Euler's identity seems baffling:

It emerges from a more general formula:

Yowza -- we're relating an *imaginary exponent* to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?

Not according to 1800s mathematician Benjamin Peirce:

It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: we must, must, **must** find an insight. Here's mine:

**Euler's formula describes two equivalent ways to move in a circle.**

That's it? This stunning equation is about spinning around? Yes -- and we can understand it by building on a few analogies:

- Starting at the number 1, see multiplication as a transformation that changes the number (1 * e^(i*pi))
- Regular exponential growth continuously increases 1 by some rate; imaginary exponential growth continuously
*rotates*a number - Growing for "pi" units of time means going pi radians around a circle
- Therefore, e^(i*pi) means starting at 1 and rotating pi (halfway around a circle) to get to -1

That's the high-level view -- let's dive into the details. By the way, if someone tries to impress you with "e^(i*pi) = -1", ask them about _i_ to the _i_th power. If they can't think it through, Euler's formula is still a magic spell to them.

**Update:** While writing, I thought a companion video might help explain the ideas more clearly:

It follows the post -- watch together, or at your leisure.

## Understanding cos(x) + i * sin(x)

The equals sign is overloaded. Sometimes we mean "set one thing to another" (like x = 3) and others we mean "these two things describe the same concept" (like sqrt(-1) = i).

Euler's formula is the latter: it gives two formulas which explain how to move in a circle. If we examine circular motion using trig, and travel x radians:

- cos(x) is the x-coordinate (horizontal distance)
- sin(x) is the y-coordinate (vertical distance)

The statement

is a clever way to smush the x and y coordinates into a single number. The analogy "complex numbers are 2-dimensional" helps us interpret a single complex number as a position on a circle.

When we set x to pi, we're traveling "pi" units along the outside of the unit circle. Because the total circumference is 2*pi, plain old pi is halfway around, putting us at -1.

Neato: The right side of Euler's formula (cos(x) + i*sin(x)) describes circular motion with imaginary numbers. Now let's figure out how the _e_ side of the equation accomplishes it.

## What is Imaginary Growth?

Combining x- and y- coordinates into a complex number is tricky, but manageable. But what does an imaginary *exponent* mean?

Let's step back a bit. When I see "3^4" I think of it like this:

- 3 is the
*end result*of growing instantly (using e) at a rate of ln(3). In other words: 3 = e^ln(3) - 3^4 is the same as growing to 3, but then growing for 4x as long. So 3^4 = e^(ln(3) * 4) = 81

Instead of seeing numbers on their own, you can think of them as something e had to "grow to". Real numbers, like 3, give an interest rate of ln(3) = 1.1, and that's what e "collects" as it's going along, growing continuosly.

Regular growth is simple -- it keeps "pushing" a number in the same, real direction it was going. 3 × 3 pushes in the original direction, making it 3 times larger (9).

Imaginary growth is different -- the "interest" we earn is in a different direction! It's like a jet engine that was strapped on sideways -- instead of going forward, we start pushing at 90 degrees.

The neat thing about a constant orthogonal (perpendicular) push is that it doesn't speed you up or slow you down -- it rotates you! Taking any number and multiplying by *i* will not change its magnitude, just the direction it points.

Intuitively, here's how I see **continuous imaginary growth rate**: "When I grow, don't push me forward or back in the direction I'm already going. Rotate me instead."

## But Shouldn't We Spin Faster and Faster?

I wondered that too. Regular growth compounds in our original direction, so we go 1, 2, 4, 8, 16, multiplying 2x each time and staying in the real numbers. We can consider this e^(ln(2)*x): grow instantly at a rate of ln(2) for "x" seconds.

And hey -- if our growth rate was twice as fast (2*ln(2) vs ln(2)), it would look the same as growing for twice as long (2x vs x). The magic of e lets us swap rate and time; 2 seconds at ln(2) is the same growth as 1 second at 2*ln(2).

Now, imagine we have some imaginary growth rate (R*i) which rotates us: e^R*i becomes imaginary and grows to "i". Well, if we double that, we get i^2 (-1) and as we keep going we just spin around the circle.

Now imagine we double that rate (2R*i). Would that spin us off the circle? Nope! Having a rate of 2R*i means we just spin twice as fast, or alternatively, spin at a rate of R for twice as long.

Once we realize that some exponential growth rate can take us from 1 to i, increasing that rate just spins us faster. We'll never escape the circle.

However, if our growth rate is complex (a+bi vs Ri) then the real part (a) will grow us like normal, while the imaginary part (bi) rotates us. But let's not get fancy: Euler's formula, e^(i*x), is about the _purely imaginary_ growth that keeps us on the circle (more later).

## A Quick Sanity Check

While writing, I had to clarify a few questions for myself:

**Why e^x -- aren't we rotating 1?**

*e* represents the process of starting at 1 and growing continuously at 100% interest for 1 unit of time.

When we write *e* we're capturing that entire process in a single number -- e represents all the whole rigamarole of continuous growth. So really, e^x is saying "start at 1 and grow continuously at 100% for x seconds", and starts from 1 like we want.

**But what does i as an exponent do?**

For a regular exponent like 3^4 we ask:

- What is the implicit growth rate? We're growing from 1 to 3 [the bottom of the exponent].
- How do we
**change**that growth rate? We scale it by 4x (^4, the top of the exponent).

We can convert our growth into "e" format: our instantaneous rate is ln(3), and we increase it to ln(3) * 4. Again, the top exponent (4) just scaled our growth rate.

When the top exponent is i (as in 3^i), we just multiply our implicit growth rate by i. So instead of growing at plain old ln(3), we're growing at ln(3) * i.

The top part of the exponent *modifies* the implicit growth rate of the bottom part.

## The Nitty Gritty Details

Let's take a closer look. Remember this definition of *e*:

That 1/n represents the interest we earned in each microscopic period. We assumed the interest was real -- but what if it were imaginary?

Now, our newly formed interest adds to us in the 90-degree direction. Surprisingly, this does not change our length -- this is a tricky concept, because it appears to make a triangle where the hypotenuse must be larger. We're dealing with a limit, and the extra distance is within the error margin we specify. This is something I want to tackle another day, but take my word: continuous perpendicular growth will rotate you. This is the heart of sine and cosine, where your change is perpendicular to your current position, and you move in a circle.

We apply *i* units of growth in infinitely small increments, each pushing us at a 90-degree angle. There is no "faster and faster" rotation - instead, we crawl along the perimeter a distance of |i| = 1 (magnitude of i).

And hey -- the distance crawled around a circle is an angle in radians! We've found another way to describe circular motion!

**To get circular motion:** Change continuously by rotating at 90-degree angle (aka imaginary growth rate).

So, Euler's formula is saying "exponential, imaginary growth traces out a circle". And this path is the same as moving in a circle using sine and cosine in the imaginary plane.

In this case, the word "exponential" is confusing because we travel around the circle at a constant rate. In most discussions, exponential growth is assumed to have a cumulative, compounding effect.

## Some Examples

You don't really believe me, do you? Here's a few examples, and how to think about them intuitively.

**Example: e^i**

Where's the x? Ah, it's just 1. Intuitively, without breaking out a calculator, we know that this means "travel 1 radian along the unit circle". In my head, I see "e" *trying* to grow 1 at 100% all in the same direction, but i keeps moving the ball and forces "1" to grow along the edge of a circle:

Not the prettiest number, but there it is. Remember to put your calculator in radian mode when punching this in.

**Example: 3^i**

This is tricky -- it's not in our standard format. But remember,

We want an initial growth of 3x at the end of the period, or an instantaneous rate of ln(3). But, the *i* comes along and changes that rate of ln(3) to "i * ln(3)":

We *thought* we were going to transform at a regular rate of ln(3) (a little faster than 100% continuous growth since e is about 2.718). But oh no, *i* spun us around: now we're transforming at an imaginary rate which means we're just rotating about. If *i* was a regular number like 4, it would have made us grow 4x faster. Now we're growing at a speed of ln(3), but sideways.

We should expect a complex number on the unit circle -- there's nothing in the growth rate to increase our size. Solving the equation:

So, rather than ending up "1" unit around the circle (like e^i) we end up ln(3) units around.

**Example: i^i**

A few months ago, this would have had me tears. Not today! Let's break down the transformations:

We start with 1 and want to change it. Like solving 3^i, what's the instantaneous growth rate represented by *i* as a base?

Hrm. Normally we'd do ln(x) to get the growth rate needed to reach x it the end of 1 unit of time. But for an imaginary rate? We need to noodle this over.

In order to start with 1 and grow to *i* we need to start rotating at the outset. How fast? Well, we need to get 90 degrees (pi/2 radians) in 1 unit of time. So our rate is "i * pi/2". Remember our rate must be imaginary since we're rotating, not growing! Plain old "pi/2" is about 1.57 and results in regular growth.

This should make sense: to turn 1.0 to *i* at the end of 1 unit, we should rotate pi/2 radians (90 degrees) in that amount of time. So, to get "i" we can use e^(i * pi/2).

Phew. That describes i as the base. How about the exponent?

Well, the *other* i tells us to change our rate -- yes, that rate we spent so long figuring out! So rather than rotating at a speed of i * pi/2, which is what a base of *i* means, we transform the rate to:

The i's cancel and make the growth rate real again! We rotated our rate and pushed ourselves into the negative numbers. And a negative growth rate means we're shrinking -- we should expect i^i to make things smaller. And it does:

Tada! (Search "i^i" on Google to use its calculator)

Take a breather: You can intuitively figure out how imaginary bases and imaginary exponents should behave. Whoa.

And as a bonus, you figured out ln(i) -- to make e^x become i, make e rotate pi/2 radians.

**Example: (i^i)^i**

A double imaginary exponent? If you insist. First off, we know what our growth rate will be inside the parenthesis:

We get a negative (shrinking) growth rate of -pi/2. And now we modify that rate *again* by *i*:

And now we have a negative rotation! We're going around the circle a rate of -pi/2 per unit time. How long do we go for? Well, there's an implicit "1" unit of time at the very top of this exponent chain; the implied default is to go for 1 time unit (just like e = e^1). 1 time unit gives us a rotation of -pi/2 radians (-90 degrees) or -i!

And, just for kicks, if we squared that crazy result:

It's "just" twice the rotation: 2 is a regular number so doubles our rotation rate to a full -180 degrees in a unit of time. Or, you can look at it as applying -90 degree rotation twice in a row.

At first blush, these are really strange exponents. But with our analogies we can take them in stride.

## Complex Growth

We can have real and imaginary growth at the same time: the real portion scales us up, and the imaginary part rotates us around:

A complex growth rate like (a + bi) is a mix of real and imaginary growth. The real part a, means "grow at 100% for *a* seconds" and the imaginary part b means "rotate for *b* seconds". Remember, rotations don't get the benefit of compounding since you keep 'pushing' in a different direction -- rotation adds up linearly.

With this in mind, we can represent any point on any sized circle using (a+bi)! The radius is e^a and the angle is determined by e^(b*i). It's like putting the number in the expand-o-tron for two cycles: once to grow it to the right size (a seconds), another time to rotate it to the right angle (b seconds). Or, you could rotate it first and the grow!

Let's say we want to know the growth amount to get to 6 + 8i. This is really asking for the natural log of an imaginary number: how do we grow e to get (6 + 8i)?

- Radius: How big of a circle do we need? Well, the magnitude is sqrt(6^2 + 8^2) = sqrt(100) = 10. Which means we need to grow for ln(10) = 2.3 seconds to reach that amount.
- Amount to rotate: What's the angle of that point? We can use arctan to figure it out: atan(8/6) = 53 degrees = .93 radian.
- Combine the result: ln(6+8i) = 2.3 + .93i

That is, we can reach the random point (6 + 8i) if we use e^(2.3 + .93i).

## Why Is This Useful?

Euler's formula gives us another way to describe motion in a circle. But we could already do that with sine and cosine -- what's so special?

It's all about perspective. Sine and cosine describe motion in terms of a *grid*, plotting out horizontal and vertical coordinates.

Euler's formula uses polar coordinates -- what's your angle and distance? Again, it's two ways to describe motion:

- Grid system: Go 3 units east and 4 units north
- Polar coordinates: Go 5 units at an angle of 71.56 degrees

Depending on the problem, polar or rectangular coordinates are more useful. Euler's formula lets us convert between the two to use the best tool for the job. Also, because e^(ix) can be converted to sine and cosine, we can rewrite formulas in trig as variations on e, which comes in very handy (no need to memorize sin(a+b), you can derive it -- more another day). And it's beautiful that every number, real or complex, is a variation of e.

