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:

Focusing on relationships, not mechanical formulas.
Seeing complex numbers as an upgrade to our number system, just like zero, decimals and negatives were.
Using visual diagrams, not just text, to understand the idea.

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.

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!

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:

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

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

$\displaystyle{1 \cdot x^2 = -1}$

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, we turn 1 into -i, and -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:

$\displaystyle{Size \hspace{2pt} of \hspace{2pt} (-x) = \sqrt{(-x)^2} = |x|}$

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:

$\displaystyle{Size \hspace{2pt} of \hspace{2pt} a + bi = \sqrt{a^2 + b^2}}$

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:

Multiplying by a complex number rotates by its angle

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, 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:

$\displaystyle{(3 + 4i) \cdot (1 + i) = 3 + 4i + 3i + 4i^2 = 3 - 4 + 7i = -1 + 7i}$

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.

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.

Posted December 21, 2007, under Guides, Math
Tags: analogy, complex, Guides, imaginary, intuitive, Math, negative, number
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Comments

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

John Kelly — December 21, 2007 @ 9:16 am
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.

JB McMichael — December 21, 2007 @ 9:30 am
It was a real breakthrough when I came to visualize that model for the first time. I really don’t understand why they don’t teach imaginaries that way!

Bryan Davis — December 21, 2007 @ 9:32 am
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).

Chick — December 21, 2007 @ 10:46 am
@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:

http://www.mth.kcl.ac.uk/events/summer_schools/summer_school2001/Alg013.html

Kalid — December 21, 2007 @ 1:34 pm
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).

Chick — December 21, 2007 @ 3:10 pm
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

Kalid — December 21, 2007 @ 4:20 pm
I really like how you explicitly relate rotation to complex numbers. It really does feel like a whole new angle (heh) to START with rotation.

I have occasionally pointed people towards http://mathforum.org/johnandbetty/ which is good for the early stages of complex numbers.

Chaz — December 21, 2007 @ 4:32 pm
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.

Kalid — December 21, 2007 @ 5:09 pm
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?).

Dave

Dave — December 21, 2007 @ 5:19 pm
I see you’ve read
Where Mathematics Comes From by George Lakoff and Rafael Nunez.

Everything said here and more, except errors, is in that book!8-))

Your pages are good publicity for these ideas. But you need to post more about analogy.

G. Lakoff — December 21, 2007 @ 5:36 pm
@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…

Chick — December 21, 2007 @ 6:05 pm
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.

bayareaguy — December 22, 2007 @ 12:22 am
@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.

Kalid — December 22, 2007 @ 2:00 am
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

abc — December 23, 2007 @ 8:13 am
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.

Robin — December 23, 2007 @ 9:53 am
@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.

Kalid — December 23, 2007 @ 10:27 am
@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” .

Robin — December 23, 2007 @ 12:06 pm
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).

Burton MacKenZie — December 24, 2007 @ 2:47 pm
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.”).

Kalid — December 25, 2007 @ 1:43 am
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.

Oliver Steele — December 27, 2007 @ 6:47 am
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

Darius — December 28, 2007 @ 1:55 am
@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!

Kalid — December 28, 2007 @ 8:07 pm
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…

Alessio — December 30, 2007 @ 4:13 am
Well Done! I share share your frustration at the fact that most high school mathematics courses do not explain complex numbers adequately.

Ivan Malison — December 30, 2007 @ 3:34 pm
@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.

Kalid — January 2, 2008 @ 8:41 pm
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.
Cheers,
Burton

Burton MacKenZie — January 8, 2008 @ 9:28 pm
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.

Kalid — January 9, 2008 @ 2:26 am
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.

Peace,
Peyton

Peyton Bland — February 6, 2008 @ 12:22 pm
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).

Kalid — February 6, 2008 @ 2:28 pm
Hello,

How can we understand e^(pi * i) = -1 ?

Osama AbuObeid — February 10, 2008 @ 2:35 am
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.

Jaloopa — February 11, 2008 @ 3:51 pm
@Osama: Great question, I’ll be covering that in a separate post.

@Jaloopa: Thanks for the info! Yep, I agree on ‘complex vs complicated’ (http://betterexplained.com/articles/combining-simplicity-and-complexity/)… but the pun may not flow as well when we write “complex numbers aren’t complicated” .

Kalid — February 11, 2008 @ 11:36 pm
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?

Alec — March 26, 2008 @ 3:16 am
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.

Kalid — March 26, 2008 @ 8:53 am
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?

Jane Messina — April 14, 2008 @ 4:58 am
Hi Jane, take a look at comments #4 and #5, they may help answer your question .

Kalid — April 14, 2008 @ 1:24 pm
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.

maheshexp — April 18, 2008 @ 4:10 am
@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.

Kalid — April 18, 2008 @ 3:22 pm
I feel smarter now.

Kenneth Rochester — April 28, 2008 @ 10:04 pm
Awesome, glad you found it useful .

Kalid — April 28, 2008 @ 10:18 pm
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

maheshexp — April 29, 2008 @ 5:00 am
Hi maheshexp, that’s an interesting observation. Yes, trig mostly deals with the raw angles, while complex numbers have you think about distances.

