

    We split 13 into “10 + 3” and 14 into “10 + 4” then combined each possibility: 10 & 10, 3 & 10, 4 & 10, 3 & 4. In a similar way, multiplying the parts of the complex number means we combine each part. When combining, if we ever get i*i, we can simplify that to -1.

  207. Ah-ha! Many thanks for your reply Kalid :)
    I’m fully with you now! If you can teach complex numbers to someone as dumb as me then you’re definitely doing something right! lol. Thanks again

  208. Hi there, I just wanted to say Thank you. I love maths but have never studied much beyond secondary school level. You have a great way of explaining things it makes it all very tangible. I am sitting back with the square root of i problem and slowly but surely getting there. I know I should see it already, but I will eventually :)

  209. Hi Ian, glad it helped! Give yourself some credit, imaginary numbers aren’t easy! It took the math field a few decades after their invention to really accept them :).

  210. Got it ! i.e what is the root of i . I may come back to you as something arose when I was trying to figure it out, that I cannot see a solution to but I will try before I bug you. Thank you again,Kalid. I will be back.
    :) from Ireland.

  211. thanks so much…I’m a beginning high school math teacher… looking to explain complex numbers so students can better understand it.

  212. well thanks a bunch! After realising i did not know what complex numbers were I thought it would be useful to learn. Excellent layout and presentation of knowledge. Now, on to prove the Riemann conjecture!

  213. Just found this by way of the Fourier article linked by a Facebook friend. Good stuff!

    Check out Isaac Asimov’s _Adding a Dimension_, if you haven’t already. You seem to have the same approach as he did. There’s even a chapter in it about complex numbers (including a conversation much like you had about negative numbers…).

    I mastered complex numbers years ago, but I’ll read this again – never know when you’ll pick up a fun factoid. (Prob’ly spend more time on the Fourier article…)

  214. Hi Paul, thanks for the comment! Haven’t seen that book yet, I’ll put it on my reading list (and man, I wish I’d seen that complex number analogy earlier in school).

    Definitely agree about the factoids. I’ve realized I’m never “done” with learning about even basic concepts. They’re like atoms, where you break them into electrons, then quarks, then strings (?) and so on. There always seems to be another level to explore.

  215. Thank you for your explanation, it makes a lot of sense. But i still am not able to grapple with the idea of some number raised to the power of i. could you explain that?

  216. I really want to congratulate you to this very nice explanation.
    My point is that the imaginary numbers as well as quaternions are just half the truth. Your explanation of i is so well anticipated because it uses the 2D space and rotations that are intuitively easy to understand. One of your first posters (Chick on 21st December 2007) wrote about the original problem that led to imaginary numbers and in fact it is a 2D problem as well.
    I have finished a masters degree in physics, and I have written i about 10.000 times in my life. I have solved hundreds of problems in my professional career using i and I always suffered that we just define it like that because ist’s so practical. And I had to become 45 years now when I finally stumbled above the _really big thing_ and that’s so awesome and mindblowing that I really want to share this here. Decovered already about 130 years ago it starts from the real side – multi dimensional problems and finds a super elegant, consistent way to describe much more than complex numbers, quaternions, and all that stuff. If you do it this way, you can handle 3D spaces as well as 5D spaces as well as Maxwell equations as well as Pauli Matrices _without_ the need of i at all but with an intuitive and consistent methodology. And best of all, when you are using this you get komplex numbers and quaternions for free. They are included, but you get an intuitive interpretation and you will see that they are just half the truth. And by the way, you will get rid of the x-product that only works in 3D space and is inconsistent. It’s so mindblowing. We really do NOT need i to describe nature in physics. So if anyone is curious – look for “Geometric Algebra”!

  217. I just want to try to give you a brief introduction into Geometric Algebra.
    In physics we learn that numbers for them self are meaningless. You always have to accompany them by the unit. 5 seconds are quite different from 5 Volts and 5 inches but also from 5 years. What Geometric Algebra does is mainly taking this second part into account. So i.e. in 2D space we can find 2 different directions e1, e2 (think here of unitary vectors that have their meaning (inches, m, ym, …) already coded in their length. e1 is i.e. 1m in one direction) These two directions we want to use as a base to describe our problem. We then can describe any point in our plane as an instruction how to go there from the origin. (go 5 times in e1 and 2 times in e2: x = 5e1 + 2e2). If we are in a nice 2D space we of course can go the other way like x = 2e2 + 5e1, so addition is commutative. You will then think about addition, subtraction and … multiplication. And then you will come to the point to ask what is the meaning of multiplication? Usually multiplication of 2 distances is something like an area and so it’s here. There appears an element that’s something like ‘weight times e1 multiplied with e2’ that I want to write as: Ae1@e2. A is just a scalar, the ‘weight’ and corresponds to the area. But what about these entities? e1@e2 is called Grassmann product of e1 and e2 and it’s anticommutative so e2@e1 = – e1@e2 what means it does matter which way around you go that area to describe it. So e1@e2 has some kind of ‘orientation’. (In physics we often describe planes with normal vectors that are oriented perpendicular to the plane – forget about the vectors but think of their properties – they can be oriented in two opposite directions to describe the same plane.) You then will find that this Grassmann product also does a rotation of 90 degrees if you multiply it with a vector in the plane and then you also understand why it’s important to know the direction. (Here you find already all the ingredients of complex numbers.) And finally you will find that squaring this aera-element corresponds to a 180 degrees rotation. Square(e1@e2) turns any vector into it’s opposite.