But utility, schmutility: the most important result is the realization that baffling equations can become intuitive with the right analogies. Don't let beautiful equations like Euler's formula remain a magic spell -- build on the analogies you know to see the insights inside the equation.

Happy math.

## Appendix

The screencast was fun, and feedback is definitely welcome. I think it helps the ideas pop, and walking through the article helped me find gaps in my intuition.

References:

- Brian Slesinsky has a neat presentation on Euler's formula
- Visual Complex Analysis has a great discussion on Euler's formula -- see p. 10 in the Google Book Preview

## Other Posts In This Series

- A Visual, Intuitive Guide to Imaginary Numbers
- Intuitive Guide to Angles, Degrees and Radians
- Intuitive Arithmetic With Complex Numbers
- Understanding Why Complex Multiplication Works
- Intuitive Understanding Of Euler's Formula
- An Interactive Guide To The Fourier Transform
- Intuitive Understanding of Sine Waves

This is a phenomenal article! Took me quite a bit of time and multiple readings to get my head around it, but now I get it. I think.

Hey cool that makes a lot of sense. I already thought of imaginary numbers that way, but the growth pulling in a circle is very straightforward.

Are there complex equations that move in this way that intersect the real set of numbers in cool ways? Like I could imagine some complex equation that loops around, intersecting the real set of numbers (the real numberline) to create the set of primes or something like that.

@Aditya: Thanks! Yes, it took me a while to really see the equation, there may be a nicer way to go back and streamline how it was presented — I’d like to avoid the need for people to have multiple readings :).

@Brandon: That’s an interesting question — actually, the Mandelbrot set is like that to some degree, where there is a complex (2d) function which gives rise to some pretty amazing patterns. I don’t know of any others off the top of my head though.

I <3 BetterExplained – keep 'em coming!

This gets easier if you've already got the hang of the physics concept that to move in a circle you must keep accelerating. If you accelerate in one direction, you will get faster and faster, but if you keep accelerating in a new relative direction, your speed will be the same, but you'll move in a circle (your velocity changes.)

Acceleration is a kind of growth, and so it logically follows that if you grow in a relative direction, you'll rotate but not speed up.

@wereogue: Thanks for the support! Yes, the physics interpretation definitely helps see this relationship, and I like the way you put it — our growth/change is really an acceleration. Our velocity is always perpendicular to our position (and acceleration perpendicular to velocity) which gives us a circle. It’s funny how much overlap there is between math and physics :).

Thank you for showing us that Maths can be easy and simple. You are an example to today’s mathematicians, what you are doing is really inspiring. I always try to think how to make things easier to teach, learn and do. But you really make it so well!

Congratulations!

Keep it doing it!

you can go very far!

Mariano

(Excuse my English)

Great article, Keep going on….

Great article and great video!

I was wondering about what you used to generate the graphics, they look great :).

Cheers from Romania,

Alex

I didnt bother to follow your argument because the topic doesnt interest me but I like your attitude that math is just logic and “common sense” and theres way too much hocus pocus and ‘mysticism” that often creeps in imho.

Thanks for making me excited about math again.

Kalid,

I’m still digesting it all, but i just have to say: “continuous perpendicular growth will rotate you” is just plain sexy. Wow, it makes sooo much sense to me Keep it up sir!

Sebastian

Kalid, this is extremely impressive. I’ve been trying to understand this for a long long time. I found it difficult to see past the numbers and symbols and wanted to understand it ‘visually’ so I knew what was trying to be achieved. You have described it all beautifully and for the first time I am really understanding how it all fits together. I so wish you were teaching me in school 25 years ago. Thanks again.

@Kalid’s Friend: It really bothered me for a long time also — Euler’s formula was used everywhere but I didn’t have a gut feel for it! I’m really happy it was able to help

@Mariano, @Mithun: Thanks for the kind words!

@stuart: Yes, I think everything should be understood / explained intuitively, and not accepted as mystic.

@mark: You’re more than welcome.

@Sebastian: Haha, I like that phrase too — whatever it takes to make it click :).

Kalid,

Thank you for this!

Would it be accurate to say that if you traced out the complex growth curve just as you did the real and imaginary growths, you would get a spiral?

By “tracing out”, I simply mean that if you are given e^(ax+bi), you simply put points at specified intervals to the answer. For example the following would give you 3 points on the way to your answer:

e^((a/4)x+(b/4)i)

e^((a/2)x+(b/2)i)

e^((3a/4)x+(3b/4)i)

Oops, on my previous post, please assume x=1.

@lewikee: Yep, you got it — imaginary exponential growth rotates you in a circle, and regular exponential growth grows you (e^a/4, e^a/2, etc.). You end up spinning around the circle but getting further and further away, making a complex spiral.

If you put

parametric plot (e^(t/4) * cos(t), e^(t/4) * sin(t))

plot on wolfram alpha

into wolfram alpha you can see an example (I separated e^ix into cos(x) + i*sin(x) to get the x and y components, and put them separately into the plot — I also scaled down the regular exponential growth to make a tighter spiral).

Wow, you are *amazing*! You explain things a way *nobody* has ever done (around me I mean). It is unbelievable: it’s like you enter my head, grab the questions inside and then explain each of them with a 100% chance of understanding!

Very nice and useful article, explains what I used to consider trivial, wow!

I love the site, please keep on publishing such wonderful articles!

Thanks a lot.

@nschoe: Thanks, really glad it helped! Yeah, I really feel it’s important to answer those “huh?” questions we get in our heads as they come up, instead of pretending that learning is one smooth process from A to B. We need to explain why we don’t want to go to C and D :).

Appreciate the support!

Hi Kalid, thanks for all these articles, I’m finding them illuminating.

I’m still struggling with this one a bit, though. I’m confused by the use of the phrase “growing instantly”. E.g. “3 is the end result of growing instantly (using e) at a rate of ln(3), i.e. 3 = e^ln(3)”

If something grows instantly, then its rate of growth is infinite, isn’t it? I don’t understand how something can grow “instantly” at a finite rate.

And then “3^4 is the same as growing to 3, but then growing for 4x as long” – but if the growth was instant before, then “4x as long” is also instant. Four times zero is zero.

And what does “using e” mean in this instance? Starting with a value of e? Why are we doing that? I thought e was a sort of fundamental growth rate, not a starting value? In your Expand-o-tron analogy in another article, the base is the desired growth rate per unit time, and the exponent is the number of units of time to grow for. But here, the exponent seems to be the rate, while the base is the starting value.

@PaulH: Thanks, great question (I need to setup a FAQ for exponents since there are so many parts that I still have to remind myself also).

You’re right — perfectly instantaneous growth would indeed be infinite. Getting at this idea of “infinitely small” is one of the problems of Calculus actually. A better phrasing may be “Growing at a compounding interval that appears perfectly smooth to us”, i.e. we cannot see the sudden, punctuated growth that pops up with compounding on an interval (like making a staircase so fine-grained that it looks like a smooth curve). For example’s sake, we can imagine compounding every nanosecond (billionth of a second).

“Using e” means figuring out the net effect of compounding as fast as possible (every nanosecond, let’s say). So taking the examples you mentioned:

>> E.g. “3 is the end result of growing instantly (using e) at a rate of ln(3), i.e. 3 = e^ln(3)”

We can think of 3 in two ways. The first is the traditional way — at the end of the year, we have 3 times as much (we start with a dollar, get 3 dollars at the end). The other way is to say “3 is the result of starting at 1.0 and growing instantly (compounding as quickly as possible) at some rate.”

If our rate is 100% and we compound as quickly as possible, we end up with 2.718… (e). But we want to get to “3″, so we need to grow a bit faster than 100% per year. The actual number is ln(3) = 109.8%, so we can say:

“We can get to 3.0 at the end of the year if we start at 1.0 and grow at 109.8% per year, compounding as fast as possible”. e is the way we work out what super-fast compounding gives us, so we can write it

e^1.098 = 3

So, we do start at 1.0, but e^x gives us the impact of starting at 1.0 and compounding at x% return as fast as possible.

In other words, e isn’t the starting point because we want to start at 2.718. We start at e because we want to compound 1.0 as fast as possible, and e is the shortcut for that (i.e., the price after shipping, handling, and taxes if you get my drift).

Hope this helps!

How would you go about solving this problem, i was asked to put ln( pi*i -ln (-e^w)) into the for a + bi, and the question states that w is a complex number. Need help for a final tomorrow.

Fantastic!!! You should do more explanations like that, you’re as good (or perhaps even better) than Sal Khan from Khan Academy!

@Rafay: I can’t really comment on specific homework problems, but break it into groups: you can see that ln(e^w) should just be “w” (since ln and e are inverses) so you get something like ln(pi * i – w). From the rules of logarithms, this equals ln(pi * i) / ln(w). The natural log of i is explained above — that should help get you started.

@Flo: Thanks! I plan to keep creating more :).

Nicely done. Another internet applied math whiz is discovered….

Wow, the last time my mind was blown this hard was when I did LSD. Only this time I can remember what it was that had caused the mind-blow (lol?) the next day.

Kalid,

I’m not sure, but I think your description of 3=e^ln(e) is a very interesting way of converting discrete mathematics to continuous. It is like saying, there is a finite rate with which to generate a number (3), you grow so much per unit time. You can use this method to compute new numbers, by “growing them”. Brilliant.

^ I meant 3=e^ln(3) :).

@Mark: Glad you liked it! Yes, it’s interesting to see regular whole numbers (like 3, 4, etc.) as just “some amount” of growth that e takes you to.

You are simply brilliant !

Please keep updating your web with new and interesting problems.

Thanks

It deserves to be called a “phenomenal” article.

Hope, to see Fourier Transform/Laplace Transform/Z-transform articles. If that happens, thousands of students would change the way they learn (I think they will like it more).

@Nasser: Thanks! I’m planning on doing the Fourier transform soon — the others, I still need to learn about :).

I’m a math major studying topology, and really really sympathize with the attitude that math is about the underlying structure, the geometry and simplicity of the problem; with the analytic formulas being a good book-keeping device, but unenlightening unless you have a picture in your head. I can’t say enough how well-written I found this article (and the one about “e”), I have never enjoyed complex numbers because it always feels like a bunch of tedious algebraic laws, but this is a wonderful explanation of euler’s formula; and after reading the small bit in “Visual Complex Analysis; I picked up a copy and its awesome.

Keep up the good work; and I also look forward to the fourier transform/laplace transform installment!

@Steve: Thanks! I really like Visual Complex Analysis too — even though I wasn’t a math major (maybe in another life :)) I have a severe interest in it and want to study it deeper. Appreciate the encouragement!

Just a comment I always considered e^ix identical to e^x, except the mapping of e^ix to the 3D complex plane created a helix that turned the runaway exponential to spiral around the x axis, but still always increasing in size.

The projection on the real and imaginary axis being the cosine and sine made it look like a simple rotation, but looking at the complete 3D picture this is not so.

This may be the most insightful article I have ever read. Finally someone care about making formulas intuitive! Very VERY well explained. Genius!

@C: Thank you — really made my day!

@Anonymous: That’s a great insight! This site has a cool diagram showing the spiral interpretation: http://www.songho.ca/math/euler/euler.html.

I love that. any number can be written in terms of e ‘growing’ at a certain rate for a certian amount of time! brilliant!

@Regan: Awesome, glad it helped!

Kalid, you are my new bestfriend.

@Loui: Math brings everyone together!

thanks!

@Anonymous: You’re welcome!

Second paragraph under “But Shouldn’t We Spin Faster and Faster?”: mustn’t it be “if our growth rate was twice that (2*ln(2)), it would look the same as growing twice as fast (2x vs x)” instead of “if our growth rate was twice that (2*ln(2)), it would look the same as growing for twice as long (2x vs x)”? Thank you.

Really helpful for the project that my group is doing in Math class! Thanks, I understand this so much more clearly now!

the comment above is from me, i forgot to write my name but i just wanted all of you math lovers to know that i LOVE math!

math is a wonderful thing

math is a really cool thing

so get off your ass, lets do some math

math math math math mathhhhh

@Paul: Great question. I clarified the sentence a little — e^rt has the cool property that increasing the growth rate (2*ln(2) vs ln(2)) would have the same effect as increasing the amount of time spent (2x vs x).

@Sam: Awesome, glad it helped!

@Jamie: Math rocks :).

This article is amazing! Reading it makes you feel like a genius.

In the section “what does i as an exponent do” it seems like you are saying that 3^4 is equivalent to ln(3) * 4. Obviously, this is wrong.

Nevermind I see what I was missing.

3^4 = (e^ln(3))^4

3^i = (e^ln(3))^i

Kalid, the intuitions your articles provide are of unmatched quality. Thanks, and keep writing!

