What is the value of the mathmatical expression e^(i*pi), if any?

Answer 1

Because e^(iθ) = cos θ + i.sin θ . . . . . called Euler's Formula,

e^(iπ) = cos π + i.sin π = -1 + i.0 = -1

It's my favourite maths result - all the well known, and important stuff in one tidy formula.

. . . . and, for the record, you can even use it to do i^i !!

just put θ = ½π and raise to the power i on both sides.

(e^i.½π)^i = {cos ½π + i.sin ½π}^i

e^(–½π) = i^i . . . . . cos ½π = 0 and sin ½π = 1

Weird . . a real number on one side and a rather difficult to imagine thing on the other.

Answer 2

"because e^(ix)=cos(x)+i*sin(x)"

is not the greatest answer. That equation is true, but it's not an explanation. In other words, you should explain what motivates the definition of the exponential of an imaginary value.

The answer is the Taylor series. It is a fact that in the complex plane, there is only one way to create an analytic (means differentiable) function that is equal to the normal exponential on the real line. This follows from the fact that any analytic function can be written as a power series.

So the complex exponential is defined by using the power series that we already know for the real exponential.

And when you substitute an imaginary value (ix) into this power series, the real part ends up as the familiar power series for cos x, while the imaginary part is the power series for sin x. *That* is where the formula comes from, and *that* is why e^(i*pi) = -1.

It is convenient to imagine exponentials of imaginary numbers as rotations in the plane. So e^(i*pi) is a rotation of pi radians, which is obviously the same as multiplying by -1. But this is just a device for remembering/understanding the complex exponential -- it is not the basis for its definition.

Answer 3

-1

Answer 4

e^(i*pi) = -1.
Because e^(i*x) is defined to be cos(x) + i*sin(x).
And as, cos(pi) = -1 and sin(pi) = 0, then
e^(i*pi) = -1 + i*0 = -1

Answer 5

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

Take x = pi

Then

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

Answer 6

e^(i*pi) = -1

Answer 7

-1

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