Intuitive Understanding Of Euler's Formula

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@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.

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“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.

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nice

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

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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.

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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!

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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.

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[…] important constant which is the base of natural logarithms. I is the square root of minus 1. http://betterexplained.com/articles/...ulers-formula/ Reply With Quote + Post New […]

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@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).

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Hi ominac, glad it helped.

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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…

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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=ipi (i is of course the imaginary number sqrt(-1)). Let’s evaluate a few of the iterations as x–>infinity.
(1+(ipi)/1)^1= 1 + ipi
(1+(ipi)/2)^2= 1 + 2*(ipi/2) - (pi^2)/4 = -1.46 + ipi (rounded)
(1+(ipi)/3)^3=-2.3 + 2i (rounded)
…
(1+(ipi)/10)^10=-1.6 + .15i
…
Now look at the picture/animation/caption at this link:

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.

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@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.

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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?

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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!

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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.

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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!

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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? :)]

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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?

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