I really like the physics interpretation wererogue mentioned (angular and uniform acceleration), which is an excellent metaphor. But for me (who thought about this topic like 19^th century Benjamin), the most important inside you provided is that (-1)^n becomes a less awkward special case if you have complex numbers. The e^(i pi n) model now feels more elegant and obviously superior

I need to read more (all) of your articles, but in case you haven’t yet: Would you please demystify residues and contour integrals ?

Doh, two! wrong words in the analogy. Although the idea of viewing the imaginary part as kind of centripetal force seems still nice, the real part is of course not uniform but proportional to the position. So x” = x works perfectly, but the analogy (with one acceleration growing exponentially, and the other one staying constant, if I got it right this time) is maybe not that nice after all.

@Rick: Thanks! I love sharing those aha moments with people

@Peter: Thanks for the comment — I’ve updated the article to be more clear in that section.

@Benedikt: Wow, appreciate the note! I’d love to do those topics as I learn more about them!

For the analogy, yes, you have to

@Rick: Thanks! I love sharing those aha moments with people

@Peter: Thanks for the comment — I’ve updated the article to be more clear in that section.

@Benedikt: Wow, appreciate the note! I’d love to do those topics as I learn more about them!

For the analogy, yes, you have to sort of pick the portions where it works best. There’s clearly some issues with scaling and rotating at once — I prefer to split them up.

So, if your exponent is e^(a+bi) then I see that as a combination of real exponential growth and imaginary exponential growth (rotation). It’s really e^a * e^bi and you get the best of both :). I’m not sure if this answers your question? If it doesn’t, let me know.

I’m still trying to grasp the concepts, and have some basic ideas.

One analogy to this movement is like a pen moving in the x direction and writing on a rotating flat disk below the pen (the disks diameter is parallel to the x axis). A (2-dimensional) spiral will be created. The spiral will be perpendicular to the pen then. This is just a basic analogy, perhaps a rotating cylinder centered on the x axis would be another analogy. The drawn figure would be a 3-dimensional spiral.

If anyone knows a similar type of analogy, let us know.

Hi Jon! You might like the spiral diagrams on this page: http://www.songho.ca/math/euler/euler.html.

Thanks Kalid. Here is a movie that shows the sine and cosine waves at right angles to each other. It seems a bit different that the others kind of descriptions.

http://upload.wikimedia.org/wikipedia/commons/8/8e/Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif

The movie was found on a page for Eulers Formula: http://en.wikipedia.org/wiki/Euler%27s_formula

Hi Kalid,

Thank you for this wonderful explanation. One question for you, however, or for anyone else who might be able to answer it: I still can’t seem to understand why it makes intuitive sense that imaginary growth is orthogonal. Could anyone explain this?

Thanks in advance, and again, a really awesome website.

–Stephen

I mean I see why the shape of the imaginary growth is a general curve, but how do you know it’s circular (ie the growth is orthogonal)?

@Jon: Thanks, that’s a great diagram! Seeing a helix is another way to interpret the formula.

@Stephen: Great question, and thanks for the kind words! For me, the key to imaginary numbers is to see an equation like

x^2 = -1

and break it down to

1 * x * x = -1

That is “What transformation x, when applied twice, will turn 1 to -1?”. A rotation of 90 degrees is one such interpretation; as long as the rotation is perfectly orthogonal, then two such rotations will result in a mirror image.

If imaginary growth had a small component in your current direction (a 89 degree rotation, say) then

1) Two imaginary rotations would not perfectly flip your direction (89 + 89 = 178)

2) Accumulating imaginary rotations could slowly grow you as you added imaginary interest (in effect, you are multiplying by a complex number, not a purely imaginary number)

But, a key principle in imaginary multiplication is that 1 = i^4 = i^8 = i^12, i.e. every set of 4 perfectly cancels. In my head, I think “multiplying by an imaginary number cannot give you any components in your current direction, otherwise that ‘boost’ could accumulate over time.”

I hope this helps! Let me know if it didn’t, I love really getting at the heart of what makes these analogies click.

Hi Kalid,

Thanks so much for your really quick reply! It really means a lot to me. You don’t know how much this concept has been bugging me haha. You’re website has really made me think deeply over the past few days….

My question doesn’t so much revolve around imaginary multiplication, but rather the complex number interest multiplication that shifts the vector, starting from 1 on the real axis, as outlined by e^(ix).

I get your point about imaginary multiplication and that because it cycles back to 1, there can be no net growth in the vector magnitude, and from there it’s reasonable to conclude that all the change is orthogonal. That make sense.

But as you said, when you’re looking at the multiplication you’re doing for e^(ix) (to achieve growth along the circle), you’re in fact multiplying by a complex number on each infinitesimal step (I guess your very first bit of interest would be all imaginary, that is, vertical). If I’m not mistaken, the multiplication would look something like (1+ix/n)(1+ix/n)(1+ix/n)….n times, with really small n’s, (please correct me if I’m wrong.) Each (1+ix/n) would cause a perpendicular change to the vector and would result in a rotation. So is there anyway to directly see how this multiplication changes the vector in a specifically perpendicular direction?

I mean, I don’t know what other path you’d take to achieve the transformation you’re looking for, other than a circular path. It makes sense…I might be chasing nothing here, but I guess I’m looking for some way to see that multiplying by a small component of (1+ix/n) with a really small n, guarantees a change in a perpendicular direction specifically, not just in a general upwards direction. After all, there is a real component to the complex multiplication we’re doing here, so does the imaginary number multiplication logic hold up here? Couldn’t we end up with say…a spiral? idk

Thanks, Kalid, for bearing with me. I really, really appreciate it.

Take care

@Stephen: You’re more than welcome, these are really fun to think about.

Ah, I think I see what you’re getting at! Yes, it’s interesting how those little minute changes add up to a perfectly circular rotation… check out this page:

http://www.cut-the-knot.org/arithmetic/algebra/Scott.shtml

There are some diagrams halfway through, but he’s plotted an example of (1 + 2i/10)^10 [i.e. taking steps of 10]. You can see how it converges on a circle. As you make the exponent higher (n=10, n=100, etc.) you can see how it “wraps” the circle more tightly.

He has more formal arguments about why the magnitude is 1 (no scaling) and the angle is exactly theta, which I need to work through and understand intuitively for myself ;).

Thanks a bunch, Kalid. The website is very helpful…I will have to go over it a few times though haha. A bit complicated….

Appreciate your help.

@Stephen: No problem. Yep, there’s a lot of gritty math there, but I found the diagrams helpful.

Hi,

Is it a right “transcription” of the original formula:

exp(pi) = ‘(square root of -1) root’ of -1

?

@erik: Hrm, I’m not sure what you mean by transcription… do you mean “How would you put the formula into a sentence?”

really good do you have any ideas about tetration to an imaginary number or irrational number?

Hi,

I love this site! Thank you!

Quick question: if e^i*pi means going pi radians around a circle (in this case 1/2 circle), does that mean I can always think of the coefficient of i as radians? Is it correct to think about it this way?

Thanks again for a great site.

@Bill: Thanks for the comment! I consider the coefficient on i as how “long” you go around the unit circle, assuming unit speed. So, pi “seconds” takes you halfway around. Another way is to think about it in terms of distance, i.e. halfway around directly (speed and time are “equivalent” here since it’s a unit speed… but I like thinking e^rate*time where the “rate” is i and the time is pi).

We’re using radians because we want measurements from the perspective of the mover (vs. degrees, which are an arbitrary measure from the perspective of the observer).

(1+x/n)^n =(1+x/(x*m))^(x*m)=((1+1/m)^m)^x | N=x*m

So taking limit n goes to infinite, also m goes to infinite

So (1+x/n)^n=e^x

Was great insight of you. Thanks for your explanation.

Was great job of you.

(1+x/n)^n =(1+x/(x*m))^(x*m)=((1+1/m)^m)^x | N=x*m

So taking limit n goes to infinite, also m goes to infinite

And n is real number. If x is imag i, m should be -i * absolute (n)

So as n goes infinite (1+i/n)^n become [(1+1/(-i * absolute (n)))^(-i * absolute (n))]^i = [e]^i

So I just wonder the expression n goes infinite applied only in real number direction. But as it shows it also work in complex number set.

as (-i * absolute (n)) goes infinite…

Am I something wrong?

If some number goes to infinite also work in 2D number set (complex number set), what is it really mean?…. ” the some number goes to some direction to the infinite”…

@Yonggook: That’s a really good question — check out this page which talks more about the limit http://www.cut-the-knot.org/arithmetic/algebra/Scott.shtml. Right now I’m not sure what it means to have a complex number go to infinity.

You claim i^i = e^(-π/2). Is it possible you might be forgetting an infinite amount of possible values? I could be wrong, but hear me out:

In general, sin([4n+1]π/2) = 1 and cos([4n+1]π/2) = 0 for any integer n

So e^(i*[4n+1]π/2) = i

Exponentiating, i^i = {e^(i*[4n+1]π/2)}^i = e^(–[4n+1]π/2)

For example, taking n = 1 yields your answer, whereas taking n = 2 (for which it seems this equation is still valid), yields i^i = e^(-5π/2). So I would conclude i^i doesn’t map to only one place on the real line.

@Christian: Good question! When talking about sine, I think about it more like this:

i^i is equal to the computation e^(-pi/2) which is approximately 0.207

The fact that e^(-pi/2) = e^(-5pi/2) is a bit like 4/4 = 2/2 = 1 — different “computations” that approximate the same real number. So, i^i still only has one spot on the number line because e^(5pi/2) = e^(pi/2).

@Kalid:

e^(5π/2) does not equal e^(π/2)

@Christian is correct:

http://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml

but to be fair @Kalid has been motivating this graphically the whole time so he used:

http://en.wikipedia.org/wiki/Complex_logarithm#Definition_of_principal_value

i wasted 4 ys of my life in the faculty of science studying magic spells

punch of idiots were there

@Christian: Whoops, my mistake! I meant to write e^(-i * 5*pi/2) = e^(-i * pi/2) but that leads right back to your point!

You are correct, if you allow the base “i” to be defined as any value e can be raised to (i*pi/2, i*5pi/2, …) then when you raise it to the “i” power you could have e^(-pi/2), e^(-5*pi/2), etc. which clearly have different values.

@Dmitry: Thanks for the clarification! Google calculator, for example, will treat (e^(i * 5 * pi/2))^i the same as (e^(i*pi/2))^i. More interesting things to read up on!

@Alqazzafi: Funny how much we can study but not really learn, right? Anyway, happy if the article helped.

Phenomenal – I thought I was the only person trying to understand intutively – superb

@suhas: Thank you!

Youre a GENIUS!!!

@Tariq: Glad you liked the article =).

Priceless insight, thank you.

@eaca: Thanks, you’re welcome.

khalid, brilliant. There is also a scenario where e^xi is a uniform helix in a 3d axes of img,real and x (x being phase angles limited to 2pi) that there use for circular electromagnetic model. It is a very good explanation of euler’s formula and imaginary no. except without the need for img no. The circle being the proj of helix on img-real plane while the x-img and x-real planes gives u the sine and cos proj.I hope u might expand on that…

@alan: Thanks for the comment — that’s a really interesting visualization I’d like to explore :).

nice

I am an electrical engineer, 74 years old, and am most impressed by your approach to such mathematical problems. I always felt to live on solid ground, the tools were useful, but now I feel to have walked on an thin ice-layer of a deep lake. And I am most fascinated by what I see under the ice.

@Helmut: Thank you for the kind words and beautiful analogy! Learning is a constant quest to venture deeper :).

There is another approach for finding i to the i. e to the ipi is -1. The square root of -1 is i. So therefore i is equal to the square root of e to the ipi or in terms of exponents it is equal to e to the ipi to the one half power. From there you simply multiply the exponents by i giving e to the iipi/2 which equals e to -pi/2 .

I do like your using i as a rotator ( precessor). Seldom is it presented as such. In fact that’s exactly what it is- a quaternion. Euler’s identity is simply (actually very profoundly) the compact form of Hamilton’s quaternions. That is: e to the ipi equals ii=jj=kk=ijk=-1

@Will: Thanks for the comment! I like that alternative derivation of seeing i as the sqrt of -1, where -1 = e^(i*pi). Glad you liked the rotator analogy, I want to learn more about Hamilton’s equations.

Thanks for this Kalid, it helped me to visualize the maths involved which is the only way that I ever end up actually learning anything. Formulas are a road-block, I need pictures.

