# Intuitive Arithmetic With Complex Numbers

Imaginary numbers have an intuitive explanation: they “rotate” numbers, just like negatives make a “mirror image” of a number. This insight makes arithmetic with complex numbers easier to understand, and is a great way to double-check your results. Here’s our cheatsheet:

This post will walk through the intuitive meanings.

## Complex Variables

In regular algebra, we often say “x = 3″ and all is dandy — there’s some number “x”, whose value is 3. With complex numbers, there’s a gotcha: there’s two dimensions to talk about. When writing

$\displaystyle{z = 3 + 4i}$

we’re saying there’s a number “z” with two parts: 3 (the real part) and 4i (imaginary part). It is a bit strange how “one” number can have two parts, but we’ve been doing this for a while. We often write:

$\displaystyle{y = 3\frac{4}{10} = 3 + .4}$

and it doesn’t bother us that a single number “y” has both an integer part (3) and a fractional part (.4 or 4/10). Y is a combination of the two. Complex numbers are similar: they have their real and imaginary parts “contained” in a single variable (shorthand is often Re and Im).

Unfortunately, we don’t have nice notation like (3.4) to “merge” the parts into a single number. I had an idea to write the imaginary part vertically, in fading ink, but it wasn’t very popular. So we’ll stick to the “a + bi” format.

## Measuring Size

Because complex numbers use two independent axes, we find size (magnitude) using the Pythagorean Theorem:

So, a number z = 3 + 4i would have a magnitude of 5. The shorthand for “magnitude of z” is this: |z|

See how it looks like the absolute value sign? Well, in a way, it is. Magnitude measures a complex number’s “distance from zero”, just like absolute value measures a negative number’s “distance from zero”.

We’ve seen that regular addition can be thought of as “sliding” by a number. Addition with complex numbers is similar, but we can slide in two dimensions (real or imaginary). For example:

Adding (3 + 4i) to (-1 + i) gives 2 + 5i.

Again, this is a visual interpretation of how “independent components” are combined: we track the real and imaginary parts separately.

Subtraction is the reverse of addition — it’s sliding in the opposite direction. Subtracting (1 + i) is the same as adding -1 * (1 + i), or adding (-1 – i).

## Complex Multiplication

Here’s where the math gets interesting. When we multiply two complex numbers (x and y) to get z:

• Add the angles: angle(z) = angle(x) + angle(y)
• Multiply the magnitudes: |z| = |x| * |y|

That is, the angle of z is the sum of the angles of x and y, and the magnitude of z is the product of the magnitudes. Believe it or not, the magic of complex numbers makes the math work out!

Multiplying by the magnitude (size) makes sense — we’re used to that happening in regular multiplication (3 × 4 means you multiply 3 by 4′s size). The reason the angle addition works is more detailed, and we’ll save it for another time. (Curious? Find the sine and cosine addition formulas and compare them to how (a + bi) * (c + di) get multiplied out).

Time for an example: let’s multiply z = 3 + 4i by itself. Before doing all the math, we know a few things:

• The resulting magnitude will be 25. z has a magnitude of 5, so |z| * |z| = 25.
• The resulting angle will be above 90. 3 + 4i is above 45 degrees (since 3 + 3i would be 45 degrees), so twice that angle will be more than 90.

With our predictions on paper, we can do the math:

$\displaystyle{(3 + 4i) * (3 + 4i) = 9 + 16i^2 + 24i = -7 + 24i}$

Time to check our results:

• Magnitude: sqrt((-7 * -7) + (24 * 24)) = sqrt(625) = 25, which matches our guess.
• Angle: Since -7 is negative and 24i is positive, we know we are going “backwards and up”, which means we’ve crossed 90 degrees (“straight up”). Getting geeky, we compute atan(24/-7) = 106.2 degrees (keeping in mind we’re in quadrant 2). This guess checks out too.

Nice. While we can always do the math out, the intuition about rotations and scaling helps us check the result. If the resulting angle was less than 90 (“forward and up”, for example), or the resulting magnitude not 25, we’d know there was a mistake in our math.

