## Interest

on Jul 17, 2007

# Stumblers Who Commented On This Page

## 33Arsenic

### Ben

@People complaining about the "proof is easy." Yes the proof is easy to understand (but I'd love to see you derive it yourself and yes that means you'll have to derive Euler's Formula), but that is not the subject of the quote. The subject is the meaning of such an identity.

The consequence of this identity (and more formally Euler's Formula) is that exponential and trigonometric functions are related. Now why that is is another question entirely.

## mellocello2003

### mellocello2003

Yes it's cool looking but when you think about it (if you know anything about math), it's really obvious.

## Coolthulhu

### Coolthulhu

Grrr...

i ≠ sqrt(-1)

i ∈ sqrt(-1)

Roots of negative numbers are equal to sets, not single numbers.

(-1)^(1/2) = sqrt(-1) = {i, -i}

If you forget about that you may get strange contradictions like:

(-1) = i * i = sqrt(-1) * sqrt(-1) = sqrt(-1 * -1) = sqrt(1) = 1

Because of that, one shouldn't assume that i = sqrt(-1).

The proper definition is i^2 = -1. Not all numbers can be squared as freely as positive real numbers.

## civver

### civver

Unifies almost all of the significant constants in mathematics in one identity.

## Kirro

### Miles

This is just pathetic. The proof is easy. To completely understand the proof, you'll probably need to have taken Calculus I and possibly Calculus II. But without that, you can still use the identity e^(ix) = cos(x) + i sin(x). So e^(pi*i) = cos(pi) + i sin(pi) = -1 + i*0 = -1. The longer explanation requires that you prove the initial identity, but that's beyond high school math, and you see it explained here: http://www.math.toronto.edu/mathnet/questionCorner/epii.html

## themoertel

### themoertel

Balls. Just...balls.

## Sheva7

### Nazim

To me, the greatest formula in mathematics.

## spwelton

### Sean

An interesting mathematical property involving pi, e, and i.

## franklinbains90

### franklinbains90

well that was anti-climactic...

## JonLandrum

### Jonathan

Stunningly gorgeous.

## Arpana-INFJ

### Arpana

Haven't a clue what this means, but find something pure, in the small, obviously meaningful statement. Unknowingness. I cavort, without shame, in my ignorance.

## elio64red

### Elio

Well, it's very useful for Fourier Transforms

## gogodidi

### Thies

I'm not gonna bother checking if Tracer-Bullet's question was ever answered, I'll just do it here.

The solution to this "problem" is through eulers relation:

a*e^(i*b) = a( cos(b) + i * sin(b))

Where b is an angle in polar form

We know this as it can be observed on the argand diagram (still, you have to imagine what genius it took to see this relationship in the first place). The proof for this relationship was made some time in the 18th century, but we now have multiple proofs for it. (Wikipedia probably has one and if you're familier with calculus should be easy enough to comprehend)

e^(i*pi):

cos(pi) = -1

i * sin(pi) = i * 0 = 0

hence:

e^(i*pi) = -1 + 0*i = -1

This is taught (or if you weren't, good luck in college) in high school and is nothing new, but it is a sensational relationship.

If you're interested in the applications of this, you may want to look into fourier series, which can me simplified into exponential form using this relation. Fourier series have several applications, among others finding solutions to partial differential equations. That is but one of many applications of it, though.

Hope that answers your question Tracer.

## avre

### Aurelia

That's possible just in maths, to prove that something is true even if you don't know what it means and can't understand it either.

## JG-NUKE

### Jack

i should have studied

## 11saga11

### John

Harvard mathematician Benjamin Peirce told a class, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

## stumbledvd

### stumbledvd

I love math

## ngenie

### Nana

The miracles of math... take a bunch of seemingly unrelated numbers and combine them to make a wonderfully simple formula.

Though I'll have to agree with Vallam, XKCD said it better:

## megamaiden21

### megamaiden21

So kewl...n true

## orochijes

## orochijes

Benjamin Peirce lived a long time ago. Since then, some ideas of interpreting this formula in an intuitive manner have surfaced. Give this article a read.

betterexplained.com/articles/intuitive-understanding-of-eulers-formula

It's not easy to understand, but it is certainly not "paradoxical".