For any complex number z, maximum value of `|z|-|z-1|` is

Options

A. 0
B. 1
C. `1//2`
D. `3//2`

Answer

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

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

### Step 2: Apply the triangle inequality
We can use the triangle inequality which states that for any two complex numbers \( z_1 \) and \( z_2 \):
\[ |z_1 - z_2| \geq ||z_1| - |z_2|| \]
In our case, let \( z_1 = z \) and \( z_2 = 1 \). Thus, we can write:
\[ |z - 1| \geq ||z| - |1|| \]

### Step 3: Substitute values
Since \( |1| = 1 \), we can rewrite the inequality as:
\[ |z - 1| \geq ||z| - 1| \]

### Step 4: Rearranging the inequality
From the above inequality, we can express \( |z| - |z - 1| \) as:
\[ |z| - |z - 1| \leq |1| \]
This implies:
\[ |z| - |z - 1| \leq 1 \]

### Step 5: Conclusion
The maximum value of \( |z| - |z - 1| \) is therefore \( 1 \).

Thus, the answer is:
\[ \text{Maximum value of } |z| - |z - 1| \text{ is } 1. \]