## Complex Division

Division is the opposite of multiplication, just like subtraction is the opposite of addition. When dividing complex numbers (x divided by y), we:

• Subtract angles angle(z) = angle(x) – angle(y)
• Divide by magnitude |z| = |x| / |y|

Sounds good. Now let’s try to do it:

$\displaystyle{\frac{3 + 4i}{1 + i}}$

Hrm. Where to start? How do we actually do the division? Dividing regular algebraic numbers gives me the creeps, let alone weirdness of i (Mister mister! Didya know that 1/i = -i? Just multiply both sides by i and see for yourself! Eek.). Luckily there’s a shortcut.

## Introducing Complex Conjugates

Our first goal of division is to subtract angles. How do we do this? Multiply by the opposite angle! This will “add” a negative angle, doing an angle subtraction.

Instead of z = a + bi, think about a number z* = a – bi, called the “complex conjugate”. It has the same real part, but is the “mirror image” in the imaginary dimension. The conjugate or “imaginary reflection” has the same magnitude, but the opposite angle!

So, multiplying by a – bi is the same as subtracting an angle. Neato.

Complex conjugates are indicated by a star (z*) or bar above the number — mathematicians love to argue about these notational conventions. Either way, the conjugate is the complex number with the imaginary part flipped:

$\displaystyle{z = a + bi}$ has complex conjugate $\displaystyle{z^* = \bar{z} = a - bi}$

Note that b doesn’t have to be “negative”. If z = 3 – 4i, then z* = 3 + 4i.

## Multiplying By the Conjugate

What happens if you multiply by the conjugate? What is z times z*? Without thinking, think about this:

$\displaystyle{z \cdot z^* = 1 \cdot z \cdot z^*}$

So we take 1 (a real number), add angle(z), and add angle (z*). But this last angle is negative — it’s a subtraction! So our final result should be a real number, since we’ve canceled the angles. The number should be |z|^2 since we scaled by the size twice.

Now let’s do an example: $\displaystyle{(3 + 4i) * (3 - 4i) = 9 - 16i^2 = 25}$

We got a real number, like we expected! The math fans can try the algebra also:

$\displaystyle{(a + bi) * (a - bi) = a^2 + abi - abi -b^2i^2 = a^2 + b^2 }$

Tada! The result has no imaginary parts, and is the magnitude squared. Understanding complex conjugates as a “negative rotation” lets us predict these results in a different way.

When multiplying by a conjugate z*, we scale by the magnitude |z*|. To reverse this effect we can divide by |z|, and to actually shrink by |z| we have to divide again. All in all, we have to divide by |z| * |z| to the original number after multiplying by the conjugate.

## Show Me The Division!

I’ve been sidestepping the division, and here’s the magic. If we want to do

$\displaystyle{\frac{3 + 4i}{1 + i}}$

We can approach it intuitively:

• Rotate by opposite angle: multiply by (1 – i) instead of (1 + i)
• Divide by magnitude squared: divide by |sqrt(2)|^2 = 2

The answer, using this approach, is:

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

The more traditional “plug and chug” method is to multiply top and bottom by the complex conjugate:

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

We’re traditionally taught to “just multiply both sides by the complex conjugate” without questioning what complex division really means. But not today.

We know what’s happening: division is subtracting an angle and shrinking the magnitude. By multiplying top and bottom by the conjugate, we subtract by the angle of (1-i), which happens to make the denominator a real number (it’s no coincidence, since it’s the exact opposite angle). We scaled both the top and bottom by the same amount, so the effects cancel. The result is to turn division into a multiplication in the numerator.

Both approaches work (you’re usually taught the second), but it’s nice to have one to double-check the other.

## More Math Tricks

Now that we understand the conjugate, there’s a few properties to consider:

$\displaystyle{(x + y)^* = x^* + y^*}$

$\displaystyle{(x \cdot y)^* = x^* \cdot y^*}$

The first should make sense. Adding two numbers and “reflecting” (conjugating) the result, is the same as adding the reflections. Another way to think about it: sliding two numbers then taking the opposite, is the same as sliding both times in the opposite direction.

The second property is trickier. Sure, the algebra may work, but what’s the intuitive explanation?

