Why does the absolute value of every real positive number raised to i equal 1?

Question

Why does the absolute value of every real positive number raised to i equal 1?

Answer 1

Yes, it is true! Because by definition, the absolute value of any number A is given as:

|A| = SQRT(A A*)

where A is the original number, whereas A* its complex conjugate. For example, if A = Re  + i Im, then A* = Re - i Im.

Next, consider a real number R. Raise it to the power of i. That is:

A = Rⁱ

The complex conjugate of this number would be:

A* = R^-i

Now the absolute value of this would be:

|A| = SQRT(A A*) = SQRT(Rⁱ R^-i) = SQRT(R⁰) = SQRT(1) = 1.

This is because any number raised to the power of 0 is 1 by definition. I could prove that too if you want, but I dont think that is necessary, unless you want me to prove. 

Answer 2

If e^i(pi) = -1  then (e^pi)ⁱ = -1.  Excuse awkward notation, which I use because the symbol for pi looks too much like small N.)

 

∴|e^i(pi)| = 1  shows that the absolute value of a particular real number, e^pi, when raised to the power i is indeed 1.

 

But is this true for every real positive number?

Answer 3

It does not.