So I ended up here because I’m trying to wrap my head around the Fast Fourier Transformation in two dimensions (image processing) and understanding Euler’s Formula is an essential stepping stone along the way. I eventually figured that out after watching this: http://www.youtube.com/watch?v=ObklYbQaX24

Now there’s a challenge for you, give the FFT the Better Explained treatment so that even a calculus flunker like me can have an “aha” moment. Massive challenge that

Hi Khalid.

Love what you do, just a remark ( correct me if I m wrong and sorry fro my poor english, native French speaker): I believe you should have wrote:

e^i =lim (n->+00) ( 1+ 100%*i/n)^n instead of e, I see it as the number growing from 1 => e^i at imaginary compounded growth.

obviously for n=1 , we are far from it as e^i is very different from (1+i), same as e^1 is very different from 2.

I believe it could be fun and useful to understand this formula by running it for n=let s say 10:

e^i ~ (1+i/10)*(1+i/10) … ntimes

by developpin : (1+i/10)*1 + (1+i/10) *i/10+ etc…

we can feel the 90 deg pull with the multiplication by i/10 ( 90 degree pull of a 1/20 of a full circle). as n tends to infinate and this pull applied quasi-instantly we are litterally and very preciselly running the cirlce circumference

Correct me if I m wrong and wish u the best!

Farid

I think I should have wrote * (1+i/10) enables you to rotate the equivalent of 1/40 of a circle… should be 9 deg . and as you said it, if you go for large numbers (1+i* 1/100000) the hypotenus remains equal ~1 at the limit ( 1/100000*1/100000 is second order)… and compounded growth rotates us in a very accurate way . Hope I m not too wrong with that insight. exponential and complex are a tricky but magic mix …. It could be cool to show an application of this in electronics as it s widely used…

Another idea: as e^x is the fastest way and ideal way to compound infinetisimal growth( 100%/unit of time), e^ix would be the ideal way to add compounded infinetisimal 90 deg pull which can be interprated as perpendiculary small vectors, I see it as a geometric vector addition:

=(1+i/10)*1+ (1+i/10)*i/10 +oo. hope I m not too wrong with that.. anyway ur article gave me plenty of brain storming moment and thanks for that! Cheers.

The real puzzle of the equation is why, out of all the numbers in the universe, the ONE number that just happens to move you around in a circle (e) just happens to be the same number that you need to integrate under the curve 1/x to get 1 unit of area.

You showed that e^i moves you around in a circle, and that sin and cos also move you around in a circle. But the mystery is why e^i moves you around in a circle in the first place. Why e???

There is some deeper relationship between exponential functions and trigonometric functions that you have not reached with this essay.

“Yowza — we’re relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?”

I think it was Gauss who said that if it wasn’t intuitively clear to you, you’d never be a first-rate mathematician. If it wasn’t Gauss, it was David Hilbert, or one of those other Germans.

@Julian: Thanks for the note! I’d like to cover the Fourier Transform eventually :).

@Farid: Appreciate the note — I’m still a little confused, but as you say, you can plot out “e” with smaller approximations for n (like n = 10) and see the imaginary interest “wrapping around the circle”. The larger your n, the closer you follow the circle [with n=1, your interest is very "chunky" and doesn't keep turning you in micro-increments]. But thanks for the note!

@mra: Great question. I actually don’t see “e” as a number by itself — it’s the result of starting with 1, and growing at 100% interest as fast as you can. The integral of 1/x can also get you to this “infinite growth” process, see this article:

http://betterexplained.com/articles/developing-your-intuition-for-math/

So instead of “why e?” think “Why is it this process of continuous growth?”. It’s important not see “e” as a magic spell :).

Sine is actually a similar process of continuous change as well, which makes it fit with e much better. See http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/

@Tim: Great quote. I can’t say I’ve really learned something unless it is intuitive to me. Otherwise, I’m just parroting facts someone else found out.

Kalid, I appreciate the response but I still don’t see it as an answer to the underlying question. You are just inviting me to rephrase the question, so I will: Why is a process of continuous growth related to a process of continuous circular movement? It’s not enough to just observe that they are both continuous! You might as well say that e is related to c because they are both speed limits of a sort.

So the mystery reamins: why does “the fastest way to grow at 100% interest” just happen to be related neatly to the irrational number whose sign is 0?

mra: No problem, let me see if I can clarify.

e^x represents continuous, never-ending change. Usually, this “change” is all in the same direction, and accumulates, and we call this exponential growth: 1, 2, 4, 8, 16, 32, etc.

But… that’s because the exponent (“x”) is assumed to be a real number. But if we let x be imaginary, something happens: our “change” is NOT in the same direction. Each instant, we are changing by 90 degrees to our current direction. This is like swinging a rope over your head… every instant, the rope is moving in a direction perpendicular to where it is pointing [one of the characteristics of a circle: the radius, where you're pointing, and the tangent, where you are going next, are perpendicular].

So, 100% continuous growth, when your interest is *always in a 90-degree direction*, will look like a circle. In other words, you can describe a circle as “start at 1, and always change 90-degrees to where you are. But as soon as you have changed a nanometer: stop, re-evaluate your direction, and grow 90 degrees to your current position.” (Continuous means you are constantly changing, not “moving for a while and then deciding to change”).

Hope this helps!

Ok, this is a very interesting and helpful reply, and it has definitely taken me at least one step closer to getting the underlying connection (which, by the way, I have been looking and looking for, and have not found anywhere but here).

I think what you are saying here – correct me if I am wrong – is that the sine curve and the exponential curve are just different manifestations of the same underlying function. If we plug in real numbers, we see an exponential curve because we are constantly pushing “up”, and if we plug in complex numbers, we see a circle because we are constantly pushing “perpendicular”. So if we do some sort of transformation from the real domain to the complex domain, an exponential curve maps to a circle, and vice versa.

Now, the next question that arises to me is, what is the deep relationship between the imaginary x and the 90 degree push. In other words, can we ask why it is that the one and only way to get 90 degree motion – which is the only motion that will give you a circle – in the complex plane is to use the same function that gives you continuous growth in the real domain? What FORCES that function to be the e^x function? Why was it impossible that an imaginary x give you a continuous 45 degree push, or an 88.63625 degree push? Why 90?

I think I might see the answer in this concept of continuity that you are using. Is it that, if you push continuously at any non-orthogonal angle, you can’t do it with continuity? If you push, say, at less than 90, then there is a component of your push that takes you away from the origin, and not only do you not make a circle, but you go spiraling off to some infinite imaginary destination. If you push at more than 90, then a component of your push is in toward the origin, and you spiral to zero. Therefore, the function that gives you maximum continuity of growth in the real plane HAS to trace a circle in the complex plane because no other “shape” could offer the same maximal continuity.

Did what I just said even make sense?

Since he did it for Khan Academy, Bill Gates should give you five million dollars so you can do this all the time.

Now let me see if I’ve got this straight. We can look at e^i as 1*e^1*i. (This little trick of yours was really helpful.) The 1*e means that we’re starting at 1, which in this case is 0 radians, our original amount, and then growing by e^1*i, where 1 is 1 radian and i is a continuous change in direction. All this is equal to cos(1) + i sin(1), a point on a circle in the imaginary plane.

I think I’ve got it now. Thanks for letting me think out loud.

@Tim: Thanks for the support — I wouldn’t sneeze at a few million :).

Happy to let you think it out: that’s pretty much how I see it. e^x is 1 * e^x, which means “Let’s grow continuously by x”. Oh wait! It turns out x = 1 * i [grow continuously and perpendicularly for 1 second]. This will rotate you “1 unit” along the circle. Turning that rotational (polar) coordinate into a linear one means you are at cos(1) + i sin(1).

I found a nice derivation of i^i on Yahoo Answers:

i^i=e^ln(i^i)

=e^i*ln(i)

=e^i*ln(o + i)

=e^i*ln(cos(pi/2 + i sin(pi/2)

=e^i*ln(e^i*pi/2)

=e^i*i*pi/2

=e^-pi/2

There may be a mistake in the above derivation, which seems to say that ln(e^i*pi/2)

is the same as e^ln(i*pi/2). Is it?

No, it’s right.

Hi Tim! Yep, that derivation is right — it’s a bit tricky without the parens. The line should be

=e^(i * ln(e^(i*pi/2))

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

In any derivation it’s important to have an intuitive feel for what’s happening. The essence of the derivation is to say “Let’s rewrite this value, i^i, as some type of growth rate for e”. Any number a can be rewritten a = e^ln(a). Thanks for the comment!

Tim sent an email I thought would be helpful (publishing with his permission):

=======

Big K–

Just from curiosity, I set out some time ago to learn about Euler’s Identity. Random searches led me to your site. But before I could assimilate your main article, I had to do some remedial learning, which your site abetted as well. And then, after I had noshed on e and pi and i and sine, I was able to able to sit down for the main course. It proved to be an imperishable feast. What made it all go down so smoothly were two key ideas: 1) any number can be converted into the e format and 2) when e is raised to a power, such as e^i, the number 1 is implicit in both the exponent and the base, yielding 1*e^1*i. This was less an insight to me than a revelation.

Many thanks.

–Tim

=======

I completely agree — the notion of seeing “e” as (1 * e, that is, we’re starting with 1 and growing continously) and i as (1 * i, that is, we’re starting at 1 and rotating) really helped Euler’s theorem click for me too.

Hi Kalid,

This is Rajesh here,

Thank you so much for such an information stuffed website and sharing with everybody.

The article about Euler’s theorem is very helpful. I need few clarifications regarding this. Can I say SINE theta as component which defines my position and COS Theta as the one which gives the rate of change of the position component, because if I differentiate SINE I’ll get COS theta.

-Rajesh

Hi Rajesh, sine & cosine are very related so there’s several ways you can look at it. On a circle, however, you have two dimensions, and sine only wiggles in one dimension (between 1 and -1), so you need both components if you want to describe a circular path. But, because circles are so symmetrical, it’s indeed the case that cosine is a “leading indicator” of what sine will do :).

mra:

I have the same issue you have. This article is wonderful, but it lacks one key insight: how do you know the raising numbers to an imaginary power will necessarily push you sideways?

I think our issue is that we try to draw parallels between imaginary and real numbers: if real growth is horizontal in the complex plane, shouldn’t imaginary growth in the CP be vertical? Well, no, because the two aren’t equatable. 1 is a powerful number; 1 to any power is 1. This is not true with i, and just this simple fact means that the two families of numbers are not as identical as we would like them to be.

But the question still remains: how do you know that i rotates numbers?

wererouge wrote “This gets easier if you’ve already got the hang of the physics concept that to move in a circle you must keep accelerating. If you accelerate in one direction, you will get faster and faster, but if you keep accelerating in a new relative direction, your speed will be the same, but you’ll move in a circle (your velocity changes.)” Then theoritically we can have perpetual motion. I dont know much about math and stumbled here trying to figure out euler’s magical formula perhaps its just what we need for a free energy. Hope you guys can decode what those crop circles mean for the good of humanity.

BTW here’s an article I think it’s partially decoded and I believe they are trying to tell us some kind of free energy in relation to euler’s. http://the2012scenario.com/2010/06/elegant-crop-circle-decoded/

Khalid, thank you so much for the info!

Can I ask you why exactly is imaginary exponential growth at 90 degree?

I mean seen one way it seems very intuitive: since there is no real growth, the length (i.e. the radius of the circle) cannot change. Therefore the e frustratingly goes in circles, unable to really grow!The more it is told that its growth is imaginary, the faster it spins around in circles.

But if I look at what’s happening to it in the Re-Im plane, it seems to be oscillating between being completely real and being completely imaginary! Even though growth is imaginary, it is possible for it to become real again, only with a minus sign! Why this oscillation, given that it is only imaginary growth?

So to satisfy my curiosity, I request you to forward me any resource you know regarding the reason imaginary growth happens at 90 degrees

Swayam—I was doing some reading, and here’s what I found.

TL;DR: The reason e^z (z is a complex number) results in curvy, circular growth is that e^z can be defined as foiling out polynomials with real and complex parts, and since i follows a nice pattern when you raise it to consecutive powers (try it yourself!), so do the results of the FOILing.

e is often defined as {lim(x->infinity) of [(1+1/x)^x]}. Similarly e^A is often defined as {lim(x->inf) of [(1+A/x)^x]}. If A is a complex number, this formula still holds.

Now, imagine letting A=i*pi (i is of course the imaginary number sqrt(-1)). Let’s evaluate a few of the iterations as x–>infinity.