The result (xy)* means:

• Multiply the magnitudes: |x| * |y|
• Add the angles and take the conjugate (opposite): angle(x) + angle(y) becomes “-angle(x) + -angle(y)”

And x* times y* means:

• Multiply the magnitudes: |x| * |y| (this is the same as above)
• Add the conjugate angles: angle(x*) + angle(y*) = -angle(x) + -angle(y)

Aha! We get the same angle and magnitude in each case, and we didn’t have to jump into the traditional algebra explanation. Algebra is fine, but it isn’t always the most satisfying explanation.

## A Quick Example

The conjugate is a way to “undo” a rotation. Think about it this way:

• I deposited $3,$10, $15.75 and$23.50 into my account. What transaction will cancel these out? To find the opposite: add them up, and multiply by -1.
• I rotated a line by doing several multiplications: (3 + 4i), (1 + i), and (2 + 10i). What rotation will cancel these out? To find the opposite: multiply the complex numbers together, and take the conjugate of the result.

See the conjugate z* as a way to “cancel” the rotation effects of z, just like a negative number “cancels” the effects of addition. One caveat: with conjugates, you need to divide by |z| * |z| to remove the scaling effects as well.

## Closing Thoughts

The math here isn’t new, but I never realized why complex conjugates worked as they did. Why a – bi and not -a + bi? Well, complex conjugates are not a random choice, but a mirror image from the imaginary perspective, with the exact opposite angle.

Seeing imaginary numbers as rotations gives us a new mindset to approach problems; the “plug and chug” formulas can make intuitive sense, even for a strange topic like complex numbers. Happy math.

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## 57 thoughts on “Intuitive Arithmetic With Complex Numbers”

1. Great explanation! Here is what i tried to understand the intuitution for multiplication:
We know that any comlplex number can be represented as z=r(cosA+isinA) where r is the magnitude and A is its angle.
Now, let z1=r(cosA+isinA) and z2=q(cosB+isinB)
z1*z2=rq(cosA+isinA)(cosB+isinB)
=rq(cosAcosB-sinAsinB+i(sinAcosB+cosAsinB))
=rq(cos(A+B)+isin(A+B))
so magnitude of z1*z2 is rq(product of magnitudes of z1 and z2) and the angle of z1*z2 is A+B(sum of angles of z1 and z2).
Again, great explanation, thanks!

2. Hi Ragavendar, thanks for the comment. You got it, by converting to polar coordinates you can see the sine and cosine addition formula emerge. I’ll be writing more on this in a later post, but thanks for providing the details!

3. If you’re interested in a _fantastic_ text on complex analysis (with gorgeous proofs and arguments), you should check out Visual Complex Analysis, by Tristan Needham.

It’s amazing. And amazingly intuitive.

4. Thanks Alec! I’ve been looking for a good book on complex analysis, I’ll have to check it out .

5. Hello, good article but I think it’s still missing the big A-ha idea with complex numbers, which is their relation to “e” for example when I multiply two exponentials I multiply the bases and add the exponents which is analogous to multiplying complex numbers. when you divide complex numbers you divide the magnitude and subtract the angles and when diving exponential you divide the bases and subtract the exponents.
I guess this connection is manifested in euler’s formula? I don’t get it either, but wish I did.

6. even though I know those rules work for all bases when multiplying or dividing exponentials not just “e” However e is used in euler’s formula
e^ix = cosx + isinx

7. Hi Jayson, yep, Euler’s formula is a great use of complex numbers. I’ll be covering it in a later article .

8. Khalid,
You are always better explained !! Though most of us knew the concepts before, reading you is an experience by itself !!

9. Hey, Here is a basic question from me.

1. Why does complex number came into existance?
2. Why we need to rotate the number?

Sometimes I go dumb and ask such question don’t mind

10. Hi Maheshexp, check out the article on complex numbers to see why they are needed.

The rotation is one way for us to visualize what’s happening, just like multiplication can be seen as “stretching” a number. We don’t need to do this, but it’s a nice learning tool.

