Question

For any complex number $Z$, the minimum value of $\left|Z\right|+|Z-1|$ is

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is B

$1$

Explanations for the correct option.

Simplify the expression

$\begin{array}{rcl}\left|Z\right|+|Z-1|& =& \left|Z\right|+|-\left(1-Z\right)|\\ & =& \left|Z\right|+|1-Z|\mathbf{}\left[\because \left|-Z\right|=\left|Z\right|\right]\\ & \ge & \left|Z+\left(1-Z\right)\right|\left[\because \left|Z_1\right|+\left|Z_2\right|\ge \left|Z_1+Z_2\right|\right]\\ & \ge & \left|1\right|\\ & \ge & 1\end{array}$

Hence, the minimum value is $1$.

Therefore, the correct option is (B).