Why is e^(pi i) = -1?

#1

Post by ulvund » Fri, Oct 30, 2009, 1:28 PM UTC

Why is e^(pi i) = -1?
math.toronto.edu

Re: Why is e^(pi i) = -1?

#2

Post by sophacles » Fri, Oct 30, 2009, 2:20 PM UTC

So to broadcast my math ignorance to all:

Why does cos(pi) + i sin(pi) == -1 ??

Why does i disappear in this?

Edit: Thanks to the replies below.

Re: Why is e^(pi i) = -1?

#3

Post by travisjeffery » Fri, Oct 30, 2009, 2:21 PM UTC

The article didn't have any mention of Euler, but this is typically called Euler's Formula and is considered one of the most beautiful formulas to Mathematicians.

Here's some more info if you want to find out some info, applications and such: http://en.wikipedia.org/wiki/Eulers_formula

Re: Why is e^(pi i) = -1?

#4

Post by rottencupcakes » Fri, Oct 30, 2009, 2:24 PM UTC

sophacles wrote: post #2

So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.

We verify in a quick Google search: http://www.google.com/search?hl=en&safe=off&q=sin(pi...

that sin(pi) = 0.

Re: Why is e^(pi i) = -1?

#5

Post by [deleted] » Fri, Oct 30, 2009, 2:24 PM UTC

sophacles wrote: post #2

So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.

[deleted]

Re: Why is e^(pi i) = -1?

#6

Post by amalcon » Fri, Oct 30, 2009, 2:24 PM UTC

sophacles wrote: post #2

So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.

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.

Re: Why is e^(pi i) = -1?

#7

Post by tom_b » Fri, Oct 30, 2009, 2:25 PM UTC

For those of you interested in a longer look at this, check out "E, the Story of a Number" by Eli Maor.

Yes, I read when I should probably be hacking.

Re: Why is e^(pi i) = -1?

#8

Post by travisjeffery » Fri, Oct 30, 2009, 2:27 PM UTC

sophacles wrote: post #2

So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.

The formula is:

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

Re: Why is e^(pi i) = -1?

#9

Post by daredevildave » Fri, Oct 30, 2009, 2:41 PM UTC

Write this as e^(iπ) + 1 = 0 and you have my favourite equation.

It relates all the fundamental mathematical numbers: e, i, π, 1 and 0.

Re: Why is e^(pi i) = -1?

#10

Post by catzaa » Fri, Oct 30, 2009, 2:42 PM UTC

sophacles wrote: post #2

So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.

If you draw a Sine graph, you will see that sin(x) is zero at 0, pi, 2*pi, ...

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.