If z is a complex number, then the minimum value of `|z|+|z-1|` is

Options

A. 1
B. 0
C. ` (1)/(2)`
D. None of these

Answer

To find the minimum value of the expression \( |z| + |z - 1| \) where \( z \) is a complex number, we can use the properties of complex numbers and the triangle inequality. Here’s a step-by-step solution:

### Step 1: Understand the expression
We need to minimize the expression \( |z| + |z - 1| \). Here, \( |z| \) represents the distance of the point \( z \) from the origin (0,0) in the complex plane, and \( |z - 1| \) represents the distance of the point \( z \) from the point (1,0).

### Step 2: Geometric interpretation
The expression \( |z| + |z - 1| \) can be interpreted geometrically. It represents the sum of the distances from the point \( z \) to the origin and from \( z \) to the point (1,0).

### Step 3: Apply the triangle inequality
According to the triangle inequality, for any points \( A \), \( B \), and \( C \) in the plane, the following holds:
\[ |A - B| + |B - C| \geq |A - C| \]
In our case, let \( A \) be the origin (0,0), \( B \) be the point \( z \), and \( C \) be the point (1,0). Thus, we have:
\[ |z| + |z - 1| \geq |0 - 1| = 1 \]

### Step 4: Finding the minimum value
The minimum value of \( |z| + |z - 1| \) occurs when \( z \) lies on the line segment connecting the origin (0,0) and the point (1,0). The closest point on this line segment to both the origin and (1,0) is the midpoint, which is \( z = \frac{1}{2} \).

### Step 5: Calculate the minimum value
Now, substituting \( z = \frac{1}{2} \) into the expression:
\[ |z| + |z - 1| = \left| \frac{1}{2} \right| + \left| \frac{1}{2} - 1 \right| = \frac{1}{2} + \left| -\frac{1}{2} \right| = \frac{1}{2} + \frac{1}{2} = 1 \]

Thus, the minimum value of \( |z| + |z - 1| \) is \( 1 \).

### Final Answer
The minimum value of \( |z| + |z - 1| \) is \( 1 \).

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