Kalid — April 29, 2008 @ 10:39 am
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!

caitie — May 3, 2008 @ 9:15 pm
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

Neeraj — May 17, 2008 @ 4:06 am
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.

Anonymous — May 19, 2008 @ 4:12 pm
@Anonymous: Thanks, glad you enjoyed it! That’s a great insight about moving around on the unit circle, I like that interpretation too .

Kalid — May 19, 2008 @ 10:54 pm
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?

Thanks.

Peter — May 25, 2008 @ 4:44 pm
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,

-Kalid

Kalid — May 25, 2008 @ 7:49 pm
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!
Weston

Weston — June 26, 2008 @ 7:52 am
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 ).

-Kalid

Kalid — June 26, 2008 @ 10:46 am
THANKS!i found your articles very very useful.

vaibhav — July 22, 2008 @ 8:46 am
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?

chuck — July 24, 2008 @ 5:33 pm
@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 .

Kalid — July 24, 2008 @ 9:45 pm
Hey, I’ve got a few questions/comments myself.

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

i
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.

Ryan Scott Scheel — August 23, 2008 @ 10:57 pm
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,
actor,
dishwasher,
teacher,
hobo,
psynurse,
sailor,
autor,
poet,
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

^-^

niko — October 27, 2008 @ 5:32 am
lol thx helped my curiousity im only in 9th grade >.> but it now my brain doesnt ponder about wat it is.

Sungmin — November 12, 2008 @ 4:32 pm
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.

Psy — November 17, 2008 @ 11:49 am
@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.

Kalid — November 18, 2008 @ 1:20 am
hi kalid,
thank you very much. i have not found any better explanation than this one on complex numbers.

– abu ihsan,
Kuala Lumpur, Malaysia.

abu ihsan — November 25, 2008 @ 8:11 am
Thank you for your beautiful explanations.

Michael, Jerusalem, Israel — December 2, 2008 @ 1:40 am
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!!!

Ti — December 8, 2008 @ 8:39 pm
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!

Daniel Prichard — December 9, 2008 @ 2:31 pm
@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.

Kalid — December 9, 2008 @ 4:08 pm
@abu, @Michael: Thanks for the kind words, glad it was useful.

Kalid — December 9, 2008 @ 4:08 pm
Hi,
Thank you for light a candle

Reza Sepas Yar — December 25, 2008 @ 3:10 pm
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.

Jules — January 28, 2009 @ 7:56 pm
Maybe more simple and nicer (at least for some people) interpretation of imaginary unit and complex numbers you can find in the paper “The simple complex numbers”
http://arxiv.org/abs/0802.0312?context=physics

Jarek — January 31, 2009 @ 10:56 am
@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.

Kalid — January 31, 2009 @ 11:44 am
Haha this article has got me all excited about math again..Wish they’d teach this way in school..

Arvind — February 4, 2009 @ 11:01 pm
@Arvind: Thanks, glad you enjoyed it!

Kalid — February 4, 2009 @ 11:08 pm
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,
V.

Val — February 11, 2009 @ 1:59 pm
@Val: Thanks for the comment — glad you were able to learn something new!

Kalid — February 11, 2009 @ 6:06 pm
thanks for the “a-ha” moment. =)

mike — February 16, 2009 @ 2:23 pm
@Mike: Of course!

Kalid — February 17, 2009 @ 4:04 am
http://www.youtube.com/watch?v=EQJBp6Ym-6A&feature=PlayList&p=EC88901EBADDD980&index=5

A relevant lecture on complex numbers with a little diversion into Eulers formula

Brian — February 20, 2009 @ 10:15 am
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.

stary — March 2, 2009 @ 5:47 pm
@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.

Kalid — March 2, 2009 @ 7:32 pm
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!

Deryk — March 6, 2009 @ 8:15 am
Awesome awesome stuff! Please keep this stuff coming!

Jeff — March 6, 2009 @ 9:55 am
@Deryk: Really glad it helped! Yeah, I don’t think the therapy stuck .

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

Kalid — March 8, 2009 @ 3:21 pm
Very interesting article.. and complex numbers dealt with with utmost simplicity and ease.

Prassanth — March 17, 2009 @ 9:00 pm
@prassanth: Thanks! I really strive for simplicity, glad it helped.

Kalid — March 17, 2009 @ 11:06 pm
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.

M.

MonkeyBoy — March 27, 2009 @ 2:08 am
@M: Thanks so much! That really means a lot .

Kalid — March 27, 2009 @ 12:56 pm
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.

free mmorpgs — April 4, 2009 @ 5:21 pm
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!
Thanks

giles — April 17, 2009 @ 12:48 pm
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.

balakrishnan — April 20, 2009 @ 7:47 am
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.

balakrishnan — April 20, 2009 @ 9:28 am
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 !!!

Eduard — May 7, 2009 @ 2:46 am
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?

John — May 17, 2009 @ 8:27 pm
@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!

Kalid — May 18, 2009 @ 10:21 pm
@Kalid: Your explanation was “i”-opening… Thanks.

John — May 19, 2009 @ 2:24 pm
@John: Awesome, glad it helped!

Kalid — May 19, 2009 @ 3:04 pm
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.

Steven — June 9, 2009 @ 7:53 am

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