Thread: Complex numbers help, please!!!

Im doing some studying and im having a little problem. Can someone explain this problem to me or give a hint please:

Let a and b be complex numbers with |a|=1

a. The direct congruence of the complex plane T(z) = az + b is a translation if a =1, and a rotation centered at b/(1-a) of magnitude Ang(a) if a does not equal 1.

Originally Posted by GreenandGold

Im doing some studying and im having a little problem. Can someone explain this problem to me or give a hint please:

Let a and b be complex numbers with |a|=1

a. The direct congruence of the complex plane T(z) = az + b is a translation if a =1, and a rotation centered at b/(1-a) of magnitude Ang(a) if a does not equal 1.

If |a|=1 then the map z → az is a rotation about the origin through an angle Arg(a). To get a rotation through Arg(a) about the point b/(1-a), write z as z = b/(1-a) + (z - b/(1-a)), and multiply the part in bold parentheses by a. Then check that the result is equal to az+b.