Question

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

• A. $1$
• B. $0$
• C. $\frac{1}{2}$
• D. $\frac{3}{2}$

Answer 1

The correct option is A $1$

$\text{Given.}$

$|z|+|z-1|=|z|+|-(1-z)|$

$=|z|+|(1-z)|$

$\text{Use triangle inequality,}$

$|z_1+z_2|\le \left|z_1\right|+\left|z_2\right|$

$|z+1-2|\le |z|+|(1-z)|$

$|z|+|-(z-1)|\ge 1$

$|z|+|z-1|\ge 1$

Hence, minimum value is 1

obtion $A$ is correct.