If two complex numbers `z_(1),z_(2)` are such that
`|z_(1)|=|z_(2)|`,
is it then necessary that `z_(1)=z_(2)` ?

Answer

To determine whether two complex numbers \( z_1 \) and \( z_2 \) are necessarily equal when their magnitudes (or moduli) are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Definition of Complex Numbers**:
Let \( z_1 = x_1 + i y_1 \) and \( z_2 = x_2 + i y_2 \), where \( x_1, y_1, x_2, y_2 \) are real numbers.

2. **Magnitude of Complex Numbers**:
The magnitude (modulus) of a complex number \( z \) is defined as:
\[ |z| = \sqrt{x^2 + y^2} \]
Therefore, we have:
\[ |z_1| = \sqrt{x_1^2 + y_1^2} \quad \text{and} \quad |z_2| = \sqrt{x_2^2 + y_2^2} \]

3. **Given Condition**:
We are given that:
\[ |z_1| = |z_2| \]
This implies:
\[ \sqrt{x_1^2 + y_1^2} = \sqrt{x_2^2 + y_2^2} \]

4. **Squaring Both Sides**:
Squaring both sides of the equation gives us:
\[ x_1^2 + y_1^2 = x_2^2 + y_2^2 \]

5. **Equality of Complex Numbers**:
For two complex numbers \( z_1 \) and \( z_2 \) to be equal, we need:
\[ x_1 = x_2 \quad \text{and} \quad y_1 = y_2 \]
However, from the equation \( x_1^2 + y_1^2 = x_2^2 + y_2^2 \), we cannot conclude that \( x_1 = x_2 \) and \( y_1 = y_2 \).

6. **Counterexample**:
To illustrate this, consider the complex numbers:
\[ z_1 = 3 + 4i \quad \text{and} \quad z_2 = 4 + 3i \]
Here, we have:
\[ |z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ |z_2| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Thus, \( |z_1| = |z_2| \), but clearly \( z_1 \neq z_2 \).

### Conclusion:
Therefore, it is not necessary that \( z_1 = z_2 \) even if \( |z_1| = |z_2| \).