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

Explanation

To find the minimum value of \( |z| + |z - 1| \) where \( z \) is a complex number, we can use the properties of complex numbers and the triangle inequality.

### Step-by-Step Solution:

1. **Understanding the expression**:
We need to minimize the expression \( |z| + |z - 1| \). Here, \( |z| \) is the distance from the origin (0,0) in the complex plane, and \( |z - 1| \) is the distance from the point (1,0).

2. **Using the triangle inequality**:
The triangle inequality states that for any two complex numbers \( z_1 \) and \( z_2 \):
\[ |z_1| + |z_2| \geq |z_1 + z_2| \]
We can apply this property to our expression by letting \( z_1 = z \) and \( z_2 = 1 - z \).

3. **Rearranging the expression**:
We can rewrite \( |z| + |z - 1| \) as:
\[ |z| + |1 - z| \geq |z + (1 - z)| = |1| \]
This shows that:
\[ |z| + |z - 1| \geq 1 \]

4. **Finding when equality holds**:
The equality in the triangle inequality holds when \( z \) lies on the line segment connecting the points 0 and 1 in the complex plane. This means that \( z \) can be any point on the line segment between these two points.

5. **Conclusion**:
The minimum value of \( |z| + |z - 1| \) occurs when \( z \) is on the line segment between 0 and 1, and the minimum value is:
\[ \text{Minimum value} = 1 \]

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

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