Abstract Algebra: Theory and Applications · Problem 19

Permutation Groups — Thomas Judson

Mathematics — Abstract Algebra · complex-numbers-properties

Problem

Prove each of the following statements. ((a))

|z| = |\bar{z}|

((b))

z\bar{z} = |z|^2

((c))

z^{-1} = \bar{z}/|z|^2

((d))

|z+w| \leq |z|+|w|

((e))

|z-w| \geq \big||z|-|w|\big|

((f))

|zw| = |z||w|