Question

If |z₁| = |z₂|, is it necessary that z₁ = z₂?

Answer 1

Given |z₁| = |z₂|

If |z₁| = |z₂| then z₁ and z₂ are at the same distance from origin.

But if arg (z₁) ≠ arg (z₂) then z₁ and z₂ are different.

So, if |z₁| = |z₂|, then it is not necessary that z₁ = z₂.

For example: z₁ = 3 + 4i and z₂ = 4 + 3i

Here |z₁| = |z₂| = 5 but z₁ ≠ z₂.