for any point, z, whose size is one, find the length of z^2

Question

for any point, z, whose size is one, find the length of z^2

Answer 1

A complex number, a+bi, can be thought of as the vector, v, from (0,0) to (a,b).  The "length" (magnitude) of the complex number is the length of the vector v.  So, by the Pythagorean Theorem, the magnitude of    

z = a+bi is √(a² + b²) = 1.

 

Squaring both sides, we have a² + b² = 1

 

z² = (a+bi)² = a²+ 2abi - b²

 

                  = (a²-b²) + 2abi 

 

So, the length of z² is √[(a²-b²)²+(2ab)²]

 

                            = √[a⁴-2a²b²+b⁴+4a²b²]

 

                            = √[a⁴+2a²b²+b⁴] 

 

                            = √(a²+b²)² = √(1)² = 1