How Does Euler's Formula e^(2*pi*i) Evaluate to 1?

[xenogizmo]

How does that evaluate 1??

e^(2*pi*i)

 

[NeutronStar]

e^(2*pi*i)

It comes from Euler's formula

e^{\left( i\theta\right) } =\cos\theta +i \sin\theta

e^{\left(i2\pi\right) } =\cos2\pi +i \sin2\pi

e^{\left(i2\pi\right) } =1 + 0

e^{\left(i2\pi\right) } =1

 

[damoclark]

Well the way you get this is to look at the infinite polynomial expansion of
e^x, cos(x) and sin(x)

e^x= 1+ x/1! + x^2/2! + x^3/3! + x^4/4! ... + x^k/k! + ...etc...

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - x^10/10! +... etc...

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11! -...etc...

This is how your calculator probably calculates sin(x) and cos(x) ...
These series can be derived quite easily, and if your curious I can explain how you can get these infinite series expressions for the functions e^x,
sin(x) and cos(x).

Now if you substitute i*x for x into e^x, where i = square root of -1.
Note that :

i = i
i^2 = -1
i^3 = -i
i^4 = 1

i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1 ect in this repeating fasion

then

e^(i*x) = 1 + i*x/1! + (i*x)^2/2! + (i*x)^3/3! + (i*x)^4/4! +(i*x)^5/5!...

=1 +ix/1! - x^2/2! -ix^3/3! + x^4/4! + ix^5/5! - x^6/6!...

This series can be spit into two infinite series

e^(i*x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8!... = cos(x)
+ i*( x - x^3/3! + x^5/5! - x^7/7! + x^9/9! ... = i*sin(x)

therefor

e^ (i*x) = cos(x) + i*sin(x)

Then if you substitute Pi for x you have

e^(i*Pi) = cos(Pi) + i*sin(Pi)
= -1 + 0
= -1

This is a very beautiful and elegant formula discovered by Euler I think.

The formula also has a very nice geometric representation when you view it in the complex plane, and this is generally an easier way to understand the formula if you don't like all the algebra.

 

[tongos]

taylor series

something that is also interesting i^i=
e^(ipi)^(i^1/2)=(i^2)^(i^(1/2)= e^(-pi/2)

 

[mathwonk]

one also knows the unique solution of f''+f = 0, f(0)= a, and f'(0) = b, is
acos(x) + b sin(x). But e^(ix) solves it with a = 1, b = i. so e^(ix) = cos(x)+isin(x).

 

[strid]

I haven't seen this before...

but though i follow the concluding above, I ran in some trouble a moment. ago...
My first thought was that it is quite close to the Eulers formula e^(pi*i) +1 =0 , so I stared to fiddle with it in hope of getting something funny out of it.. but i found this...

e^(pi*i) + 1 = 0
e^(2*pi*i) = 1

e^(pi*i) = -1
(e^(pi*i))^2 = 1

- (e^(pi*i)) = (e^(pi*i))^2

SQRT( -(e^(pi*i)) ) = e^(pi*i)

The problem now is that on the left side there is a SQRT of a negative number (about -23,14) which means it is a complex number... while on the other side we have a real postive number (about 23,14)...

Have I made some mathematcial error, or how can this be...?

 

[hypermorphism]

...

SQRT( -(e^(pi*i)) ) = e^(pi*i)

The problem now is that on the left side there is a SQRT of a negative number (about -23,14) which means it is a complex number... while on the other side we have a real postive number (about 23,14)...

Have I made some mathematcial error, or how can this be...?

Two things. The designation -x does not mean that the quantity -x is a negative real number. Secondly, the f(x)=x^2 function is not globally invertible, so you cannot always assume that if x^2=a^2, then x=a. Thus, in your example, SQRT( -(e^(pi*i)) ) = SQRT(-(-1)) = SQRT(1) = 1. 1 is one square root of (e^(pi*i))^2 = 1. The other square root is -1.

 

[strid]

ok... got it now...

guessed already in before-hand that Euler probably knows more than me :)