(1+(i*pi)/1)^1= 1 + i*pi

(1+(i*pi)/2)^2= 1 + 2*(i*pi/2) – (pi^2)/4 = -1.46 + i*pi (rounded)

(1+(i*pi)/3)^3=-2.3 + 2i (rounded)

…

(1+(i*pi)/10)^10=-1.6 + .15i

…

Now look at the picture/animation/caption at this link:

http://en.wikipedia.org/wiki/Exponentiation#Complex_exponents_with_positive_real_bases

The final point of each blue “arm” is the solution to one the polynomials we calculated above (or would have calculated if we had done all the numbers).

Though it’s hard to fully explain, the arithmetic and FOIL-ing doesn’t lie; it definitely approaches -1 + 0i.

As for any number, A=i*k where k is a real number, the same principles apply. Somehow, the FOILing just works out.

@swayam: Great question — you’ll want to check out the article on imaginary numbers in this series. If a negative multiplication is a 180-degree flip, then imaginary numbers (i * i = -1) must be a 90-degree flip each (so two flips gets us to 180). The reason for the “oscillation” is the imaginary growth is constantly turning us perpendicular to our current position. Eventually, we circle around to pure imaginary (north), pure negative real (west), pure negative imaginary (south), pure positive real (east). Imagine putting a rocket sideways on a wooden board, and nailing the center down in the middle. What will happen? (The imaginary growth is the rocket ship, pushing us sideways to our current position).

I suspect that Euler’s eq and complex analysis in general are related to flux compactification in string theory. (The flux is considered to be a higher-order kind of EM flux). This flux is seemingly able to bend branes (or dimensions) into tiny particles like the Calabi-yau compact manifolds using a e^z algorithm where z is complex such that a inward spiral is obtained. Likewise, a value of z to result in an outward spiral may account for inflation although that is more difficult for me to see. By analogy I suspect that the unfolding and folding of biological molecules like DNA may operate by means of the same math principles. However the physics principles of how the algorithm is manipulated seems to be unknown, e.g., string theory postulates that flux compactification must hide the extra dimensions beyond 4D spacetime, but the Calabi-Yau compact manifold is the endpoint with no suggestion as to how it got there.

Great article – thanks for taking the time to explain so much. But I still don’t see intuitively why i as an exponent causes rotation? It is intuitive to me why multiplying by i gives rotation (because multiplying by -1 gives rotation of 180 degrees). In your article you say “Imaginary growth is different — the “interest” we earn is in a different direction! It’s like a jet engine that was strapped on sideways — instead of going forward, we start pushing at 90 degrees…” I just don’t see how we can jump to saying that imaginary growth pushes at 90 degrees? Can you help me out?

Thanks!

Something else. I think the terminology of imaginary numbers and real numbers really hinders students. For a long time (20 years!) i thought imaginary numbers actually were ‘unreal’ in some sense and out there in some twilight half-world of semi-reality. But of course they’re not. They shouldn’t be seen as mysterious at all. I think if mathematicians had stipulated from the outset that the ‘imaginary numbers’ just had a different kind of polarity it would still be easier for students today e.g. when negative numbers were first stipulated we gave them the ‘negative’ polarity and said they run backwards at 180 degrees from the positive numbers. We didn’t use some new symbol, say ‘n’ to represent this new mysterious ‘unreal’ number ‘-1′… instead we just kept ’1′ and we stipulated a new polarity of ‘minus’ that we appended to 1. Similarly if we had just said that the negative roots had yet another kind of polarity (say a superscripted leading circle-plus and circle-minus… picture a leading superscripted plus and minus in a circle… instead of +i and -i) then we could have avoided inventing a new mysterious number ‘i’ and avoided the term ‘imaginary’. Just like we avoided inventing a new number ‘n’ to mean -1. We could have said that this new polarity runs perpendicular from the ‘real’ line. The circle (in this alternate circle-plus and circle-minus scheme) could have represented the ‘o’ in ‘orthogonal’ indicating that these are orthogonal numbers. No new number ‘i’ and no talk of imaginary and real numbers… just new interesting and REAL orthogonal numbers!

I’m interested to hear comments from your readers on how this alternate imaginary number scheme may have worked out down the centuries – if we had never had ‘i’. Where ever see +i and -i today we would have had instead circle-plus and circle-minus … obeying all the same rules as +i and -i. So 7i would have been circle-plus 7 and -3i would have been circle-minus 3…. and the complex number 4+7i would have been 4 + circle-plus 7.

Hi Steve, great question. Let me see if I can clarify.

One interpretation of exponential growth is “earning interest”. What does 2^x mean? It’s (1 + 100%)^x

That is, you start with a number, earn 100% interest, and repeat that process x times. So 1 earns 100% interest (becoming 2), 2 earns 100% interest (becoming 4), 4 earns 100% interest (becoming 8), and so on.

What happens if our interest is imaginary? Well, 1 earns 100% interest and becomes… 1 + i. This isn’t in the same direction, it’s growing perpendicular! (Think about the endpoint… you went from (1, 0) to (1, 1), i.e., you grew “due north”).

However, getting 100% “all at once” is a chunky type of growth, and not continuous. e^x (and therefore e^ix) is about taking tiny slivers of interest and applying those in sequence. You take the smallest sliver of interest you can, and apply that (so you grow due North, or perpendicular). Then you take the next sliver, and grow perpendicular to your current location (originally, you started East and grew North. Now you’re headed East and very slightly North, and head East and a little bit more North). This process constantly repeats, leading to circular motion (there’s a diagram in the post showing the slivers of interest constantly rotating you). Hope this helps!

Hi Steve, totally agree about the language. In the article on imaginary numbers, I noted the name “imaginary number” was meant to be an insult! It’s crazy that we blithely introduce the “unfathomable” numbers without mentioning this historical footnote (at the time, imaginary numbers were considered “impossible”, just like zero, and irrationals, and negatives were before them).

As you say, a better name would be “2d-numbers” or “orthogonal numbers” similar (although the 2d interpretation didn’t arrive until decades after imaginaries were discovered). Or, perhaps a better notation with a comma: 1,1 [written with a comma to separate the real & imaginary part, vs. an explicit 1 + i... we don't write 3 + .4, do we? :)]

I had forgotten about all this subject. You brought it back.

thanks very much

Hi ominac, glad it helped.

I was hoping you’d share your opinion of what ‘i’ means to you.

Mathematicians say that imaginary numbers system is orthogonal to real numbers systems but i see no suggestions about an inherent meaning of arriving at a square root -1 when deriving an equation related to a ‘real world’ phenomenon (e.g., electricity).

Further, cos(x) + sin(x) is all that one requires to ‘rotate’ around a circle, point by point.

The inclusion of ‘i’ as a coefficient of sin(x) does not induce a rotation.

however, multiplying 2 COMPLEX numbers does result in rotation: “multiplication by a complex number of modulus 1 acts as a rotation.” – wikipedia

So, I ask, are you convoluting the vector form of the equation of a circle [cos(x) + sin(x), where x stands in for an angle of rotation] which gives the path of rotation around a point (a circle) versus the general form of the Euler equation which includes i as a coefficient…which isn’t the same as ’1′ as a coefficient….i.e., it changes the meaning of the equation.

I ask, not accuse, because it has been many years since i finished my engineering degree, and so haven’t played with these in a while.

Hi Mike, I see i as a rotation.

For something like cos(x) + sin(x), note that you need a 2nd dimension otherwise the numbers run together. Try plugging in x=45 degrees: you’ll get cos(45) + sin(45) = .7 + .7 = 1.4. Which isn’t “45 degrees around the circle” like we’d expect. We need a way to track the vertical component separately from the horizontal one (i gives us an option here).

Kalid, your work is really a masterpiece. Finally I found a site (better late than never) that’s really helping me a lot. I couldn’t get this kind insights when I used to study.

Thanks a lot.

I will recommend your work to all my friends.

Hi Charles, thanks so much for the comment and support. Once I first experienced what a math insight could feel like, I had to keep looking for them to share!

Hi Kalid, thanks again for your great articles. I have one question on this topic.

I understand that imaginary growth rotates around the unit circle, and I understand that using sin/cos we can do the same. What I don’t understand (maybe I’m missing a point here) is that the rotation we get by plugging a number into e^i.x gives us the exact same rotation as plugging in the number into cos(x)+i.sin(x). Why do these two ways of rotating are exactly ‘in sync’, rotate at the same ‘speed’?

I hope you understand my question, basically: why are we sure that for example e^i.2 = cos(2) + i.sin(2) ? I see this must be true for the special case of euler’s jewel, where we plug in pi, but that does not guarantee that this is the case for the general formula (‘the intermediate steps’ so to speak). Might be a dumb question but the reason why this is true doesn’t click for me currently. Thanks!

Hi Jeroen, great question. Intuitively, e^ix and cos(x) + i*sin(x) are two ways to describe the same act of “start at 1.0 and rotate by x radians”, just like 2^3 and 2 * 2 * 2 are both ways of describing the same act of “multiply by 2, three times”.

Why does this work? We saw that e^ix represents rotation, and cosine/sine are defined to be the horizontal and vertical coordinates as we rotate on the unit circle.

But that may not be satisfying enough :). Analytically, cosine and sine can be defined with an infinite series. For example, sin(x) = 0 + x – x^3/3! + x^5/5! + … [more here: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/. Cosine can be defined a similar way [cos(x) = 1 - x^2/2! + x^4/4! - ...].

And, what do you know, e^x has a series definition too [e^x = 1 + x + x^2/2! + x^3/3! + ...].

If you plug in “ix” for x in e^x, you’ll see the series match up: series for e^ix = series for cos(ix) + series for sin(ix).

This is a very rigorous mathematical justification; I prefer to focus on the insight that e^ix and cos(x) + i*sin(x) are referencing the same point on a circle.

“3 is the end result of growing instantly (using e) at a rate of ln(3). 3 = e^ln(3)”

This line reads confusingly. It’s two sentences but could be read as (ln(3).3 – e^ln(3)

I’m sre some people could interpret the dot as an operator such as multiplication.

“3 is the end result of growing instantly (using e) at a rate of ln(3):-

3 = e^ln(3) “

is a lot clearer.

Sorry- there’s no way to edit these comments – that should read as follows:-

“3 is the end result of growing instantly (using e) at a rate of ln(3). 3 = e^ln(3)”

This line reads confusingly. It’s two sentences but could be read as (ln(3).3 = e^ln(3)

I’m sure some people could interpret the dot as an operator such as multiplication.

“3 is the end result of growing instantly (using e) at a rate of ln(3):-

3 = e^ln(3) “

is a lot clearer.

e represents the process of starting at 1 and growing continuously at 100% interest for 1 unit of time.

Shouldn’t it be – over an infinite amount of time. e raised to anything approximates to the expected result only when the time approaches infinity. Not nitpicking – I am just trying to reconcile this with your other article on e. Great stuff, keep posting – now I can explain these things to my daughter without having to hide behind mysticism.

@Pete: Thanks for the feedback, I’ll clarify that part!

@George: No worries, great question. e is just a number, about 2.7. So we can rephrase it like this:

If you have 100% yearly interest, but compound as *fast* as you can (i.e., get your interest every microsecond and add it to the principal), you’ll turn 1.0 into 2.7 at the end of the year.

Now, we can repeat this process for any number of years (not just 1), and that becomes 2.7^x (where x is the number of years we compound). So if we let this system run for 2 years we’d get 2.7 * 2.7 = 7.29, and so.

e is the exact amount of interest we get (a tad bit more than 2.7), and we wrap it up as a letter (just like pi, so we don’t have to worry about our decimal accuracy, we can say “Ok, that exact amount of interest you get, let’s just call it e and work from there”).

Hope that helps clarify! We can of course wait any number of years, but e (2.7-ish) is what we have after waiting a single year. (I’ve said year here, but it’s really whatever time period we’re looking at. If you get 100% continuous interest per month, then e is your growth after 1 month).

pardon me, that’s horrendous. why not start by reading what euler wrote about the topic?

euler’s identity has a much more fundamental origin than the trig function formula which generalizes it. one could, for instance, demonstrate that the derivative of [exp(ix)/{cos(x) + i sin(x)}] is zero, with a pretty minimal set of assumptions and zero hand waving.

i^i has an _infinite_ number of distinct values. that fact (and it is a fact) ought to make you rethink some things. i strongly recommend beginning with powers (and especially fractional powers) in the complex plane. it’s a precalculus topic and your misapprehensions there are, for want of a better phrase, compounding.

misanthropope, I appreciate the content, though not the tone, of your suggestions (I hope you come to realize that snarkiness does not lend wisdom or gravitas to a message). If the title didn’t make it clear, the post isn’t for those exploring the deepest nuances of the equation (see: Wikipedia, MathWorld, your typical professor, etc.). Sure, we can substitute i = i^5 = i^9… or explore the equation with calculus, baffle newcomers, and feel macho and clever. My goal was to be helpful.

