In http://boards.straightdope.com/sdmb/showthread.php?t=581447, Indistinguishable provides a link (http://pleasantfeeling.wordpress.com/2010/08/03/eulers-theorem-while-true-is-overrated-and-is-also-understood-by-small-children/) that explains Euler’s equation. I am posting this question here as I don’t want to distract the discussion in the original thread.
In the link provided in the other thread, r is defined as 1 radian and R as one full revolution (360 degrees). Later the following equation is stated: r^(2*pi) = R. At leas thats the way it looks. When my mouse hovers oveer the picture, the formula is given as r^(2/pi) = R. Typo perhaps? Either way it makes no sense to me.
If r is 1 radian then 1 to the power of anything is still 1. If they are measuring r in degrees then it still doesnt work, as far as I can tell. What is going on here?
Am I stupid or is this actually that confusing?
I can’t answer fully, but note that the mouseover text says:
r^{2 \pi} = R
the slash is a backslash, and is part of the LaTeX symbol “\pi” - it does not indicate division.
Edit: and rsup[/sup] is correct. As I understand it, r is not “one radian”, it is “rotation through one radian”. Rotation through one radian, raised to the power of 2 pi, results in rotation through a full circle (i.e. 2 pi radians).
I’m not entirely sure why it is raised to the power rather than multiplied, but the discussion about logarithms implies it must be taken as read…
Thanks, I did actually notice the mouseover text, but it only made things more confusing.
tbh, I don’t see how one radian is any different to rotation through one radian. Also, 1 radian * 2 * pi does actually equal 360 degrees, or 1 full rotation. Which is why I thought it was a typo, perhaps the equation was meant to be r2pi = R.
If that is not the case then I would love a fuller explanation of how r^(2*pi) = R.
I don’t really understand it, but the other thread has some discussion of why rotation is an example of an exponential function.
Yes, the backslash is part of LaTeX code behind the scenes that shouldn’t be visible to you; it should indeed be read as 2 * π, not 2/π. And yes, r is not the number 1, but rather the number “rotate by one radian”, which is a different thing entirely. (Perhaps you will ask “Huh? ‘rotate by one radian’ can be thought of as a number?”. Yes, it can, the same way the number -1 means “rotate by 180 degrees” (and note that -1 and 180 are completely different numbers). If you haven’t read this post, it may be helpful.)
I understand that this may all be confusing if you haven’t seen things presented this way before; you’re not stupid for having some difficulties understanding it, and I’ll be happy to answer any questions you have about it. I hope that in the end, you’ll come to see that it’s all extraordinarily simple, once you learn to think about arithmetic the right way. [And please, continue to let me know whatever in my explanations is confusing]
As for why some things are multiplication, some things are exponentiation, etc.:
Any function f with the property that f turns additions into additions (i.e., distributes over addition; i.e., is “linear”) can be thought of as multiplication by some constant. That is, if f is continuous, f(0) = 0, and f(x + y) = f(x) + f(y), we can think of f as multiplication by some constant. In particular, if f is the operation on 2d vectors which rotates them by some fixed amount, then f will satisfy these properties. [Thinking of vector addition as given by triangles, the observation that rotation distributes over addition is the observation that when you rotate a triangle, it’s still a triangle]. Thus, rotation by the angle θ is multiplication by some constant, which we can call rot(θ) for now.
Just like we can add vectors to other vectors, we can also add linear operators to other linear operators, just by performing the addition pointwise. That is, if f and g are functions as above, then we can define a new function f + g, using the rule that (f + g)(x) = f(x) + g(x).
So what? Well, we can also combine linear operators in another way: given two linear operators f and g, we can compose them together into their cumulative effect; that is, we can define a function g * f by the rule (g * f)(x) = g(f(x)). Why am I using an asterisk here? Well, because composition of functions is also a kind of multiplication. If you think about it, it also distributes over additions (on either the left or the right), so, by our rule above, we can also call it multiplication.
Finally, any function f with the property that f turns additions into multiplications can be thought of as exponentiation with some base. That is, if f is continuous, f(0) = 1, and f(x + y) = f(x) * f(y), we can think of f as exponentiation with some base [with f(1), the 1st power, giving us information about what the base is].
In particular, consider the function rot(), which sends an angle to rotation by that angle. Well, rot(x + y) = rotation by the angle x + y = the cumulative effect of first rotating by the angle y and then rotating by the angle x = rot(x) * rot(y). Thus, rot is an exponential function; we can think of rot(x) as meaning rot(1)[sup]x[/sup].
And so, using the letter “r” as shorthand for “rot(1)”, we have that rotation by the angle x = r[sup]x[/sup].
In general, we will identify a number with the action of multiplying by that number. So, we’ll think of the number 2 as meaning the same thing as the action of multiplying by 2, which is to say, “Become twice as large”. And we’ll think of 8 as meaning the action “Become 8 times as large” and so on. And any nice action [one which distributes over additions, as in the above post] can be thought of as (multiplication by) some number.
Consider the difference between “Become 180 times as large” and “Rotate 180 degrees (aka, π radians, aka 1/2 a revolution)”. The former is multiplying by 180, while the latter is multiplying by -1. But 180 (and π, and 1/2) is very different from -1.
Similarly, “Become 7 times as large” is very different from “Rotate by 7 degrees”. “Become 7 times as large” is multiplying by 7. “Rotate by 7 degrees” is something rather else. There’s a 7 in its description, but that doesn’t mean it is itself the number 7. It’s a very different number. What number is that? Well, it is what it is; there’s no actually better way of describing it than as the complex number “Rotate by 7 degrees”.
And finally, “Become 1 times as large” is very different from “Rotate by 1 degree”, which is also different from “Rotate by 1 radian”, which is also different from “Rotate by 1 [whatever random unit of angle]”. “Become 1 times as large” is multiplication by 1. But “Rotate by 1 degree”, “Rotate by 1 radian”, etc., are different numbers that just happen to be described with a 1, without themselves being 1.
If you insist on presenting it more traditionally, “Rotate by 7 degrees” is cos(7 degrees) = sin(7 degrees) * i. But there’s really no need to split it into components like that. It’s perfectly fine to just call it “Rotate by 7 degrees”.
Similarly, the r = “Rotate by 1 radian” which is confusing you is equal to cos(1 radian) + sin(1 radian) * i. It’s just that particular complex number, whose effect is rotation by 1 radian. Note that cos(1 radian) + sin(1 radian) * i is very different from just 1.
There’s a typo here; that = should be a +. “Rotate by 7 degrees” is cos(7 degrees) + sin(7 degrees) * i.
[This is because rotating a vector by 7 degrees is the same as constructing the hypotenuse of a right triangle whose base points in the same direction and is scaled by a factor of cos(7 degrees), and whose opposite leg (which is turned 90 degrees) is scaled by a factor of sin(7 degrees). Ah, but this is a mess, isn’t it? Much nicer to just say “Rotate by 7 degrees” and leave it at that, unless one has some real need to break it down this way.]
I suspect I have probably failed to actually make my explanation of Euler’s equation clear to the OP and Colophon and whoever else may be reading this thread. But I would like to hear more from anyone who was confused by my explanations (including the linked posts) any comments about particularly confusing parts or unaddressed issues, suggestions for improvement, etc.
As I just said in the other thread, I would really like to help bring the world to a point where mathematicians stop presenting the ideas around Euler’s theorem in such extraordinarily silly, obfuscating ways as they almost always do, but to succeed in this plan, it would be helpful to have feedback on what is and is not working as I try to explain the arithmetic of rotation.