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

Answer

To determine whether it is necessary that \( z_1 = z_2 \) given that \( |z_1| = |z_2| \), we can analyze the situation step by step.

### Step 1: Understanding the Modulus of Complex Numbers
The modulus of a complex number \( z = a + bi \) (where \( a \) and \( b \) are real numbers) is defined as:
\[ |z| = \sqrt{a^2 + b^2} \]
This represents the distance of the complex number from the origin in the complex plane.

### Step 2: Given Condition
We are given that:
\[ |z_1| = |z_2| \]
This means that both complex numbers \( z_1 \) and \( z_2 \) are at the same distance from the origin.

### Step 3: Example of Two Complex Numbers
Let’s consider two specific complex numbers:
- Let \( z_1 = 3 + 4i \)
- Let \( z_2 = 4 + 3i \)

### Step 4: Calculate the Modulus of \( z_1 \)
Calculating the modulus of \( z_1 \):
\[ |z_1| = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Calculate the Modulus of \( z_2 \)
Calculating the modulus of \( z_2 \):
\[ |z_2| = |4 + 3i| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 6: Conclusion from the Example
From the calculations, we see that:
\[ |z_1| = 5 \quad \text{and} \quad |z_2| = 5 \]
Thus, \( |z_1| = |z_2| \) holds true. However, we also observe that:
\[ z_1 = 3 + 4i \quad \text{and} \quad z_2 = 4 + 3i \]
Clearly, \( z_1 \neq z_2 \).

### Final Conclusion
Therefore, it is not necessary that \( z_1 = z_2 \) even if \( |z_1| = |z_2| \). The condition only implies that they are at the same distance from the origin but does not guarantee equality.