To the author:

You have a great way of presenting information… You clearly have a deep understanding of the subject, so much so that you can explain it in such ‘simplified’ and physical forms. You would make a great teacher (if not already).

@Appreciative: Thank you! Very glad the material helped :).

that’s great

Hey Kalid, Thanks for the pointing to this article. I’m going to need to go back a couple of times! But this was exactly what I was looking for. Thank you.

I’m currently working for a Grant tasked with enhancing calculus education for engineers (by making things less abstract, something I know you care about…). If you are interested, I’d love to hear your thoughts. (You should have my email from this comment). But either way, you’re very good at what you do, keep it up!

Thanks JD, very glad it helped! I just reached out to your email :).

Very nice job, helped me quite a bit to understand better what is going on!

Hello Kalid, Happy 2014 I have often thought that many things can be considered as a metaphor for human existence, or at least an aspect of human existence. So as I was reading your post, I “imagined ” the inside of my skull to be a “3D” complex plane if you will ( a spherical 4 pi r^2 version ) where the Complex plane has a Real axis , and an Imaginary axis, but also possesses a third axis ( similar to X Y Z ) , and along this third axis can be plotted a point that is part real, part imaginary and part part awareness. As a metaphor………many times

Sorry Kalid, I accidentally hit the “post comment’ button, so I will start again

Hello Kalid, Happy 2014 I have often thought that many things can be considered as a metaphor for human existence, or at least an aspect of human existence. So as I was reading your post, I “imagined ” the inside of my skull to be a “3D” complex plane if you will ( a spherical 4 pi r^2 version ) where the Complex plane has a Real axis , and an Imaginary axis, but also possesses a third axis ( similar to X Y Z ) , and employing this third axis can be plotted a point that is part real, part imaginary and part awareness. As a metaphor………many times I find myself taking the real and mixing it together with the imagined , but when I apply real concentration / awareness to what is going on, rather than focusing on those two only , I find myself focusing on the “actual” within me, and this “rotates” me within even further, and I experience “growth” that is neither linear nor circular. Neither sensation nor thought. I once read that true learning has occurred, if the student comes away feeling lighter than before he/she felt before the lesson. I know I have extrapolated “away” from the mathematical topic being discussed, but all learning should hopefully bring us closer to our heart”s desire. May I just say a big thankyou To you Kalid, for taking the time to share your mathematical insights in not only a clear and good humored fashion, but also the quality of your presentations are first class. I will leave you with a quote….” The essence of mathematics lies in its freedom.”

Sincere Regards Bluetone

G’day Kalid,

I like your explanations – very much actually, and I enjoyed the read – thanks for that!

There is just one thing that I do not get, and it seems to me to be central to the whole issue.

Your explanation flubbed one very important point – Why is the complex magnitude fixed when raising real numbers to purely imaginary powers?

Your explanation of purely imaginary powers reads, in part: “Surprisingly, this does not change our length …This is something I want to tackle another day”.

It is also very surprising to me, and it seems to me to be the key to understanding complex numbers intuitively.

Let me restate what I am after in my own words:

“What is an intuitive way of saying that ANY real number (say 2) raised to ANY purely imaginary power, say 2i, MUST lie on a perfect circle, centred at the origin, on the complex plane?”

Your article has focused on the real number, e, raised to different, purely imaginary exponents, like (i x pi/2), and (i), and (i x 2 x pi) etc. Fair enough. But if I use ANY other real number instead of e, like 2, or 2 million, and raise them to the same imaginary exponents, then they ALSO lie (at different places) on the SAME UNIT CIRCLE on the complex plane! Wow! Intuitively, why must this be the case?

Of course our starting point must be that any number raised to the zero exponent, including (i x zero), must be the real number 1, plus I also accept, intuitively that a purely imaginary exponent will rotate just like any complex multiplication.

But here is the crux: what stops the resulting magnitude from increasing from away from 1, (or decreasing below 1) to create a spiral on the complex plane?

It ‘feels’ like the action of purely imaginary exponents is on a fixed-length ‘rod’ of length 1 tied to the origin, but I lack any intuition for why the rod’s length cannot vary as it rotates. I understand intuitively why it starts at 1, and why it rotates around the origin, but the surprising part is that it does not ever move away from or towards the origin in a kind of spiral, instead of the perfect circle.

The standard explanations I have found seem to prefer Euler’s proof using the power series expansion for e: relating it to the power series expansions for sine and cosine. For my taste, a power series is not ‘intuitive’ – it’s hard for me to get a good ‘intuitive meaning’ derived from a sum of the infinitely differentiated functions. Possibly you agree?

Can you please help me here? Why is the action of raising to purely imaginary powers a ‘rod-like’ rotation around the origin? Why is it, intuitively, that the magnitude cannot be anything but 1, regardless of the real number we choose, or the real number we multiply by i to raise the exponent?

My current thinking is that it must relate to the subtle difference between multiplication and exponentiation. These closely related concepts are not the same, and the ‘special nature’ of exponentiation somehow forces a fixed magnitude when using purely imaginary powers. Not much to go on so far!

I sincerely appreciate your input on this problem.

For complete disclosure – I am an old Electrical Engineer, and all this maths stuff I have been doing for years. I have been trained to apply the rules, but in some cases I just behave like a robot instead of using my intuition, and your article has inspired me to fill in the gaps in my intuition.

Many thanks in anticipation.

G’day again Kalid,

Further to my previous question – perhaps the answer to my own question lies in the meaning of any complex multiplication by purely imaginary units (i times something).

It seems intuitive to me that just multiplying by i causes a ‘pure’ rotation around the origin, and continuing this pure rotation defines a perfect circle.

Further, exponentiation (raising to a power) is a ‘kind-of’ multiplication, so it seems reasonable to conclude that exponentiation by purely imaginary numbers will retain the purely circular characteristic – purely circular rotation around the origin must be what happens when we multiply or raise to the power of a purely imaginary number.

The subtle difference between multiplying by i and raising to the power of i is simply this: Multiplying rotates in a perfect circle, retaining the original magnitude, but while exponentiation retains the purely circular characteristic, it also forces the magnitude (radius) of the circle to be exactly 1, because anything raised to the zero power is 1, and this must always be one point on the resulting perfect circle.

Euler’s equation simply means that the ‘special’ real number ‘e’ is the rotation that repeats every 2 times pi times i. Other real number bases higher that e rotate ‘faster’, and numbers lower than e rotate ‘slower’ than 2 times pi times i, but still on the same circle. The real base determines the rate of rotation.

Euler’s identity is simply the special case when we use the base e, raised to purely imaginary numbers, it rotates through real the axis at -1 (going half-way around the circle), at exactly [pi times i]: e^i.pi = -1.

Or, said another way: all real numbers raised to purely imaginary powers rotate on a circle that repeatedly cuts through 1 and -1 on the real axis – it’s just that the base e first returns to the real axis when the exponent is exactly i times pi, and begins to repeat then circle when the exponent is exactly i times 2 times pi.

This ‘intuition’ is a sort-of cheat – I have defined my way out of this problem by saying that purely imaginary multiplication means rotation in a pure circle. Complex multiplication in general is the combination of this pure rotation for the complex angle, plus a translation for the magnitude.

So, do you feel that I have answered my own question? Does my explanation count as ‘intuitive’ in your view?

At least I have avoided referring to a power-series definition of e (not very intuitive), plus I have been more general, in that I am not restricting myself to imaginary powers of e, but discussing the ‘shape’ of all imaginary powers of any real number.

Hi Paul! Thanks for the comment, great question.

That’s something that bothered me as well, give the section “Example: 3^i” one more gander. I wanted to work through why a real number to an imaginary exponent always stayed on the unit circle.

Here’s my intuition. It might sound mystical, but there’s one circle, the unit circle. Every other circle is just a scaled variation of it (larger or smaller). Fine. Similarly, there’s only one number, 1.0, and every other number is a scaled / rotated version of it. Also fine.

How about growth? Well… there’s only one type of growth: e^x. Every other type of growth is just a slowed-down or sped-up version of this “unit” growth (e^x represents 100% continuous growth).

When we have a setup like 2^x, we’re saying “Ok, let’s grow a little slower than e”. We can substitute 2 = e^ln(2), and realize 2^x is the same as e^x, but it grows more slowly (at ln(2) = 69.3% interest, instead of e’s 100% interest).

An imaginary exponent modifies your growth rate, turning it perpendicular (which forces you to spin in a circle — you’ve strapped the rocket on sideways). Making 2^x grow perpendicular just means “Well, now you’re growing in a circle, but at a rate of 69.3%, compared to e^x’s 100%”. And as you might guess, 2^i will go around the circle, but a shorter distance than e^i would have. In fact, e^i will go 1 unit around the circle, while 2^i will go .693 units around the circle.

Hope this helps!

Thanks heaps Kalid, for taking the time to discuss this issue.

After posting my second comment, my feeling is that my intuitive understanding leads me to look for something unitary, simple, or ‘pure’, to define complex multiplication and exponentiation. That seems intuitive.

So, how about this:

We could think of multiplication as being an operator with two ‘sub-operations’ – the ‘Real’ ‘operation’ that scales the magnitude, PLUS the ‘imaginary’ operation that adds to the angle – or rotates.

To apply this multiplication in two parts it is convenient if the two ‘operations’ that need to be performed are ‘orthogonal’ operations – where orthogonal simply means that they do not affect each other, and thus can be computed independently and the two results combined to get the final answer.

In this view of multiplication, the imaginary rotation operation is orthogonal to the familiar scaling of real multiplication – by DEFINITION – it’s the way we must change the angle while leaving the magnitude unchanged.

Similarly, the scaling of the magnitude is DEFINED as the operation that does not change the angle, but only scales the magnitude.

Together the two are combined to perform the familiar operation of complex multiplication.

Therefore, our ‘pure’ imaginary ‘operation’ must trace a perfect circle – only because we have defined the operator as only capable of adding to the angle, but never changing the magnitude – a fixed radius is a perfect circle by definition!

But, I do not feel that the ‘unit circle’ is the one ‘mystical’ circle (in your words), rather it is just the consequence of repeatedly using the purely imaginary operation – a perfect circle with fixed radius that goes around for infinity with increasing angle, analogous to the way that the real axis goes on to infinity with increasing scale.

However, I found that thinking about ‘growth’ when looking at e^i caused me to go off-track – and that word ‘growth’ caused me to incorrectly intuit that the graph of increasing powers of i would be a spiral, not the perfect circle.

Now, the way I am thinking is this: the ‘circle of any radius’ is intrinsic to the multiplication by the ‘pure’ i, because we define that to be ‘adding the angle but leaving the magnitude alone’.

Further, the very special ‘unit circle’ actually has nothing to do with the special number ‘e’, rather it is derived independently by the property that anything to the power of zero must be 1. So the circle is locked to a length of 1 unit as a natural property of exponentiation. This works regardless of the real number we choose as the base, not just by choosing e.

In fact, thinking about e as the base wrecked my intuition of what was happening when we raised real numbers to multiples of i. Perhaps the unit circle is ‘deeper’ concept than ‘e’. However the base ‘e’ is still special because it relates the other special number ‘pi’ by way of rotation FREQUENCY: [i times 2 times pi] for each full rotation.

I hope that makes sense.

Please do not take offense if I appear to contradict you. All I am really discussing is my own evolving intuition of complex powers, and ‘pi’, and ‘e’. Of course this may be different to your, or anyone else’s intuition.

Hi Kalid,

… and another thing …

I’ll admit that it seems like a cheat to to use a rotation definition to avoid explaining why circles and especially unit circles appear on Argand diagrams for purely complex exponents, but I have a background thought process that led me this way.

Allow me to explain:

We are accustomed to thinking of the multiplication of real numbers as a pure scaling operation, and it makes perfect sense to do so, BUT, if we think of real multiplication in a slightly different way, then maybe we can intuitively understand complex multiplication without fearing the apparently enormous gulf between real and complex numbers.

Specifically, look at multiplying by a negative number, say -2. Our old-fashioned brains think of this is just ‘going the other way’ on the real number line. Fair enough.

BUT, if we think of a real number as a vector, or an arrow from the origin to a point on the real number line, it is intuitive that the arrow of the negative number is made to ‘point the other way’ using the already familiar concept of rotation by 180 degrees (or pi radians).

So if we think of multiplying by 2 by -2, it could be broken into two distinct (orthogonal) steps. Step 1 – changing the magnitude: scale 2 by 2 to get an arrow pointing from the origin to 4, then Step 2 – rotation around the origin: rotating the new arrow by 180 degrees, leaving the arrow pointing to -4.

