Intuitive Understanding Of Euler's Formula

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.

The movie was found on a page for Eulers Formula: http://en.wikipedia.org/wiki/Euler's_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?”

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

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.

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/

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.

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"…

(1+x/n)^n =(1+x/(xm))^(xm)=((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?

(1+x/n)^n =(1+x/(xm))^(xm)=((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.

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 1e^1i. 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.