Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

2. |z|² = |z²|

Solution:

Let z = x + yi

|z| = |x + yi| and |z|² = |x + yi|²

⇒ |z|² = x² + y² .....(i)

Now, z² = x² + y²i² + 2xyi

z² = x² - y² + 2xyi

$|\text{z}^2|=\sqrt{(\text{x}^2-\text{y}^2)^2+(2\text{xy})^2}$

$=\sqrt{\text{x}^4+\text{y}^4-2\text{x}^2\text{y}^2+4\text{x}^2\text{y}^2}$

$=\sqrt{\text{x}^4+\text{y}^4+2\text{x}^2\text{y}^2}$

$=\sqrt{(\text{x}^2+\text{y}^2)^2}$

So, $|\text{z}|^2=\text{x}^2+\text{y}^2=|\text{z}|^2$

So, $|\text{z}|^2=|\text{z}^2|$