Q.If $|z_1|=|z_2|$, is it necessary that $z_1=z_2$?

Concept: Modulus vs. equality of complex numbers

The modulus $|z|$ measures only the distance from the origin in the complex plane. Two complex numbers can lie at the same distance from the origin yet point in entirely different directions.

Reasoning:

1. For any complex number $z = a + ib$, the modulus is $|z| = \sqrt{a^2 + b^2}$.

2. Consider $z_1 = 1$ and $z_2 = i$. Then $|z_1| = 1$ and $|z_2| = 1$, so $|z_1| = |z_2|$. …

Why modulus alone doesn't determine a complex number

The modulus $|z|$ of a complex number $z = x + iy$ measures its distance from the origin in the complex plane: $|z| = \sqrt{x^2 + y^2}$. This is a single real number that captures how far $z$ is from zero, but it tells us nothing about the direction in which $z$ lies.

A complex number has two degrees of freedom—its real part and its imaginary part. The modulus collapses both pieces of information into one number (distance), so we lose the angular information. Many different complex numbers can share the same distance from the origin while pointing in completely different directions.

The geometric picture

Think of the complex plane as a coordinate system where the horizontal axis represents real parts and the vertical axis represents imaginary parts. The condition $|z_1| = |z_2|$ says that $z_1$ and $z_2$ are equidistant from the origin. Geometrically, this means both points lie on the same circle centered at the origin.

A circle contains infinitely many points. Knowing that two complex numbers lie on the same circle tells us they share a common distance from the origin, but they could be at entirely different positions along that circle.

Concrete counterexamples

Let's construct explicit examples where $|z_1| = |z_2|$ but $z_1 \neq z_2$:

1. Simple example: Take $z_1 = 1$ and $z_2 = i$.

   • $|z_1| = |1| = 1$
   • $|z_2| = |i| = 1$
   • Clearly $1 \neq i$, yet their moduli are equal.

2. Another example: Let $z_1 = 3 + 4i$ and $z_2 = 5$.

   • $|z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$
   • $|z_2| = |5| = 5$
   • Again, $3 + 4i \neq 5$ despite equal moduli. …