    1) Geometric Algebra treats numbers AND their entities (units, directions, … their meaning)
    2) _when it comes to multiple dimensions (multiple units, directions, meanings)_ i can be associated with the result of the Grassmann product i = e1@e2 which by itself is the unit element (an oriented plane) of the multiplication of _the entities_ of the two different dimensions.
    3) The Grassmann product e1@e2 has a clearly understandable meaning and i has been revealed as a powerful workaround that we already could get rid of.
    4) We could and should introduce a new aera of ‘meaning’ and ‘understanding’ into physics by application of the Geometric Algebra that’s already well defined and widely used in computer graphics. But we are already on the run. So that’s what’s really mindblowing!

    PS: I used a unusual notation here to avoid confusion. e1@e2 is usually written as e1^e2. As this might be confused with ‘to the power of’ I have chosen my unusual notation.

  218. This makes me wonder why we are taught to use “y” in the coordinate system instead of “i”. Y just seems extraneous since i basically means y. Is it just because people are avoiding the use of imaginary numbers or is there some reason to still use x-y instead of x-i?

  219. Hi Harmony, great question. “x” and “y” are the generic descriptions of two different dimensions (such as distance vs. time, income vs. time, etc.), and “i” is a specific interpretation. In fact, the interpretation of i as a new dimension wasn’t discovered until many years later. But, it would definitely help to have i used as the “y” dimension in a lot of problems — then it wouldn’t seem so strange!

  220. Hi Harmony,
    an important difference between y and i is. y*y=1 in classical vector algebra like x*x = 1 while i*i = -1. x-y diagrams are way more intuitive to the unexperienced pupil like x-i diagrams.

  221. Thanks Oliver, that’s a great point. x and y are identical, whereas i implies a rotation (so indeed, i*i = -1).

  222. This was really helpful, Kalid. Thank you for helping me to get a much better understanding of imaginary and complex numbers, something that has puzzled me for a long time. You are doing a valuable thing here.

    Vegard, Norway :)

  223. This is a good explanation .You have used the Argand’s Diagram for representation of complex numbers.
    Another use of ‘Imaginary Number’ , is to reduce the computation involving Infinite Power Series . Try to find out ” Sin^2(x)+cos^2(x) = 1 ” , using Infinite series representation of Sin(x) and Cos(x) .It becomes much easier to represent by the Complex Form—Exp(i*x) = cos(x) +i*sin(x) and Exp(-i*x) = Cos(x) – i*Sin(x)
    The Product Exp(ix-ix)=exp(0) = 1 = cos^2(x)+ Sin^2(x) ,gives the desired solution in a compact manner.

  224. I graduated with a BSc in electrical engineering and I haven’t understood complex numbers like I do now
    Open a university
    Great thanks

  225. @Vegard: Thanks!

    @Brajabasi: Thanks, Euler’s formula is a great application.

    @Karl: Very glad it helped!

    @Stan: The problem with that reasoning is you’ve excluded the possibility of numbers beyond positive and negatives from the start. Similarly, someone might argue that a decimal like sqrt(2) doesn’t exist because 1*1 = 1, and 2*2 = 4, and therefore no number can be squared to get 2. Why must we limit ourselves to numbers that can be shown on our hands, or along a single dimension?

  226. You have not showed how to rotate a vector about an arbitrary angle.
    Also, if the goal was simply to have a way of find negative numbers, why not something simpler, such as a flip flop?
    You also have not defined Complex and imaginary numbers operations, even though you used them (I saw that you linked it but you should still touch on that).

    Apart from that, good article, thanks.

  227. I’m pretty horrible with math in general, I tend to think its in part due to my brain shutting down because of the way its taught (rules to memorize of short-cuts that are sometimes counter-intuitive taught by teachers who say “why? because that’s the way it works– that’s why”). I’m pretty horrible as in have at best a feeble grasp on algebra, and while doing some signal processing coding I of course came into fast Fourier transform which I had to sit down and read into in order to help wrap my brain around them, which like a lot of this stuff became a recursive process for me and started at complex numbers.

    In trying to understand the ‘why’ of it I came across this explanation, which for me, was well ahead of anything else I had read and of course, the humor kept me smiling through the process. I’d be lying if I said I totally got it still, but the idea of rotation and additional dimensions cemented the very beginnings of it.

    Thank you quite a bit, I look forward to reading your other writings. Please keep it up!

  228. if
    multiplying by i rotates the vector counter-clockwise
    multiplying by -i rotates the vector clockwise
    how about multiplying by +(1/i) or -(1/i)?
    in what direction vector shall go?
    will it change direction only?
    magnitude as well?

  229. @jamal if u find out 1/(-i)=i & 1/i=-i( i.e by conjugation multiplying & dividing numerator & denom by i) u will be the same magnitude but only the direction of rotation is opposite to that what is mentioned…

  230. Wow. This was amazing. I’ve just subscribed. I’m in love with maths but I often have no idea where to start; this is now my favorite resource right next to MathIsFun.com and Khan Academy on YouTube. Thanks!

    (and oops I was using the question/feedback form up above to post this comment. Ignore that!)

