## Tuesday, November 28, 2017

### Math Puzzle

Consider the famous formula eiπ = -1. Here i is the complex number, the square root of -1.

This is a special case of the more general

eiθ = cos θ + i sin θ
Thus, not only is eiπ = -1, but ei2π = +1.

So far, so good.

Let's evaluate the following: (ei2π)π. That is, the quantity ei2π raised to the π power.

On the one hand, since ei2π is equal to 1, this is just 1 raised to the π power, which is still just one.

On the other hand, using simple algebra (power times exponent) we can write (ei2π)π= ei2π2. Now using the formula given above, this quantity is cos(2π2) + i sin(2π2), which is most assuredly not equal to one.

What gives? Where did error creep in?

#### 2 comments:

1. One raised to any power is not "still just one".
For fractional powers (p), things get tricky. For example, 1^(1/2) (i.e., the square root of 1), could be 1, or it could be -1. Similarly, 1^(1/4) could be i, or -i, or 1 or -1.
The formula cos(2\pip) + i sin(2\pip) gives one of these solutions in both of these cases [i.e., cos(\pi) + i sin(\pi) = -1 and cos(\pi/2) + i sin(\pi/2) = i] and provides a solution in the case of p=\pi

1. Ahh... I've been found out!