For any complex number `z`, the minimum value of `|z|+|z-1|`

Options

A. Re(z) `lt0`
B. 1
C. 2
D. 0

Answer

To find the minimum value of \( |z| + |z - 1| \) for any complex number \( z \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Geometric Interpretation**:
The expression \( |z| + |z - 1| \) can be interpreted geometrically as the sum of distances from the point \( z \) to the points \( 0 \) and \( 1 \) in the complex plane.

3. **Applying the Triangle Inequality**:
By the triangle inequality, we know that for any two points \( A \) and \( B \) in a plane, the distance from \( A \) to \( B \) is less than or equal to the sum of the distances from \( A \) to any point \( C \) and from \( C \) to \( B \). In our case:
\[ |z| + |z - 1| \geq |0 - 1| = 1 \]
This means that the minimum value of \( |z| + |z - 1| \) cannot be less than 1.

4. **Finding the Minimum Value**:
To achieve this minimum value of 1, we need to find a point \( z \) such that the distances \( |z| \) and \( |z - 1| \) add up to exactly 1. The point that achieves this is the midpoint between the points \( 0 \) and \( 1 \), which is \( z = \frac{1}{2} \).

5. **Verification**:
Let's calculate \( |z| + |z - 1| \) at \( z = \frac{1}{2} \):
\[ |z| = \left| \frac{1}{2} \right| = \frac{1}{2} \]
\[ |z - 1| = \left| \frac{1}{2} - 1 \right| = \left| -\frac{1}{2} \right| = \frac{1}{2} \]
Therefore,
\[ |z| + |z - 1| = \frac{1}{2} + \frac{1}{2} = 1 \]

6. **Conclusion**:
Thus, the minimum value of \( |z| + |z - 1| \) is indeed \( 1 \).

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

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