  231. Hi Kalid, tried to post a comment without any luck. Hopefully this works.

    Thanks very much for the guides, I particularly enjoyed this as well as the guide on exponentials. e has always given me such a headache! Using what you taught us there (namely that e is the limit of rate of growth…did I get that right?), what can we say about Euler’s Identity (\displaystyle{e^(i*pi)=-1})? Is there an intuitive physical meaning?

    Again, thanks so much, you’re very good at what you do!


  232. The way I like to think about it is that “i” allow us keep the reverse of an exponent just like negative numbers allow us to keep the reverse of an addition. For instance,
    let’s say x=sqrt(9). I can express x as being equal to sqrt(-3*-3). Then I should have the right to write x=sqrt(-3)*sqrt(-3) just as I can write x = sqrt(3)*sqrt(3) no?. But without imaginary numbers, this doesn’t work. I can’t “store” a negative square root, just as we didn’t have a way to “store” a negative result before negative numbers came in existence. So in order to keep algebra consistent under the exponent function, we had to come up with the imaginary numbers. Otherwise, we would have ended up with inconsistent results, as in this exemple, depending of the actual operations used to solve the problem.

  233. This is just a mad random thought.

    Could we not use the idea of the square root of minus nine to describe, more completely, the idea of the number 3 looking at itself in a mirror.

    When I see myself in a photo, I realise that my hair is parted on the opposite side when I look at myself in a mirror.

    In a mirror I appear identical, except that I am a rotated mirror image.

    Martin Escher’s drawing of a “hand drawing itself” may also be relevant.

    Also, what happens when I hold a mirror in front of a mirror? The recursive image disappears to infinity.

    As an only child, at about 2-3 years of age, I remember thinking that I had found my sibling in the mirror. Why wasn’t he behind the mirror when I looked? Was that my “square root of a negative number” moment?

    I just love your website. It reminds me of how little I currently actually know and conversely how much there is still to learn and understand.

  234. Hi. Thank you for your explanation. I actually found your site after searching for information on Pareto’s 80/20 idea, but I just could’nt help browsing your explanations of things, and boy am I glad I did. This notion of a complex number has given me nightmares. I love your take on the subject. I especially love what you said about Euler and him not even understanding negatative numbers. Did you know that when he wrote his papers he did it in a style so that he ‘did’nt have to argue with those of lesser understanding’? From what I’ve read, those are actually his words…That guy has given me too many headaches for too long. Thanks again.
    By the way, and this is a bit unrelated to the topic, but I was wondering if you had any idea how that other genius of the age, Einstein, figured out how E=M.(the freakin speed of light)^2…How did he know E was proportional to lightspeed^2? When I try to relate the formula to something I know a bit about, I think of nuclear material breakdown, and it works, but how on earth did he know the formula depended on c^2! Apart from complex numbers, this is the last thing I need to know before I expire, so your assistance would be much appreciated.

  235. This article can do without the incessant exclamation marks, childish outcries, and baseless foundations such as the Romans not understanding division.

  236. @Brian: Thanks, glad it’s piquing your curiosity!

    @Vernon: Awesome, glad you’re enjoying it. Many times, topics are presented in an overly formal or complex way — to appear impressive? In a misguided attempt to “wow” the student? I’m not sure.

    Einstein’s original paper is only 3 pages: http://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

    And especially this statement: “If a body gives off the energy L in the form of radiation, its mass diminishes by L/c^2”

    (Note, he used L and not E to represent energy originally). He basically realized that as energy was transferred, the mass changed by m = E/c^2. After some re-arranging, you get the famous E = mc^2. There’s more to it, but his original formulation was a different form of what we recognize today!

    @ACG: One my favorite parts about the internet: to each their own!

  237. Good question raised by Vernon, It is well know Einstein’s passion of Maxwell’s equations in electromagnetic field and energy. Add to that his obsession of speed of light since he was a kid.
    I think it is the energy (or more specific, kinetic energy) that led him to link mass to speed of light.
    This video http://www.youtube.com/watch?v=hW7DW9NIO9M explains why (m) suddenly appeared in Einstein’s equation. Or you may hear it from Einstein himself http://www.youtube.com/watch?v=vb1EO6aoaFQ

  238. I love this and I will be sharing it with my high school students even if it makes some of them glaze over and hurt their brains. It is an excellent representation of i. thank you for creating this.

  239. May I suggest you edit your discussion of negative numbers with the following observation: According to the http://en.wikipedia.org/wiki/ article on Double-entry_bookkeeping_system, it was fully in place (by the inventor) in 1300. Today, the confusing (to me, anyway) system of ‘Debit’ accounts and ‘Credit’ accounts seems to be an obvious holdover of a primitive and awkward understanding mathematics (at least from a modern point of view).

    Imagine for a moment how much easier it would be to learn accounting if accountants learned (or accepted) ‘modern’ (post 1759) mathematics!

  240. I really used to hate complex numbers. Now I’ve started loving them. Infact complex numbers aren’t really complex at all.

