Chapter 6.6, Problem 98E

To determine

Whether the statement is true or false. Justify the answer.

Answer to Problem 98E

The given statement is true.

Explanation of Solution

Given statement:

The product of two complex numbers is zero ony when the modulus of one (or both) of the complex number is zero.

Let $z_1=r_1(\cos {\theta }_1+i\sin {\theta }_1)$ and $z_2=r_2(\cos {\theta }_2+i\sin {\theta }_2)$ are two complex numbers, then their product is

  $z_1{z}_2=r_1{r}_2\left(\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right)$ .

If the product is zero, then

  $\begin{array}{l}{r}_1{r}_2\left(\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right)=0\\ \Rightarrow r_1{r}_2=0\\ \Rightarrow r_1=0\text{or }{r}_2=0\end{array}$

Therefore, the modulus of one or both of the complex number is zero. Thus, the given statement is true.