11. After reading your article (which was much appreciated considering I’m in Calc 3 and still couldn’t comprehend complex numbers past their arithmetic consequences), I got to thinking of a real world example that could help substantiate my thoughts. My ‘a-ha moment’ occurred in the following fashion:
How can you describe an object’s (i.e. a bike) position from you?
Before negatives:
“The bike is 4 feet from me in that direction” and you point in the direction of the bike.
After negatives:
“The bike is -4 feet from me in that direction” and you point in the opposite direction.

My aha moment, considering the use-fulness of imaginary numbers occurred when I pointed in a direction other than the 2 aforementioned ones. If you pointed 45 degrees to the right of the bike, you could say “the bike is 1+i feet away from me in that direction”.

I think what throws people off is the addition sign. Perhaps saying “1i1″ would make more sense conceptually even if it’d complicate operations performed on complex numbers.

12. Hi Tim, that’s great — thanks for sharing your insight!

Yes, negative and complex numbers are a different way to talk about where things are. Even decimals are like this: we write 2.3, not 2 + .3, even though we could. Similarly, writing 2i3 might be easier to make sense of than 2 + 3i.

Again, appreciate the notes! I love seeing the different ways people look at the same topic.

13. There is already a way to write complex numbers without addition in a nice form that avoids addition and even “i”, but it goes under the category of “interesting but useless”: the Quaternary Imaginary number system, base 2i. Donald Knuth created it, and it’s an interesting theoretical idea, but quite useless.

On another note, I’m all for rewriting “a+bi” as aib. It makes it more obvious that it’s one number, and not two separate numbers.

14. @Zac: Great point — I think I ran across Knuth’s system a while ago, and agree it’s interesting but not very practical. I’d love it if we’d rewrite a+bi into something more “combined” to show that they are really two parts of the same number (just like 3 + 1/2 is better expressed by 3.5).

15. @Paulodic, Roger: Thanks!

16. @Reza: You’re welcome!

17. thnks. a lot. ive been looking for a website like explaining math intuitively. i hope that one day you will come up with an article explaining fourier transform in a similar fashion. again thank a lot.

18. @Luis: Thank you for the comment, really glad it was helpful. I think the Fourier transform would be a great topic, I need to study it more to move beyond an “academic” understanding into an intuitive one. But once that happens I’ll be writing about it .

19. In the division portion, why is (3+4i) divided by the magnitude squared of (1+i) and simply just the magnitude?

20. @Steve: Great question — I should make this more clear.

==========
We can approach it intuitively:

* Rotate by opposite angle: multiply by (1 – i) instead of (1 + i)
[Note: When we multiply by (1-i) we cancel out the angle just like dividing by (1+i) would. However, we end up scaling the number by the size of (1-i).]

* Divide by magnitude squared: divide by |sqrt(2)|^2 = 2

[We need to divide twice: first, by the size of (1-i), because we multiplied by it above, and second by the size of (1+i), which was part of the original division. Both have the same size (just reflections of the same angle)].

So, we have to divide twice: once for the original (1+i), and again to cancel the “side effect” of multiplying by (1-i) in the first step.

If we wanted, we could just multiply by (1-i)/sizeof(1-i) to remove the angle and keep the same size, all in one step. Then we could divide by the sizeof(1+i) as we intended in the beginning.

Hope this helps!

21. Thanks for the explanation – very easily understood. Anyone have any ideas on how to visualize taking a number to the power of i? I’m trying to better understand Euler’s identity (e^(i*pi)+1=0). Thanks again,

Micah

22. @Micah: Thanks, glad you enjoyed it. Euler’s identity is a great use of complex numbers — one way to visualize it is seeing e^i*x as saying you need to change by x% in the “i” (perpendicular), vs in the real dimension. Constantly changing in a perpendicular direction will move you in a circle, leading to the full formula. I’d like to write more on this in a future article .

23. Kalid, Delicious work…
When calculating the magnitude |a + bi|, why is the “i” term left out of SQRT(a^2 + b^2)? I know it would give the wrong answer, but isn’t “i” part of the original complex number?

24. @John: Great question! The “i” in the original number is more of a direction designation, and says you’re going b units in the i direction.