    Great job man… :)

  241. Wow, great article. This might look incredible, but it turns out that I actually learned of imaginary and complex numbers for the first time by actually asking a question about rotations. I was asking my mom what does it meant to have a negative velocity, and she told me that it was traveling the same speed but in the opposite direction. I wanted to throw a challenge at her and at the same time satisfy my curiosity, so I said: ” okay, so traveling at -5km/h North is the same as traveling at 5 km/h South. But how about the relationship between North and East? If I’m traveling at 10 m/s East, at what velocity I’m traveling towards the North?”. She stared at me for a while, but then she said: “Solve this equation, and if you make it I’ll tell you, and she wrote in my notebook: \displaystyle{x^2+1=0}. Immediately I told her that there is no number such that when squared you get a negative number, and she said “Oh yes, there is, and that number is ‘i’.” I was like “what??! What are you talking about?”. Then she introduced to me the new numerical system of the complex numbers (which is not really that new). I kinda felt disappointed because I really thought she wasn’t going to be able to answer that one, and that was what I really wanted, but then at the same time it felt good I knew the answer. The next day I went like crazy to my science teacher and I asked her the same question, but this time I was testing her rather than just looking for an answer. She told me that there is no such a thing but I told her: “You’re not telling me the truth, Miss. I know you know the answer.” Then I explained to her and she started laughing and said “Oh okay you got me here.” Nowadays I spend looking for answers to all sort of different questions about complex numbers like “What’s the sine of i?” “How do I take the logarithm of a complex number?” “What meaning does ‘i factorial’ have?” or “Is i=-i?”. I already found the answer of some of them, others, not quite yet. I know the sine of i equals \displaystyle{(e^2-e)i/2e} and that in order to take the principal logarithm of a complex number you have to use polar form. However, I still don’t know what is i factorial or don’t know whether -i=i, because -i and i cause some ambiguities. What is so funny about this is that I started dealing with complex numbers when I was 10. I’m 14 and I’m still looking for many answers. In fact, looking for these answers led me to this blog.

  242. In your video, there were the 3 follow up questions in the end, do you have the solutions to those? Thanks and love your articles :)

    Btw, “Or anything with a cyclic, circular relationship — have anything in mind?” Didn’t quite get that…

  243. Hi,

    Why don’t we just call these numbers by x and y instead of real and imaginary. For example say 3x+4y instead of 3+4i.

  244. If the concept of the number line embraces zero, negatives and irrationals, and indeed even makes the relationship between them obvious, then why not extend the number line to the number plane to include imaginaries.

    I am troubled by the invocation of rotation as being essential to understanding complex numbers…it seems like pulling something out of your hat. Algebraic operations along the number line simply move back and forth along the line. Why not consider operations in the complex plane simply operations that move about the plane, with rotation simply being one manner of moving about the plane

  245. Hi Raj, great question. In hindsight it seems we should just use x and y, but the notion that imaginary numbers were 2-dimensional wasn’t thought up until decades after their initial discovery. The name when these strange new numbers came about (“imaginary”) stuck, unfortunately.

  246. Really cool…..i don’t understand how my teachers couldn’t get this through like this. Thanks so much for the site. It’s really easy to understand.

  247. So I’ve read comments down to number 100… I’m not sure if the explanation would have worked for me. I prefer this:

    1. There is no square root of -1 in the real numbers.

    2. You can define laws of “addition” and “multiplication” to operate on ordered pairs or tuples (a, b) of numbers in such a way that tuples with b=0 behave exactly as the corresponding real numbers do, and (0, 1) x (0, 1) = (-1, 0). i.e. complex numbers.

    3. Of course you can plot these tuples as points in two-dimensional Cartesian geometry. And of course this looks just like your diagrams.

    4. Polar coordinates.

    5. Everything else about complex numbers.

    This possibly isn’t so good for students whose mathematical intuition is geometrical. I like numbers. I like that Cartesian geometry does all the things that Euclidean geometry does because tuple coordinates and number sets -are- “points” and “straight lines” which fulfil the axioms of Euclid and therefore fulfil all their consequences, theorems and all that, although it may be unfortunate that geometry then is just a matter of algebra which you can still do without a special geometric insight. But, back to the positive, it disposes of anxiety – mine anyway – about just insisting that an equation has a solution when it doesn’t (in real numbers), and the apparently meaningless difference between i and -i, which looks like it should be hugely important. For me, good mathematics is logical, and you can’t have things just because you want them. So, my way, you just observe the properties of (0, 1), and call it “i”.

    For quite a while, it felt to me as though someone could discover an argument that completely destroyed complex number mathematics. I’m much happier starting basing it on Cartesian geometry and good logical rules for operations.

  248. Why is the number one multiplied in the 1*X^2=9 problem? It seems uber-redundant and something that is meant to confuse people needlessly. To me, this whole imaginary number thing seems much like religious doctrine, which uses peer pressure to get people believing in something that was made up. Even after going through many different websites trying to find different and more thorough explanations of imaginary numbers, I am still seeing it as bullshit and the more you try to explain it, the more it seems like tooth fairy B.S. or like Jesus turning a couple fish into food for thousands. It also seems that even the guy in the video doesn’t really understand it, as there are far too many dead-ends in the explanation. My running theory is that everything is just made up and forced down people’s throats until they just accept it as truth. Saying (“just pretend i exists”) is not far from pretending that the tooth fairy exists and me, as an adult, put a tooth under my pillow and wake up to find my tooth still there. I don’t see the “i”, note do I see the need to include the number one in the equation x^2 =9. I am not trying to bash your explanation-I just really want to understand. Nobody has delivered a satisfactory explanation thus far. Thanks for trying though.

