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 canât 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.
âStephen
I mean I see why the shape of the imaginary growth is a general curve, but how do you know itâs circular (ie the growth is orthogonal)?
@Jon: Thanks, thatâs 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 âWhat transformation x, when applied twice, will turn 1 to -1?â. 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
- Two imaginary rotations would not perfectly flip your direction (89 + 89 = 178)
- 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 âmultiplying by an imaginary number cannot give you any components in your current direction, otherwise that âboostâ could accumulate over time.â
I hope this helps! Let me know if it didnât, 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 donât know how much this concept has been bugging me haha. Youâre website has really made me think deeply over the past few daysâŚ
My question doesnât 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 itâs reasonable to conclude that all the change is orthogonal. That make sense.
But as you said, when youâre looking at the multiplication youâre doing for e^(ix) (to achieve growth along the circle), youâre 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 Iâm not mistaken, the multiplication would look something like (1+ix/n)(1+ix/n)(1+ix/n)âŚn times, with really small nâs, (please correct me if Iâm 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 donât know what other path youâd take to achieve the transformation youâre looking for, other than a circular path. It makes senseâŚI might be chasing nothing here, but I guess Iâm 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 weâre doing here, so does the imaginary number multiplication logic hold up here? Couldnât we end up with sayâŚa spiral? idk
Thanks, Kalid, for bearing with me. I really, really appreciate it.
Take care
@Stephen: Youâre more than welcome, these are really fun to think about.
Ah, I think I see what youâre getting at! Yes, itâs 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 heâs 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 âwrapsâ 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 helpfulâŚI will have to go over it a few times though haha. A bit complicatedâŚ
Appreciate your help.
Hi,
Is it a right âtranscriptionâ of the original formula:
exp(pi) = â(square root of -1) rootâ of -1
?
@erik: Hrm, Iâm not sure what you mean by transcription⌠do you mean âHow would you put the formula into a sentence?â
@alan: Thanks for the comment â thatâs a really interesting visualization Iâd like to explore :).
Kalid, your work is really a masterpiece. Finally I found a site (better late than never) thatâs really helping me a lot. I couldnât get this kind insights when I used to study.
Thanks a lot.
I will recommend your work to all my friends.
wererouge wrote âThis gets easier if youâve 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 youâll move in a circle (your velocity changes.)â Then theoritically we can have perpetual motion. I dont know much about math and stumbled here trying to figure out eulerâs 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 hereâs an article I think itâs partially decoded and I believe they are trying to tell us some kind of free energy in relation to eulerâs. 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 âstart at 1.0 and rotate by x radiansâ, just like 2^3 and 2 * 2 * 2 are both ways of describing the same act of âmultiply by 2, three timesâ.
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! - âŚ].
And, what do you know, e^x has a series definition too [e^x = 1 + x + x^2/2! + x^3/3! + âŚ].
If you plug in âixâ for x in e^x, youâll 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"âŚ
(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 infiniteâŚ
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 Kâ
Just from curiosity, I set out some time ago to learn about Eulerâs 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.
âTim
I completely agree â the notion of seeing âeâ as (1 * e, that is, weâre starting with 1 and growing continously) and i as (1 * i, that is, weâre starting at 1 and rotating) really helped Eulerâs theorem click for me too.