Question

If z is a complex number, then ______.

Options

• |z²| > |z|²

• |z²| = |z|²

• |z²| < |z|²

• |z²| ≥ |z|² 

Answer 1

If z is a complex number, then |z²| = |z|².

Explanation:

Let z = x + yi

|z| = |z + yi| and |z|² = |x + yi|²

⇒ |z|² = x² + y²    ......(i)

Now z² = x² + y²i² + 2xyi

z² = x² – y² + 2xyi

|z|² = `sqrt((x^2 - y^2)^2 + (xy)^2)`

= `sqrt(x^4 + y^4 - 2x^2 y^2 + 4x^2 y^2)`

= `sqrt(x^4 + y^4 + 2x^2 y^2)`

= `sqrt((x^2 + y^2)^2`

So |z²| = x² + y² = |z|²

So |z²| = |z|²