  249. ***I meant “nor” do I see, not “note” do I see…in the third to last line of my comment.

  250. Hi Alex, the reason for writing x^2 = 9 as “1 * x * x = 9” is that it makes it clear that each “x” is a transformation on the number 1.

    In a similar way, we might write “-3” as “0 – 3” to realize that we are “three below zero”. Once you are familiar with imaginary numbers you don’t need to do this, but the first time it’s helpful to untangle what is happening.

  251. Thanks for your answer. I get how we might write “-3” as “0-3” to realize a position relative to zero, but I don’t get where the number one comes from in “1 * x * x = 9 “. I don’t know what a “transformation on the number 1” is. Am I missing some rule that didn’t get explained to me earlier on? I think it’s possible that I am lacking an awareness of some such rule(s), as math teachers I’ve had either did not understand the maths themselves or purposely left out very important bits of info. I vividly remember going through college level math courses (just 3-4 years ago) and the math those teachers taught is completely different from what I’m learning on your site and the Khan Academy.

  252. No problem, happy to clarify. A lot of math education just gives you facts without really sharing ways to get the ideas intuitively, which can be really frustrating.

    I see any number, like 14, as a scaled-up version of 1. 14 is the same as 1, it’s just 14 times bigger. -5 is the same as 1, except it’s pointing the other way (negative) and is 5 times bigger.

    This article explains more about seeing numbers as “transformations” on the number 1:


    When we see an equation like

    x^2 = -1

    it looks really strange until we walk through what it means. Implicitly, every multiplication means we start with the number 1 and do something to do it.

    So, x^2 is really “1 * x^2” just like “3” is really “0 + 3”. We want to remember that we had a starting point.

    x^2 is really just x * x, so another way to put it:

    “x^2 = -1” is asking the question “If we start at 1, can we make two transformations, and arrive at -1?”

    I didn’t say “make two multiplications” because that wording makes us think of the changes we already know. The real question is whether there’s any transformation (type of change) that can turn 1 to -1 in two steps. A rotation is one such change that would let this happen. We aren’t used to rotating numbers, sure, just like a bug walking on a wire isn’t used to moving in a different direction. It doesn’t mean we can’t use that dimension though :).

    Check out the visual arithmetic article above to see if it helps things.

  253. Hi kalid.
    I figured out something and wanted to share with you. I used to always wonder why complex numbers come into electronics. Am an Electronics and Telecommunication student.
    I figured out that complex numbers come into maths whenever one force splits into two. Rotation is just one thing that makes one force splits into two. One becomes the real part and the other…the imaginary part. A capacitor for example does this….it exerts two forces on the electrons….it delays them by storing them like a dam and the leaking it after a certain threshold…..plus it exerts a force of its own internal resistance.
    The resistance of the capacitor is the real part of the capacitor and its storing abilty is its imaginary part. It took time for me to understand it since there was nothing rotating in this whole process.

    Have written in detail in the following article.


  254. You are my hero! As many others, I used imaginary numbers through college and after reading your blog I realized that I never really understood them until today, your site is freaking awesome!


  255. Thanks, this was very interesting! So basically numbers now have angles, multiplying numbers is an operation involving the addition of angles, and negative numbers have square roots. Thanks for the ‘ah-ha’ moment!

  256. My favorite equation using (i) is i^i = e^((-Pi)/2)), in other words that i^i is in the set of all real numbers. Use Euler’s formula to prove it to yourselves.

  257. From your earlier post:

    “The real question is whether there’s any transformation (type of change) that can turn 1 to -1 in two steps.”

    How about two translations of -1 along a number line?
    Couldn’t you also transform 1 to 9 by two translations of adding 5?

    Why would a high school aged student see a rotation as an intuitive transformation to go from 1 to -1? You did not say “make two multiplications”, but that assumption is necessary for students to follow along your train of thought.

  258. Dear Kalid, first of all congratulations on all your amazing work, i’m glad someone else thinks that the approach we use to teach maths is bad, anyway i wanted to ask you something because i can’t find an answer anywhere on google: basically we define i as the square root of a negative number because the rules of math don’t allow it, hence immaginary, but my question is this: is there an ulterior motive for defining it as a negative square root? like for example could we define i as ln(-e)? being that negative logarithms are also “against the rules”. I think finding that the answer might provide additional insight on immaginary numbers. What do you think?

    Sincerely, another math lover

  259. @Matt

    I had the same question last year and though about it a lot. I think we don’t need it because a negative logarithm does have a solution when you consider complex numbers. In fact, every functions (sin, cos, log, …) can be expressed as a complex exponential. So there is no need to define as ln(-e) as it would be redundant.

    Note that the square root is a power function. Power functions are pretty common compare to logarithm so it makes a lot more sense to define the imaginary part from this function. I think we could define the imaginary part from the logarithm function but it would end up more complex. After all, we use power functions much more often then logarithm .

  260. This is a great article. It has explained a lot of what goes through my mind about complex numbers. How would you describe a 3-dimensional system of numbers? What new numbers would you use and what would their role play?

  261. @KT: Great question. The problem with a fixed translation is that the amount varies: going from 1 to -1 in two operations requires steps of 1, but going from 9 to -9 requires steps of 9.