The new way is still the correct answer, and it seems a bit weird, but it subtly introduces the concept of complex number multiplication. The only remaining step is to consider rotations that leave the vector pointing to ‘a position not on the real number line’, and draw that position on the diagram I know as the Argand diagram.

Clearly, any rotation that isn’t 180 degrees will end up in this new ‘space’, but now it becomes a simple exercise to demonstrate that squaring a new ‘special point’ we can now locate at a magnitude of 1 and an angle 90 degrees off the real axis, results in the real number -1. Logically this ‘special point’ must be the square root of -1, or the famous ‘i’!!!

Using this thinking maybe ‘i’ makes a sort of intuitive sense, that is hinted at by thinking of multiplying by our usual negative numbers as being done by a special ’180 degree rotation’ operation on vectors along the real axis.

I will admit that since I am an Engineer, it is very easy for me to introduce things that others might find incomprehensible, such as vectors, so I was wondering if you felt that this explanation meets your own standard of ‘intuitive’ for explaining the existence of i, the famous square-root of -1, and the multiplication of complex numbers?

Thanks again for paying attention to my ramblings, I look forward to any comments you may have.

Kind regards.

After doing a lot of thinking and reading and playing with my (complex number) calculator, it seems that the ‘magic’ of Euler’s formula does not lie in its circular shape, but in it’s rotational frequency – Euler’s formula means that ‘e’ raised to all the powers of the complex unity ‘i’, goes around EXACTLY 2 times pi per revolution. Instead of the growth along the real axis that is characteristic of ‘e’ to real powers, complex powers produce infinite revolutions with a period exactly related to ‘pi’.

You see, any real number raised to the same complex powers of x used in Euler’s formula, also go round-and-round the unit circle. From that perspective, the unit circle is not as special as I though it would be, because it is just in the general nature of complex exponents to go round in a circle, regardless of the chosen base.

Any other real number base still produces sinusoidal real and complex components, its just that by only choosing the base ‘e’ makes those sinusoids have the ‘natural’ period of 2 times pi (in radians).

All the discussions I have just been reading about why the limit definition of e: (1 + z/n)^n converging on a unit circle are actually beside the point. Of course it traces a circle – that is given already – built-in, so to speak, to the way complex numbers are multiplied, not a special property of the special number ‘e’.

It only added to my confusion when I read about those limit definitions of e, instead of being clearer and more intuitive.

It is actually quite interesting to note that bases smaller than ‘e’ rotate slower than 2 times pi per revolution, and larger bases rotate faster.

The base ‘e’ gives us the natural sweet-spot frequency – the simplest mathematical expression for revolution in terms of the familiar sine and cosine waves – so maybe this could be intuited as the ‘unit of frequency’, constructed from the real unit 1, the complex unit ‘i’, and the two special numbers ‘e’ and ‘pi’? Very special indeed!

Perhaps the expression e^ix, where we make x the time variable: ‘t’, can be thought of intuitively as the ‘unit of all oscillating things’ with every other oscillation just being scaled to different frequencies of that unity?

Aha, now I get it…!

I just revisited the famous Fourier transform using my new intuition about e^ix being the ‘unit of all oscillating things’.

The Fourier transform makes intuitive sense now because I can see the oscillator, e^ix, in the definition.

What the Fourier transform achieves is the decomposition of a waveform, any waveform, maybe piece of a Mozart symphony or a Lady Gaga song – it doesn’t matter, into it’s constituent ‘pieces’, where each ‘piece’ is a pure tone.

The pure tone at middle-C is said to be about 256 Hertz (cycles per second), and this is expressed by the formula e^ix, by plugging 256 times pi times 2 into x;

Pure middle-C (256 Hertz) = e^i.2.pi.256

Hitting middle-C on a piano keyboard produces a tone (partly) expressed by the complex exponential formula with 256 plugged in. But it also produces a bunch of softer tones nearby, certainly at the harmonic tones 512 hertz, 1024 hertz and other frequencies too.

If we recorded the striking of the middle-C key on a piano, and express that recorded waveform using the Fourier transform, we would get the breakdown of the piano’s middle-C sound in terms of all its characteristic tones. Of course this would include the loudest tone located at 256 hertz, but also lots and lots of softer tones at other frequencies.

The waveform of a harp playing middle-C, or even a human voice singing, can be broken down into a similar set of tones that are characteristic of that harp or that singer.

Relating this back to our complex exponential function, the Fourier transform expresses the tones that make up any waveform, and their relative ‘loudness’, as a sum of all of those ‘units of oscillation’, or A.e^ix, scaled with different frequencies (x = 2.pi.f in Hertz) and amplitudes (A).

Hi Paul,

Awesome, glad things are clicking! Yes, that’s it: the Fourier Transform gives the amplitude at each possible frequency. You might like this video:

http://www.youtube.com/watch?v=121DoSs62eY

(He plays a middle C on the piano, you can see the spike in the amplitude at the desired frequency. As you move up an octave [double the frequency] the spike moves. That piano isn’t particularly tuned, so the spike is noisy, but the idea is there)

As you mention, a harp vs. human voice has different characteristic subtones — a pure 256Hz sine wave sounds very bland, like an emergency broadcast tune (http://www.youtube.com/watch?v=21ZELdFob38).

Hi Kalid,

Until now, the Fourier transform definition just looked like a butt-ugly integration formula. I could ‘do the math’ like a robot, in your words, but before now I didn’t understand why it worked.

Long, long ago my university professors told me that the Fourier transform would break a waveform down into it’s constituent frequencies, and I just took their word for it without really understanding why it worked.

Now that very same integration formula actually looks like something straightforward and intuitive – a sum of oscillators – because I see e^ix for what it actually is – the unit oscillator!

Exactly! I had the same struggles when first learning. Another way to put it: The Fourier Transform projects a signal onto the unit oscillator (e^ix) and runs through every possible frequency (coefficient on x), and seeing what the overlap is there. This is similar to taking a dot product of a vector, projecting one vector onto another, and seeing how they overlap. If the projection is 0, there is no overlap, i.e. no component at that frequency.

That’s exactly what I see the Fourier transform as now – a kind of scanner that tells us how much of a given frequency is found in a sample – the total of what we find in scan of at a specific frequency tells is the ‘amount’ of that frequency in that sample.

Repeating for different frequencies gives us the full picture – the relative amount of each frequency in the sample waveform we are analysing.

Naturally, here in the real world, we use a software library or a semiconductor chip to actually do the scanning for us – FFT libraries etc. I am not so crazy as to actually do those integrations by hand!

Now… onto my other old friend – the Laplace transform…

I like the “scanner” analogy. Ah, I’m actually looking at the LaPlace transform now too. My key intuition is that it’s just a generalization of the Fourier Transform to spirals, not circles.

A spiral has both a rotational frequency and a decay (or growth) factor. This can be represented by a complex number (a + bi, which means growth factor ‘a’, rotational speed ‘b’) compared to the Fourier Transform, which assumes a growth factor of 0 (i.e., the circle components never change size). One advantage of this general approach is you can model signals which decay (or grow to infinity), especially physical systems (like a spring which oscillates while losing energy to friction).

Hello Kalid,

I am trying to be intuitive ….so my question is this

Is it mathematically valid to distinguish between the ” imaginary zero ” that is the origin of the imaginary axis as opposed to the “real zero” which is the origin of the real axis ? They magically occupy the same place at the same time on the complex plane. As it stands , Euler’s identity is written in such a way that it regards the origins of the Imaginary axis and real axis to be identical. But how can they, when one origin is real and the other is imaginary ? If one was to write Euler’s identity with respect to ” both” origins ….We would have a “shadow ” of the identity ………….

(е^πі/ 0!)^2 ⁼ 1 ( which reads( Eulers identity over Zero factorial Squared is equal to 1. As you know 0! is 1. and usually denotes 1 x ” the potential” of something to occupy a particular space, however in this case, we are actually factoring “two” zeros, the real zero and the imaginary zero, and taking advantage of the fact that 0! equals 1. As a result, this “reflective shadow ” of the identity, it is equal to 1 not -1. This is perhaps an unforeseen consequence of mixing ” dimensions”. The imaginary is a different dimension to the real dimension is it not ? And even if this “equation” is perhaps “token”, and somewhat redundant, it is at least acknowledging that we now have an exception to the rule that says ” two things cannot occupy the same place at the same time. ” In the complex plane, we can cleverly combine real with imaginary (a + bi) , however if the imaginary number in a complex number is zero and the real number in that complex number is also zero, then the complex number should be equal to 0! not just plain old zero, because when both are zero, there is no longer a “combination” of real and imaginary parts, and for a point on the complex plane to “exist”, it must be a combination of both real + imaginary numbers. Kalid, in your view of things, does this line of reasoning have any mathematical merit and is my equation mathematically sound ?

Regards Bluetone

Regards Bluetone

@Bluetone: Great question! I think it comes down to a matter of definition; I found a discussion here: http://www.physicsforums.com/showthread.php?t=206108.

My theoretical math knowledge isn’t very well refined, so I’m not sure of the formal / analytic definition of the reals, imaginaries, etc. You may be aware, in some systems there is the concept of a “signed” zero (is zero positive or negative?), which has some of the similar issues: http://en.wikipedia.org/wiki/Signed_zero. I also found the idea of http://en.wikipedia.org/wiki/Semi-continuity which might be related (if you consider the real and imaginary axes overlapping, which one gets the filled-in dot? :))

In general, the definition that leads to the most practical conclusions is the one we usually go with (such as defining 0! = 1).

Thank you for your reply Kalid.

How do we teach our kids to be “intuitive” I wonder ?

Kind Regards

Bluetone

Hello again Kalid,

I believe that the key to understanding the transform methods (Fourier, Laplace and Z-transforms) lies in their motivation – we need to ask – why were they invented, and what do they do for us?

They are not like the exponential function taken alone, which is something more akin to a discovery, like discovering an natural element, say carbon. The transforms are more like tools or processes – they are there to be used as the need might arise. Fashioning useful things using or containing carbon, in my analogy.

The Fourier transform has probably the simplest motivation to understand: it is used whenever we want to decompose a signal into it’s constituent parts. This could be for many reasons, like we want to design a mobile phone or a GPS satellite system for example.

The Laplace is another great workhorse transform used just a much as the Fourier transform, but for more complex, and maybe less intuitive reasons.

Now that I think about it, it is worthwhile discussing some the reasons I think that the Laplace transform is used:

1. The Laplace transform converts calculus into algebra. Transforming a differentiation becomes a multiplication operation in the Laplace transform, and an integration becomes just a division. So even massive systems of differential equations are reduced to mere algebra with the magic ‘s-variable’ and these can be quickly simplified and factorised.

2. Directly related to this, the Laplace transform gets rid of those ugly, ugly transcendental functions like sines, cosines, hyperbolics, logs etc. Once again we are dealing with mere algebra with the s-variable, instead of apparently intractable differential equations.

3. Probably of most importance to me is related to System Theory: When the fully factored Laplace functions are graphed, we can easily see how close a system is to instability (the vertical axis) – an oscillation that keeps growing out of control. The fairly methodical process of deriving the Laplace transform equations for a System can allow an engineer to see what changes could be made to ‘pull-back’ from the axis: to make a system relatively less prone to instability.

That last one seems like a lot to handle, and it probably counts literally as ‘rocket science’ – because that’s what rocket scientists (and engineers) have to do – keep the rocket (system) under control even though the environment around changes – and sometimes it can change radically.

You see, the things that are used to build springs, walls, rockets etc all change their characteristics as they get hotter or colder for example. Having a manageable set of equations, or a least a way of deriving the equations, is crucial to answering questions like: will something shake apart when the rocket nozzle reaches it’s designed max temperature? What will happen as the metal walls stiffen in the dead cold of space? etc.

So, here goes my first guess for an intuitive understanding of the Laplace transform in terms of its relationship to Euler’s famous equation:

1. Maybe Laplace reduces calculus to algebra because ‘e’ to the power of anything equals itself after differentiation and integration – it ‘survives’ the process. e^x is always there, regardless of how many times calculus operations are performed.

2. Because Euler’s equation lets us write any sine, cosine, hyperbolic function etc in terms of the base e, the ugly transcendentals disappear and are replaced with the more manageable ‘e^x’ – so even calculus with transcendentals can be reduced to mere algebra in the s-variable.

3. Both 1 and 2 allow us to have systems of equations, manageable with algebra and expressed with s-variable, which engineers and scientists are trained to use for analysis and design of whole systems.

Understanding 1. and 2. above require an intuition for how e^x works, and 3. provides us the motivation for even bothering to go through the steps to derive each Laplace transform.