If I said I walked 3 blocks East and 4 blocks North, I might write it as:

3E + 4N

With our normal numbers, we assume everything is a real number (East/West) so can just write

3 + 4N

And if I wanted to find the total distance traveled, using the pythagorean theorem I’d do sqrt(3^2 + 4^2) = 5.

Note that N (or i) is just a way to designate that the quantity is in a different, 90-degree direction. The pythagorean theorem assumes this (in fact, requires it; it only works in right triangles) so we don’t need i in the calculation. I hope this helps!

25. Haha! This is just great! I’ve used Cnum’s when coding fractals, but never really understood the concept. I’m a better man now. You explain really, really well. And i love your humour.

26. @Dennis: Thanks, glad you enjoyed it!

27. GR8 man! Vivid explanations!!!I wish u ‘d been my Math tutor @ college….I ‘d taken up math as my career! Nevertheless…I m enjoying it now!!! Thanks man!!!

28. The traditional difficulty in understanding the complex numbers is a man-made one. It reflects intellectual shortage in the course of definitions and thought construction. The term “imaginary” component in the definition of complex numbers is misleading. Furthermore, the definition of radicals or square roots in the field of Algebra has its own flaw where “negative” numbers become hard to define a radical for. Much of that is due to the historic and gradual accumulation of mathematical knowledge which characterizes mathematicians more as a cult—a culture with rich inheritance of thought, history, and terminology. The term “imaginary” for instance should have been called “rotational” instead. Imaginary things are understood as things that has no real reflection and exist only in the human mind. This is not a true case with complex numbers and their “imaginary” components. Complex numbers have real applications in physics when rotational physical phenomena arises—things that has a cyclic nature such as an alternative current in electricity. All what an imaginary component in a complex number means is that the magnitude of the complex number is specially aligned at a certain angle relative to the conventional positive real line axis. Again the real line is a misname for real numbers. So really, the complex numbers are nothing but real numbers in two-dimensional space with some closure field property pertaining to the roots of negative numbers. With this idea of rotation in mind, the study of complex numbers can be much easily understood.

29. Thank you for your explanations. Can you do an article on imaginary exponentiation? I get that multiplication by an imaginary is like rotation, but what about multiplying a real by itself an imaginary number of times!?

30. Err, quite elegant for the most part, however I’m hung up on the division. Why do we take the conjugate of the denominator and not the numerator (obviously, algebraically this is obvious, but intuitively?) And why do we WANT to shrink the denominator by its modulus, why not shrink it by 56 or something arbitrary?

31. Here is a crisp and dynamic word for complex numbers, hand both as a verb and a noun: twirl.

I’ve used it with my eight year old son, thinking of a video camera over a plane that can be both zoomed and rotated. It did not take him long to convince himself that zooms and rotations can be combined, in arbitrary order, without affecting the result. The combination of a zoom and a rotation is a twirl (imagine the twirling trajectory of the point 1+0i).

Unfortunately, it is a useful term only for multiplication, not adding. But perhaps that is a feature rather than a bug, helping keep focus on the operations rather than the numbers.

May I say that I love this site, and wish I had discovered it before today!

32. @Largo: Oh, I like “twirl” and the video camera analogy — it gets the idea across! (I love, love, love seeing how other people see these topics). And well, adding is like sliding the camera up and down (moving left/right/up/down is adding numbers in the real or complex plane).

Thanks for the kind words, happy you’re enjoying the site!

33. Dear Kalid,
Your blog is really mind-bending! I am sure one of your readers will be the next Great! (of course I am included in that set of potential ones ) Well, on sincere lines, I am really grateful to you for making this intuitive work available to all of us readers.

When it comes to naming, one thought came to my mind – people call non-existing things like ghosts and spirits as supernatural things and they call numbers which are as real as Real ones imaginary ! Just coz they are kinda intangible for the time being and in another dimension!

Well, even great mathematician Fibonacci called zero as ‘sign zero’ ! So, I think its only a matter of time before we realize the realness of these so-interpreted ‘non-real’ numbers

34. @Harish: Glad you enjoyed it — yes, I really hope that in 100 years (or 50, or 20!) imaginary numbers will be seen just as “real” as 0 or -1.