    With a rotation, the size of the input doesn’t matter: we turn 90 degrees with each step, and always face backwards. It’s a good point and I’ll see if I can clarify that. (Move from forward to backward in two steps, no matter how large the original number is.)

    @Matt, @Alex: Interesting question. There might be a way to define i as the solution to a negative logarithm.

    i = e^(i * pi/2), so

    which is a bit of a recursive definition but perhaps can work? I haven’t taken formal Algebraic structures classes to know exactly how these are all constructed, but there may be a way. But technically, yes, the logs of negative numbers can be solved with complex numbers.

    @Joseph: Great question. You can extend to 3d and more (4d are quaternions). In some dimensions problems arise (in 3d there’s the problem of “Gimbal Lock”) but it’s a fun thing to think about!

  262. Hi Kalid,

    Your posts are fantastic. This is the way math should have been taught in my high school!!. It’s a bit late now, but I am trying to catch up! A basic question for you.

    In your example, how does multiplying by (1+i) rotate something by 45 degrees? Since multiplying by i rotates by 90 degrees, I am thinking we should multiply by i/2 to rotate something by 45 degrees. What is wrong in my thinking? thanks

  263. Hi Kumar, great question. i/2 is really

    \displaystyle{\frac{1}{2} i }

    which is two operations: scale by 1/2, then rotate by i. But what we really want is for the rotation in i to be divided by 2, not to reduce the size of the number.

    The square root splits an operation into 2 equal steps. For example, 9 represents a jump from 1 to 9. The square root of 9 is 3, which represents two equal steps: 1 to 3, and 3 to 9.

    So, if we want to jump to i (90 degrees) in two steps… we’d take the square root of i! Whoa, what a weird concept:

    \displaystyle{\sqrt{i} = 45-degree turn}

    We can visualize this with the number (1 + i), which is a 45-degree angle (it’s equally strong on both sides, a perfect diagonal, just like a 45-degree angle).

    Technically, the number (1 + i) is a scaled up version of the square root of i, since its magnitude is sqrt(1^2 + 1^2) = sqrt(2). But the direction is the same.

    This is definitely a tricky thing to keep straight, you might enjoy this article:


  264. Hi Kalid – love the explanations!

    Something which vexed me though was the following paragraph:
    “If we multiply by -i twice, we turn 1 into -i, and -i into -1. So there’s really two square roots of -1: i and -i.”

    Looking at the figure, this doesn’t fit in with my mental image of the situation at all – in my mind multiplying by -i “turns the dial” 90 degrees – so this would (per the figure) turn 1 into -1 rather than -i.

    When you say “What transformation x, when applied twice, turns 1 into -1?”, I completely agree that this would be two transformations of either i or -i, but this seems to be in conflict with the first statement I quoted.

    I’d really appreciate a clarifying response :-) (keep up the good work!)

  265. Hi Nicolay, thanks for the comment. Positive i is a 90-degree turn counter-clockwise, and negative i is a 90-degree turn counter-clockwise. So, in both cases, it takes 2 steps to go from 1 to -1. Hope that helps!

  266. Thanks for the response, Kalid! :-)

    “Positive i is a 90-degree turn counter-clockwise, and negative i is a 90-degree turn counter-clockwise.”

    Yes, I completely agree with that. What I don’t understand is how you can do TWO turns counter-clockwise and go from 1 into -i [1]. In my head, that would go from 1 to -1, not from 1 to -i.

    1: “If we multiply by -i twice, we turn 1 into -i”

  267. Ah! Now I see. That sentence wasn’t clear, 1 to -i is the result of the first rotation by -i. The second rotation by -i would turn -i to -1. [The full sentence is: “If we multiply by -i twice, we would first turn 1 into -i, and then -i into -1.”]

    I’ve updated the text to:

    “If we multiply by -i twice, the first multiplication would turn 1 into -i, and the second turns -i into -1.”

    Thanks for the feedback!

  268. Hahaha, I can’t believe I spent so much time in frustration due to a comma. Thanks a lot for clearing that up! :-)

  269. The problem for me is the justification for the complex plane and the rotation concept in the first place. The number line was a logical outgrowth of the discovery of zero, negatives, rational, and irrational numbers and just filled in our ‘blind spots’. It is coherent and intuitive and corresponds to our experiences in everyday life.

    The complex plane does not have that intuitive feel. Rather it feels like a gimmick that happens to work for describing cyclic phenomena. I have a feeling that this may be a ‘blind spot’ for me and I am waiting for the lights to go on.

    All the best,


  270. Hi there, Kalid. First off, I just wanna thank you for the great article, it demystified a lot of the hocus-pocus surrounding complex numbers. It did raise a couple of questions in my mind, however:

    1) Multiplication by i. I understand the concept of it rotating a number by 90 degrees (or pi/2 radians!), but what does its coefficient do? For example, say we had a complex number z = 3 + 4i. What does the 4 do? My raw intuition tells me that it would represent a full rotation. Is this correct? If not, what does the coefficient actually mean?

    2) Is it possible to combine 3 complex planes at right angles in order to be able to express rotation in any direction? Again, my intuition tells me it’s possible, but I can’t reconcile it with the fact that a garden-variety 3-D system uses three axes, since a complex plane contains a real and an imaginary axis; it would seem that there wouldn’t be enough “room” (I hope that conveys my meaning properly. Probably not.) to include the necessary 3 imaginary axes and the 3 real ones.