But I think I will need to chew on this problem further for a while…

Hi again Kalid,

There is an excellent video on youtube with clear graphics, showing visually what the the Laplace transform does, how it relates to the Fourier transform and control systems theory.

http://www.youtube.com/watch?v=ZGPtPkTft8g

I am still digesting it’s contents, plus other videos, as I try to get a better intuitive grasp on how the Laplace transform works…

@Bluetone: Excellent question — I think people are naturally intuitive, we just need to encourage it and show that the rigor is a supplemental way to understand something, but not a replacement for really understanding it yourself. (It doesn’t mean “go with your gut instinct”, it means “be sure you truly understand it, and put a concept into your own terms”).

@Paul: Thank you, this is great! I like your description of the breakdown, it’s exactly the high-level insights I need when diving in. Looking forward to that video too!

Is there a practical (by that I mean real life application) use for Euler’s Identity? I’m looking to use it as the basis of my mathematical modeling project but I need something to model and also something to test and gather data. Any ideas would be appreciated!

@Dylaan: Euler’s identity is often used whenever you need to model a circular path or repeating pattern, since it’s a convenient way to build a circle in a single function (instead of having to separate out sine and cosine).

As an example, the Fourier Transform uses Euler’s formula to decompose signals into repeating cycles, which is used to analyze data (http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/).

Hi Dylan, Kalid,

I am hunting for the nearly same thing – the ‘real life application’ that demonstrates the truth of Euler’s equation (not Euler’s identity, which is closely related).

The nearest I have come up with is they physical example of a ruler held to a desk, with about half of the ruler poking over the edge of the desk – when the ruler is ‘struck’ it vibrates for a second and settles back to motionless.

Technically this physical motion of the ruler is called a ‘damped sinusiod’, and this can be mathematically shown as a simple exponential [e^-kt] multiplied by a sinusiod [sin(wt+p)].

The oscillation – the sine part, can be expressed using Euler’s equation, and this type of solution is commonly found in mass-spring-damper systems studied by engineers.

BUT – saying that solutions to the equation of motion for a vibrating ruler is based on Euler’s equation is a long, long, way from being what I would call intuitive. By itself this is not what I wanted to demonstrate a ‘real-life’ example of Euler’s equation.

(Actually I am cheating a bit – the ruler’s motion also contains harmonics of the principle frequency (w) – but that is not really the problem I have with using it as an example.)

I feel that the same is true for the Fourier transform example – it is not intuitive, even though I know it is true. As an Engineer I have lived and worked with the mathematics for a very long time, so what is self-evident to me is definitely not what a layperson would call intuitive.

The real ‘surprise’ of Euler’s equation is that any oscillation we find in nature, which we can express as the familiar sine wave, can also be expressed using the imaginary number ‘i’. Writing the word sine just ‘hides’ the imaginary part, and rewriting the same sine using the complex exponential formula [e^it] reveals the true nature of every sinusoid.

Euler’s equation tells us that all sinusioids are essentially complex exponentials [e to the power of i] but while it is mathematically true, the hard part is to demonstrate this idea intuitively.

Thanks Paul. For me, the truth of Euler’s Equation emerges once I realized that it was just another way to build a circle.

Method 1 is to laboriously compute the grid coordinates of every point on the circle (using sine and cosine). Method 2, Euler’s Formula, is to simply rotate a line around the center.

The tools needed to create this rotation are the number 1 (our starting point), continuous growth (e^x) and rotation (i)… and putting them together in the proper order :). [Remembering that radians measure distance moved, so using them instead of degrees, etc.]

That made things click for me, and I didn’t need a deeper meaning (beyond that it can also make a circle), since circles are applicable to most problems. Many trig facts, such as sin(a+b), become easy to compute with Euler’s Formula vs. grinding through the trig identities. I hope to write on this soon.

You are my hero. i envisioned the universe to be a spiraling spiral, and you have shown me that this is what this equation is. i feel i have found the end of the rainbow. thanks.

Actually a shrinking spiraling spiral. This is how i think of eternity in the universe.

Existence is a byproduct so to speak of the great nothing. If you think about it, the only thing there really is is the great nothing that can never be measured or understood. because it can’t be understood it has to be ever changing randomly. Maybe it could be represented as pi. I see an ever shrinking spiral that keeps spiraling so that it doesn’t touch itself. if it touches itself, then there would be two points the same, which means it could be understood if measured at those two points, but because that’s impossible, it just keeps shrinking.

pi will never go to where it’s been. This is the same as a spiral that never touches itself. I suck at math, i wish somebody could use this equation to solve for pi.

I appreciate the insight you are sharing and it has helped a great deal. One thing i am still in the dark about is what is the point of the e operator in the first place when having an imaginary exponent.

Basically, why doesn’t 1 ^ i rotate me counter clockwise around the unit circile.

Hi Dan, great question. e is a good choice of base because it represents 100% continuous growth: e^i will rotate you 1 radian around the unit circle.

Any other base will also rotate you, but by different amounts. 2^i will rotate you slightly less than 1 radian around (ln(2) = .693, so you’ll move .693 radians around the unit circle).

1 as a base is interesting, because it doesn’t represent any growth (1 to any power = 1). So, 1^i = 1. (Another way to put it: when saying you have 1 as your base, you are saying your growth rate is ln(1) = 0, which means you aren’t changing at all).

I see what you’re saying. Thanks Kalid.

This was a great audio and follow up. I haven’t listened to or read the entire stream yet, but I will. I wanted to contribute this. I teach AP Calc at the HS level here in the US. I have some pretty bright students. While two of them were in PreCalc with me, after we had done some work with DeMoivre’s theorem, they posed this question of what the function f(x)=i^x would be like (what is base i ?). This question led to one of the most robust and interesting explorations that I, and they, made. One young man programmed the solutions to show that they were like the spiral helix of a DNA strand. Really cool.

Thanks John! I love hearing how other people (especially teachers) are exploring it. I’ve come to see i as the epitome of rotation, so when we use it as a base we end up rotating around. Great stuff.

Incredible article!

I found that trying these ideas out with other imaginary numbers really helped to make clear what is so special about the definition i^2=-1. By other imaginary numbers I mean what if you take i^2=0 or i^2=1 and then everything between and beyond. When it equals 0 instead of falling back into a circle you just keep going up from 1; and when i^2=1$ you start growing upwards and then accelerate out more and more and you get a hyperbola (hence the hyperbolic sine and cosine). There’s a really neat wolfram demonstration here: http://demonstrations.wolfram.com/TransformationsOfComplexDualAndHyperbolicNumbers/

Basically as i^2 gets really negative the e^it flattens out like a pancake, when it gets close to 0 the circle bows up into two verticle lines at 1 and -1, and from there it stretches out into a hyperbola. I hope that made some sense.

More than 30 years before Euler published his famous equation, Roger Cotes wrote essentially the same formula, except he expressed it using the natural logarithm (ln) instead of the exponential function.

From Roger Cotes’ notes:

ln (cos x + i.sin x) = ix

Now, Cotes’ equation is truly extraordinary, but I cannot see how Cotes came to this conclusion, more that 30 years before Euler (in fact while Euler was still a small boy in a far off land).

This earlier version is not as simple as Euler’s, and Cotes did not offer any proof of his result (Euler provided that proof later). Also, Cote’s version does not evoke any intuition for me at all – it just seems to appear out of nowhere, without any motivation.

This anomaly also highlights the fact that, historically speaking, logarithms were discovered before exponentials, even though exponentials are the more obvious concept, being merely repeated multiplications.

I wonder how this happened. Why did Cotes’ strange version of Euler’s equation appear more than 30 years before Euler’s?

It is a pity that Roger Cotes died so young – what would the history of mathematics be if he had instead lived to be an old man?

I made an animated, geometric proof of Euler’s formula, using the intuitions that I learned from this website, and from the amazing slideshow at http://slesinsky.org/brian/misc/eulers_identity.html (already linked above I believe), and from some basic rules about vectors (basic physics) and limits (basic calculus).

Check it out if you enjoy being enlightened as much as I do, and believe that most ‘complex’ concepts, once the abstractions are removed, are really not all that complicated. I ‘understand’ Euler’s formula now as well as I believe anyone can ‘understand’ anything, I can explain it in terms of other things that I ‘understand’, and ultimately on things that are so familiar to my experience that I can take for granted.

https://www.desmos.com/calculator/ohwnsgnef2

Hi Paul, really neat question. In my head, I see logarithms as giving the “input” that leads to some effect. So, ln(10) is asking “What amount of time is needed to grow from 1 to 10, assuming 100% continuous growth? (ln(10) = 2.302 time periods)

In my head, when I see

ln(cos(x) + i.sin(x)) = ix

I think “What input would lead to the pattern cos(x) + i.sin(x)?”

My intuition is that only an imaginary growth rate (i*x) would lead to circular motion (cos(x) + i.sin(x)). It is interesting that in general, logarithms were discovered before exponents: I see this as thinking about the cause (time, or an interest rate) before working out the effect (what the growth actually looks like).

Hi Kalid,

I understand your intuition for ln(cos(x) + i.sin(x)) = ix, which is essentially knowing Euler’s formula, and understanding that ln(x) is the reverse function of e^x, and therefore cos(x) + i.sin(x) therefore would be what what ’causes’ ix to appear as the result.

My personal amazement is that apparently Roger Cotes could see that too, long before Euler’s elegant equation was published.

It seems that in Cotes’ era (early 1700′s), logarithms were heavily employed by mathematicians, but there was much less understanding of the exponential function, until Euler shed his intellectual light on it decades later.

So, how did Cotes understand his natural log version of the equation? Today it is seen as the reverse of Euler’s equation, but what intuition could Cotes possibly have?

We are lucky to have any of Cotes’ notes, published by his relative after his early death, but I cannot help but feel that Cotes understood much, much, more than we have on the public record.

I feel this particularly because his natural log version of Euler’s equation only makes sense when we see it is the reverse of Euler’s. Can we reasonably conjecture that Roger Cotes already knew and understood that e^ix = cos(x) + i.sin(x), long before Euler?

No less than Sir Isaac Newton, a friend of Roger Cotes, wrote of him:

“If he had lived we would have known something.”

The early death of Roger Cotes was more than just a personal tragedy – it could well have set-back mankind’s advances in science and mathematics by many decades.

The contents of this site should be taught in schools parallel to “standard” course.

I knew for a long time that the trigonometric terms of Euler’s formula represent helical motion is ‘x’ is taken as an axis, which simply reduces to circle if ‘x’ is considered and angle.

But I never took the time to analyze the exponential part in this way, Great insight.

This one makes me cry.

Thank you that’s the closest anyone’s ever come to explaining it to me I think I’ll reread at my leisure! Good to know the identity has practical uses and isn’t just magic!

When we consider only addition the entire maths can fit into a single dimension. We pour in multiplication and it gets 2D. Couldnt there be a 3D model? What growth would be more superficial than that of multiplication? Moreover can we think of addition as dealing with the value of observation, multiplication as dealing with the frequency of observation and all the other operators as just functions. negative number corresponds to imaginary number.Negative numbers taught us to extend the number line behind the origin. Imaginary numbers taught us to extend the number line above the origin in another dimension. but somehow I feel like that maths isnt absolute instead it too is relative. How could negatives and imaginary numbers exist if there was only one observation??? We get the concept by comparing several numbers which cant fit into any known pattern so we introduce(-x) and sqrt of (-x).

Please answer my question, could we have 3d maths????

@Arkadeep I look at dimensions this way, if you multiply two things, it’s best to imagine it as square. If you multiply three things, it’s best to imagine as cube.

for example, 2×2 = 4 , corresponding to vertices of a square in cartesian plane.

extending it, 2x2x2 = 8, corresponding to vertices of a cube in 3-D.

even more, 2x2x2x2 = 16, corresponding to vertices of a hypercube (look up 4D cube on wikipedia, that object is very amusing.)

In short, i think of multiplication as not just 2D, but operating in as many dimensions as there are numbers being multiplied, multiplying is like stretching an object into higher-dimensional object.

point(0) -> line(1) -> square (2) -> cube/cuboid(3) -> hypercube like objects (4) .. and so on.

If we represent numbers using such objects (orthogonal in ‘n’ dimensions, cuboid like) prime numbers will always be lines, composite ones could be represented by ‘n’ dimensional objects depending on count of factors (except 1).

This is a way of looking at multiplication, as extending or stretching an object whose count of vertices represents a number to a whole new dimensions, not just 1D to 2D, but from N-D to (N+1)-D.

And of course there is higher dimensional mathematics, and actually it’s a subject of current mathematical research due to it’s role in String theory and it’s derivatives. read up kaluza-klein theory, it actually is based on 11-D objects.