# Complex number calculation

1. Nov 28, 2012

### cummings12332

1. The problem statement, all variables and given/known data
Evaluate {(1+i)^(1-i)} and describe the set{1^x} when x is a real number, distinguish between the cases when x is rational and when x is rational. for now considering the complex number.

2. The attempt at a solution
i dont know how to start with,for firest part i just write it into e^((1-i)log(1+i)) then get the number with e to the power which inculding i , and the secound part , for 1=e^(i2npi) then 1^x is e^(2inxpi) then how to consider the case for rational and irrational here？？？？？

2. Nov 28, 2012

### Staff: Mentor

start by breaking the factor into two factors (1+i)^1 * (1+i)^(-i)

3. Nov 28, 2012

### cummings12332

i can do the first part now, many thanks ,but i dont understand the secound part of the question.should it be if 1=exp(2ikpi) then 1^x= exp(2ikpix) then consider the rational and irrational case on this form. but whats the differences , i have no idea

Last edited: Nov 28, 2012
4. Nov 28, 2012

### haruspex

Yes. Think about whether different values for k can produce the same value.

5. Nov 29, 2012

### cummings12332

if rational, then 1^(p/q) is the q complex roots of 1, if x is irrational then 1^x=exp(2ikpix) then k can be chosen infinitely many values then there are infinite points

6. Nov 29, 2012

That's it.