  271. @Dan: Good question. At the lowest level, the complex numbers are an algebraic relationship that are consistent. I.e., the rules of multiplication, division, square roots, etc. work out, and don’t lead to contradictions.

    However, as humans, that isn’t always enough to accept the concept :). The notion of a complex plane, a 2d geometric metaphor, maps to the properties of the complex numbers. However, it isn’t the only way to see them — there are other metaphors (like matrices) that might work better. My brain stops itching when I see the complex plane interpretation but others may work for you. If you find them let me know!

    @Marcus: Thanks, glad it clicked.

    1) For multiplication, the complex parts rotates you, and the real part scales you. So when you multiply by “4i” you are saying “scale up 4x, and rotate”. The real coefficient is just a scaling factor. If you want to rotate twice, you’d multiply by i^2. (Which is -1, a 180-degree rotation.). Check out this article for more:


    2) Great question. Yep, it is possible to represent 3d rotations, but you you actually need 4 dimensional numbers (3 imaginary axes and one real one). These numbers are called quaternions, and I’d like to do a follow-up on them.

  272. [Square root of -1] like this.

    Kalid- I’m elated I stumbled upon your blog. I am a person who questions everything and once I started learning complex numbers, my natural reaction was: Egad, Aren’t there enough concepts and theorems and whatnots to CHOKE our minds- and as if it weren’t enough already, we have to learn imaginary concepts that don’t even EXIST? What the hell man. (*cough cou-*) (….)Wow you just silenced the choking of my mind! xD

    My teacher had actually mentioned about the rotation thingy and I was a little mind blown. Now all that’s left has exploded into a thousand jittery pieces, and- wait am I making this too ‘complex’?
    I just noticed something.
    So does this mean that complex numbers and vectors are not only related, but very much interconnected- so much so that they almost mean the same thing? Just that they seem so far apart in our minds that we can’t see the big picture?
    And one more thing.
    I’m ‘tan’kful ‘co’s I ‘sec’retly ‘cot’ on a lot of meaning as’sin’ed to this topic. (Seems like you’re not the only with a thing for puns)
    Can’t find anything with cosec. *racks my brains* *boggles my mind*
    *cough* – Uh- My mind started choking again!
    *cough* ‘*co-* * 1 sec’ I think I found it!
    AHA!! x’D

  273. Thank you – this was helpful, and I agree; I wish my math teachers would have explained this and other such topics better without glossing over them.

  274. Thanks for the post it was really helpful. I thought you might be instrested in another take on the intuition it is a little hard to comprehend buy very inuitive. https://youtu.be/1rVHLZm5Aho He also has a video on Euler’s formula.

  275. Hi, Kalid!

    I’m translating this post (to http://www.energiaeletrica.net, as I did with your Calculus Course). What do you mean by “Thinking we’ve “figured out” a topic like numbers is what keeps us in Roman Numeral land.”? I couldn’t catch the idea.

    By the way: could you update the translation page? I’ve translated http://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/ in: http://www.energiaeletrica.net/blog/repensando-a-aritmetica-um-guia-visual/


  276. Thanks Frederico! Just updated. For this phrase:

    Thinking we’ve “figured out” a topic like numbers is what keeps us in Roman Numeral land.

    I meant if we believe we’ve found the “final” form of numbers, we would have stuck with Roman numerals. We have to be open to a newer, better way to represent numbers. (Some people fight the notion of imaginary numbers, just like some Romans might have fought the notion of decimal numbers.)

  277. Being a Berkeleyan I can give an example of the ‘tangibleness’ of a negative quantity. Suppose you are used to drinking a cup of milk every morning. Now suppose that a shortage in milk made it necessary for you to drink only 1/2 cup of milk instead. The negative 1/2 cup that you can’t drink is just as tangible as the 1/2 cup that you do drink. You miss that 1/2 cup. The point is that everything is relative to your perception. And negative amounts are just as tangible as positive amounts. Only that which is perceptible is real. String theory is a delusion because it deals with dimensions that are not perceptible. IMHO 😉

  278. Dear Khalid,

    Thanks for the wonderful post. I never came across such an intuitive explanation before!

    Could you kindly elaborate more on how 1 + i is 45 degrees??? I just didn’t get it…and it did not seem to connect from the earlier discussion too. May be I am too naive…

    Thanks in anticipation,

  279. Hi Haris, you can consider 1 + i as making the sides of a triangle (1 unit “East” and 1 unit “North”). This triangle makes the diagonal of a square and is 45-degrees (half of a 90-degree angle). For other numbers (1 unit East and 3 units North) the angle has to be worked out with trigonometry. It might help to draw it out to see. Hope this helps!

  280. Sir I have a question For U .
    Which Family is bigger Real Numbers family Or imaginary Numbers family(Dont be confuse to take imaginary as complex)

  281. Although, I learnt about complex numbers in my high school. This post just refreshed my lost memory instantly. Could you please also do the same on quaternions?

  282. Thank you! I have a teenage daughter in High School Algebra II wondering why she has to learn this stuff. I have a degree in Physics and Electrical Engineering and was really having a hard time finding a non college level physics or engineering example as to why she should know this other than making it easy to derive trigonometric identities via Euler’s formula!

