Why is e^(pi i) = -1?
math.toronto.edu
math.toronto.edu
Exponentiation is to do with growth at a speed which is a multiple of how big you are already.
i is the multiplication which turns you through ninety degrees.
If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude.
Pi is how long it takes you to turn through a half circle.
So if you grow at right angles to yourself for time pi, you are pointing the opposite way.
By the time I got to university it was just taken as a given, so nobody explained it at all.
My explanation seems to have attracted a lot of positive votes in a few minutes. I've written a more detailed version here if anyone doesn't get it:
http://johnlawrenceaspden.blogspot.com/2009/10/ei-pi-1.html
If it gets much attention I might draw the diagrams that go with it.
I'm always bewildered how many scientific types (and even some mathematicians) find complex numbers mystical. They were originally discovered by mystic methods, but Argand showed us what was really going on, and they're no more weird than 2-dimensional vectors.
Actually a lot less weird than the Reals, which really are magical and mysterious, but which everyone seems to be quite happy with!
http://betterexplained.com/articles/intuitive-arithmetic-wit...
The first link was posted here a while ago, I think.
I really like Euler's Formula, but feel many explanations focus on the raw bits (showing the series are equal). The physics explanation "90-degree rotation" is a good one. Here's how I explained it to my mom:
"The formula is saying there's two ways to get to a place on a circle. One is to go across (cos) and up (sin), and the other is to go out (1.0) and rotate (x radians)."
This is a beautiful result which ties together radians, exponential growth, and complex numbers -- I have a post drafted that I want to share on this :).
It's the only way to get to the complexes without doing anything spooky! Defining i to be "the square root of minus one" is about as sane as defining it to be the square root of the colour blue.
The first people who thought about it followed that approach, and were rightly scared stiff and confused by it. Even Euler made trivial mistakes.
Argand came up with the right way of thinking about it:
Take all the tuples (x,y). (Where x and y are just integers). Define on them addition and multiplication rules.
Oh look! There's a big system of these things, and embedded within it is a sub-system which works exactly the same as the integers and their rules.
Since they're exactly the same for all practical purposes, we may as well forget about the difference and say that we'll write (a,0) as a and (0,b) as ib, and (a,b) as a+ib
And although non of the pairs in that subsystem square to be (-1,0), also known as -1, there are two things that do! So now we know that both (0,1) and (0, -1), otherwise known as 1i and -1i, or i and -i, are square roots of -1.
No slight of hand or magical thinking necessary.
As it happens, we can do the same thing with the reals, to form something that embeds the reals. But the reals really are dark and mysterious and need to be brought about by a kind of magic.
http://www.math.toronto.edu/mathnet/answers/imagexist.html
I'll gratuitously paste the relevant part here, but the summary is that to multiply two tuples, you simply take the cross product, as you would a binomial.
---
Remember that any collection of objects for which there is a definition of what the objects are and when two objects are equal, there is a rule for how to add two objects, there is a rule for how to multiply two objects, and these rules obey familiar arithmetic laws like commutativity, associativity, and distributivity, is, by definition, a number system.
These properties are all satisfied by complex numbers.
We have a definition of when two complex numbers are to be considered equal: they are equal if and only if they are the same pair of real numbers.
We have a rule for adding two complex numbers (which, remember, are nothing more than pairs of real numbers):
(a,b) + (c,d) = (a+c, b+d)
(a,b)(c,d) = (ac-bd, ad+bc)
Therefore, complex numbers form a number system.
Within this number system, is there an object which, when squared, gives -1? Yes. It is the pair (0,1). When you square it using the above rule of multiplication, you get
(0,1)(0,1) = ( (0)(0) - (1)(1), (0)(1)+(1)(0) ) = (-1,0).
However, complex numbers of the form (a,0) behave identically to the way ordinary real numbers a behave. They add and multiply in exactly the same way that ordinary real numbers do:
(a,0) + (b,0) = (a+b,0)
(a,0)(b,0) = (ab,0)
This is actually a slightly tricky point, because it requires the direction of growth to change as you grow. If you grow by a finite amount all at once, then you do get slightly longer. So you might expect an outward spiral.
But a point moving round in a circle is always moving at right angles to the radius connecting it to the centre.
And in the same way, an arrow whose tip is moving at right angles to its shaft isn't extending. If the length is changing, then the direction isn't 90 degrees.
To reason about this properly I think you need some sort of theory of infinitesimals and continuous motion.
I.e. you have a bank account and put in a dollar. Its a great bank, they give you 100% interest. So, after a year, you have (1 + 1) = 2 dollars.
But wait, instead, they compound twice. That is, every 6 months, you get 50% interest. So now you have (1 + 1/2)^2 = 2.25 dollars.
Instead of compounding twice, they compound 4 times. So now you have (1 + 1/4)^4 = 2.44 dollars. Each time you get interest, its given in proportion to what you have in the account at the time. So, if they are compounding 4 times:
1 --> (1.25 * 1) = 1.25 --> (1.25 * 1.25) = 1.56 --> (1.25 * 1.56) = 1.95 --> (1.25 * 1.95) = 2.44.