35. Thank you so much …. Reading your article has given me a good reason to start liking complex numbers … Math would be real fun if all concepts are taught this way in school ….

36. Thank you!!!!!! I’ve banged my head against the wall for years over complex numbers. I now see the light!!!!

37. @Joe: Awesome! I totally hear you, complex numbers bothered me for years ever since first encountering them.

38. Not knowing how anything works above 2 dimensions I’ve tried imagining the 3rd dimension as the imaginary dimension to help visualize things. I’m only in high school so this might not make much sense but…

say y = f(x) = square root (x)
if x can be any real we would have 3 dimensions and this seemed to satisfy my curiosity for 3 dimensional graphs. How ever if x is also allowed to be a real AND an imaginary/complex number I have to add a 4th dimension. x-real, x imag, y real, y imag are the 4 dimensions. The real x “axis” becomes a x “plane” and same thing for the y axis.

My question is can I actually define a 3 dimensional function just with something along the lines of
“y plane position”= f(x) = some function where x is uni-dimensional
or y = f(x) where x is the plane and y is uni-dimensional
simply with the help of complex numbers?
For me this is much easier to understand than something like z = yx

As far as I know, vectors, complex numbers, parametric/ polar and 3d functions are usually taught separately. I’d love to see an article where you can somehow unify them all (or anything in 3 dimensions really).

Sorry for the wall of text and thanks for explaining things so awesomely.

39. @Alex: Interesting question — I think you’re asking whether x must be real, but y = f(x) can be complex (real / imaginary), which gives 3 dimensions? I hadn’t thought of that before — unfortunately, I’m not familiar with the rules of defining functions, but that seems interesting. At the minimum, you could start talking about matrixes and say you are considering matrixes of the form [x, y(Re), y(Im)] and do transformations on that. It would be cool to merge vectors, complex numbers, and polar as they are all ways of describing multi-dimensional coordinates.

You might be interested in Quaternions — 4-dimensional numbers. From what I’ve read offhand, it seems when you extend complex numbers directly, you end up needing 2, 4 or 8 dimensions for reasons of symmetry.

Really interesting thoughts!

40. Hi Kalid, I may have realized an answer to the first question I wrote you about in my previous e-mail. If i^2 = -1, then this says (something1 x something1) has a magnitude of 1 (and it just so happens to be in a negative direction). If I recall, the only time when (something1 x something1) would have a magnitude of 1 is when the (something1) itself has a magnitude of 1 . Since (something1) = i, then i has a also has a magnitude of 1. Aside from some weird situation that I am unaware of where a multiplication of two numbers results in a magnitude of 1, yet at least one of the two numbers is not 1, would this be a fair explanation ?

Thanks

P.S. I am still curious on your thoughts to my second question.

41. Alright, how about this one. A friend mentioned that the a x b = 1 so ‘a’ and ‘b’ must be 1 applies to real numbers, and said “but we are not dealing with real numbers”. Sigh…of course not. So what about the sqrt(a^2 + b^2) ? If i^2 = 0 + 1i^2, and a= 0 and b=1, then sqrt(0^2 + 1^2) = sqrt(1) = 1. Any better ?

Thanks

42. Dang it ! That only gives the magnitude for i^2…not i. So I give up. No more postings from me…on this question that is .

Thanks

43. @Alonzo: Thanks for the comments! I left a detailed reply on the other article [on imaginary numbers], hope it helps!

44. Thank you for this brilliant article. Can magnitudes of “real” things be imaginary? Like, number of apples. If so, what does it mean?

45. Big K–

I used to be in the laser business where I learned about something called optical phase conjugation, which provides a good analogy to the math concept. The conjugate of a light wave is basically a reflection, but it doesn’t bounce off at an angle. Rather, it is “an exact, time-reversed replica” of the incoming wave. That is, it comes back on itself, a backwards reflection being analogous to a backward rotation.

–Tim

46. @Tim: Wow, that’s an awesome visualization, thanks for the example. I love real-world use cases that aren’t super contrived. Especially ones that get a new metaphor in place (a reflection or rotation).

47. Yes, it’s pretty cool. If you looked in a phase-conjugate mirror, all you would see is your eye.