Chapter 7.3, Problem 22E

(a)

To determine

To prove: $\left|\overline{z}\right|=\left|z\right|$, where $\overline{z}$ denotes the complex conjugate.

(b)

To determine

To prove: $z\overline{z}={\left|z\right|}^2$, where $\overline{z}$ denotes the complex conjugate.

(c)

To determine

To prove: If $z\ne 0$, then $z^{-1}=\frac{\overline{z}}{{\left|z\right|}^2}$

(d)

To determine

To prove: If $z_2\ne 0$, then $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$

(e)

To determine

To prove: $\left|z_1+z_2\right|\le \left|z_1\right|+\left|z_2\right|$