Question

For any complex number \(z\), the minimum value of \[ |z|+|z-1| \] is

• A. \(1\)
• B. \(0\)
• C. \(\dfrac{1}{2}\)
• D. \(\dfrac{3}{2}\)

Hint:

For expressions involving moduli of complex numbers, interpret them geometrically as distances in the complex plane and apply triangle inequality whenever possible.

Answer 1

The Correct Option is A

Step 1: Interpret geometrically.
Let \[ z=x+iy \] Then, \[ |z| \] represents the distance of the point \(z\) from the origin \((0,0)\).
Similarly, \[ |z-1| \] represents the distance of the point \(z\) from the point \((1,0)\).
Therefore, \[ |z|+|z-1| \] is the sum of distances from two fixed points: \[ (0,0)\quad \text{and}\quad (1,0) \]

Step 2: Apply the triangle inequality.
Using triangle inequality, \[ |z|+|z-1| \geq |1| \] since \[ z-(z-1)=1 \] Thus, \[ |z|+|z-1|\geq 1 \]

Step 3: Check when equality holds.
Equality in triangle inequality holds when the points are collinear and in the same direction.
Take \[ z=t \] where \[ 0\leq t\leq 1 \] Then, \[ |z|=t \] and \[ |z-1|=1-t \] Hence, \[ |z|+|z-1| = t+(1-t) = 1 \] Thus, the minimum value is attained.

Step 4: Final conclusion.
Therefore, \[ \boxed{1} \]