    Well done Sir!

  283. @Swoorup: Glad it helped — yep, quaternions are a planned topic :).

    @John: That’s great to hear, I struggled for ages to find a more accessible use case for imaginary numbers than phasors, voltage/current, etc. Making trig identities easier is a topic I’m working on now actually. Appreciate the note!

  284. Thank you so much for that wonderful article explaining the confusion my mind was in!!!!! I googled and went through many sites but only yours helped me really understand about complex numbers. Why aren’t you my maths teacher!!!!! Keep posting. Great effort and work. Sharing this with my whole class!!

  285. Wow what a CLEAR and BRILLIANT post!!! I admire your patience in answering the many questions you receive.
    Here’s one more question: Why are the ‘real’ and ‘imaginary’ parts of complex numbers attached with a PLUS SIGN?? What does this plus sign represent? Where does addition come in???
    Thank you for your clarity, time, and patience.

  286. Thank you for such a wonderful article. I really like your approach to learning and teaching. I am really interested in science, so I also want to learn as much about maths (the “language of the universe”) as possible, so I can learn more about the mechanisms behind various phenomena, and your article really helped by explaining something that was quite counterintuitive to me!

  287. Hi Kalid,
    I can’t begin to describe how much your articles have helped me. I am a college senior and I always knew that I didn’t *really* get what I learned in my math and physics classes. I just mechanically solved problems and took the tests…but that doesn’t fly in the electromagnetism class I’m taking now. But I think with following your tips on learning via analogies I have started to actually learn to retain this material that is really challenging for me. Thank you so very much for your amazing teaching.

  288. Hi,
    I am still waiting for a reply to my question (#400):
    Why are the ‘real’ and ‘imaginary’ parts of complex numbers attached with a PLUS SIGN?? What does this plus sign represent? Where does addition come in???
    Thank you for your clarity, time, and patience.

  289. @Mary Wiltz.
    The real part is the parallell component of a force/object/ influence etc.

    And the imaginary part is the perpendicular component of the force /object / influence etc.

    Everything in nature usually is slightly horizontal (parallel to the ground) and slightly vertical (perpendicular to the ground)

    Cos(angle) = parallel component of the object/force etc.
    i.Sin(angle) = perpendicular component of the object/force etc

    You can also see it this way.
    i^(0) = 1 ……means no rotation
    i^(90) = i ……..means rotation by 90 degree.

    But what if an object rotates by only 60 degree.
    Lets say the length/force of the object involved is 5 metres.

    So the horizontal influence(real part ) of this stick = 5cos(60)

    Its vertical influence (imaginary part) of this stick = i.sin(60)

    Together the position of the stick = 5cos(60) + 5.i.Sin(60)

    5cos (60) = top view of the stick.
    i.5.sin(60) = side-view of the stick.

    I wrote some articles on this on my blog
    One is as below.



  290. If the question is about complex numbers being written as “a+bi”, where a and b are “real numbers”, it actually is addition.

    However, the addition and multiplication symbols are also used in alternative systems of mathematics in what is called “a field”. A “field” is any set of “numbers” in which operations -similar- to addition and multiplication – and division – can be done.

    One approach to inventing complex numbers – contrary to the topic of this article – is to say, “What if negative numbers do have a square root? For instance, i is the square root of minus 1”, and, it turns out that what happens is a two-dimensional geometry of numbers.

    Of course you can add number together, including these ones, so it is quite legitimate to write “2+3i”, which is merely the number which is 2 real plus (3 real x i). Some more work can be done to establish that this is a definite number.

    Another plan which is mainly this reader’s own, is to get away from the paradox of “square root of a negative number” which is clearly nonsense, and to think of a complex number “n” as being an ordered pair or “tuple” of two real numbers (a, b). You may see where this is going if I call them East and North – like geometric coordinates. For instance, (a, b) isn’t the same number as (b, a), and you can have (a, a) as an ordered pair.

    Along with these tuples are rules of addition and multiplication: (a, b) + (c, d) = (a+c, b+d), i.e. you “add” two tuples by separately adding the first parts and adding the second parts.

    (a, b) x (c, d) = (ac – bd, ad + bc). If I’ve got that right (which is uncertain), it looks goofy, but, take it from me, these rules are going to work.

    If you look at the tuples with the second part 0, these rules are saying that (a, 0) + (b, 0) = (a+b, 0), and (a, 0) x (c, 0) = (ac, 0). That is, the tuples containing any real number followed by 0 behave exactly like the real numbers on their own. “Adding” the tuples is the same as adding the numbers.

    The tuple (0, 1), on the other hand, is i – because (0, 1) x (0, 1) = (-1, 0).

    And that’s how to make complex numbers without believing that -1 really has a square root. :-)

    For the addition part of this, as you may already see, you can also think about “vectors”. What? Well, for instance, the location of the nearest pharmacist to your home is a vector – so far east (or west), and so far north (or south). Or, of course, a straight line direction, and a distance. Then, from the pharmacist to the post office, another vector. But what if you want to go straight from your home to the post office? You add the vectors together. That means adding the two east-west components, and adding the two north-south components. (It’s a lot harder in the “polar coordinate” system.)

    Although strictly this works better on a flat map, than on a round planet.

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