(Please note: Rounding error. I just typed in to two places after the decimal point).
Why 2 or 4? Lets just call it compounding n times. As n gets larger, you get more money, but it doesn't go on forever...
For n times, the formula for how much money you have is (1 + 1/n)^n. What if you want to continuously compound? I.e. at every instant, growth is proportional to the current value? Well, then you would want something like (1 + 1/infinity)^infinity. But, we can't really do that, so instead we say:
lim (n --> inf) of (1 + 1/n)^n
And thats what we call e. Start putting in larger and larger values of n and you'll see that it converges to 2.718... The point is that e is pretty much defined as the number you get when you take a number starting at 1 and have it grow in proportion to itself continuously. Instead, if you use 5 instead of e, your growing too fast. In terms of johnaspden's explanation, you'd move too far around the circle, past (-1, 0).
Also, johnaspden, your explanations are truly great, really helped me understand Euler's formula.
Edit: Small change.
It should be e^(pi i) = 1, but pi was unfortunately defined at half the appropriate value in the 17th century.
And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.
It was programming a lot that brought me to realize this. I got in the habit of always questioning the primitives I'm given, in the hope of simplifying the definitions built on them. It was a little boggling to notice that pi, something I'd grown up with, deserved the same kind of questioning.
This is actually my favorite example of why pi is wrong—it's the "exception" that proves the rule. To see why, set τ = C/r = 2 pi, and then consider the following chart of common quadratic forms:
integral of u 1/2 u^2
kinetic energy 1/2 m v^2
distance fallen 1/2 g t^2
spring energy 1/2 k x^2
triangular area 1/2 b h
circular area 1/2 τ r^2
N.B. I made a previous comment along these lines, but put it in the wrong place. If you're still a "pi is wrong" skeptic, considering the improvement in radian angle measure may yet convince you:
The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc.
In these terms, Euler's formula would be recast as
e^(i τ) = 1
That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
I've thought pi is horrible... and adopted using a loopy pi as 2pi. I loop the first vertical line over the horizontal, right, and then make it the horizontal, and so on. If that makes any sense.
I'm thinking of making a LaTeX package containing it... (and also some other personal conventions)
Also, to strengthen your point, 2pi makes the radian system a lot more natural.
N.B. I have a secret master plan to spread the use of τ, but this comment is too small to contain it. ;-)
So you're getting the area by integrating a linear thing.
Exactly. In symbols, this reads dA/dr = C ∝ r. Setting τ = C/r = 6.2831853…, we have dA = C dr = τ r dr => A = ½ τ r². Q.E.D.
dA = C dr = τ r dr => A = ½ τ r².
Here's some more info if you want to find out some info, applications and such: http://en.wikipedia.org/wiki/Eulers_formula
It relates all the fundamental mathematical numbers: e, i, π, 1 and 0.
sin 2A = 2 sin A cos A
cos 2A = (cos A)^2 - (sin A)^2
If you are happy using sine of small x is x and cosine of small x is 1 as your base cases, you can write sine and cosine as mutually recursive functions:
(defun sin (z)
(if (small z)
z
(* 2
(sin (half z))
(cos (half z)))))
(defun cos (z)
(if (small z)
1
(- (square (cos (half z)))
(square (sin (half z))))))
It had never occurred to me to do exponential the same way, as
(defun exp (z)
(if (small z)
(+ 1 z)
(square (exp (half z)))))
(exp (complex 0 pi)) => #C(-1.0012055113842453d0 6.167817939221069d-7)
In these terms, Euler's formula would be recast as
e^(i τ) = 1
That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
1- http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
2- http://books.google.com/books?id=YXv6SEjTNKsC&pg=PA432&#...
Yes, I read when I should probably be hacking.
Why does cos(pi) + i sin(pi) == -1 ??
Why does i disappear in this?
Edit: Thanks to the replies below.
that sin(pi) = 0.
I wouldn't feel ignorant or anything. The article left out that information seemingly on purpose, presumably so that people would try to figure out where the i went.
e^(ix) = cos(x) - i sin(x)
Since sin(π) == 0, i * 0 == 0 so you have cos(π) left which is -1.
So in sum:
e^(i π) = cos(π) - i sin(π) == -1 - 0 == -1
If you draw a cosine graph you will see that it is -1 at pi. Therefore in the above sum the sine part just falls away.
It is easy to see that the above is true when you recall that cosine(theta) = adjacent side/(hypotenuse). Since pi=180 degrees, adjacent will be -1 hypotenuse 1 and opposite side 0.
I'm sure almost everyone has seen this anyway, but why not post.
Another good description is discussed in "Visual complex Analysis" by Needham.
It contains all the "celebrities" of the math world.
It's quite cool actually.
Given how much harder MathML is to author, the jsMath solution is always going to be more popular among random math folks. Particularly since they're already used to writing TeX.
(pi/2)^2e