# Question 2, Exercise 1.2

Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

### Question 2

Use the algebraic properties of complex numbers to prove that $$ \left(z_{1} z_{2}\right)\left(z_{3} z_{4}\right)=\left(z_{1} z_{3}\right)\left(z_{2} z_{4}\right)=z_{3}\left(z_{1} z_{2}\right) z_{4} $$

**Solution.**

\begin{align} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \quad \text{Multiplicative assocative law}\\ =&z_1\left(z_2 (z_3 z_4) \right) \quad \because\,\, z_5=z_3 z_4 \\ =&z_1 \left((z_2 z_3) z_4 \right) \quad \text{Multiplicative assocative law}\\ =&z_1 \left((z_3 z_2) z_4 \right) \quad \text{Multiplicative comutative law}\\ =&z_1 \left(z_3 (z_2 z_4) \right) \quad \text{Multiplicative assocative law}\\ =&(z_1 z_3) (z_2 z_4) \quad \text{Multiplicative assocative law} \end{align} That is, we have proved $$(z_1 z_2)(z_3 z_4)=(z_1 z_3) (z_2 z_4) ... (i)$$ Now \begin{align} &(z_1 z_3) (z_2 z_4) \\ =&(z_3 z_1) (z_2 z_4)\quad \text{Multiplicative commutative law} \\ =&z_3 \left(z_1 (z_2 z_4)\right)\quad \text{Multiplicative associative law} \\ &z_3 \left((z_1 (z_2) z_4\right)\quad \text{Multiplicative associative law} \\ &z_3 (z_1 z_2) z_4 \quad \text{Multiplicative associative law} \end{align} That is, we have proved $$(z_1 z_3) (z_2 z_4)=z_3 (z_1 z_2) z_4 ... (ii)$$ From (i) and (ii), we have the required result.

**Remark:** For any three complex numbers $z_1$, $z_2$ and $z_3$, we have
$$z_1 (z_2 z_3) = (z_1 z_2)z_3 = z_1 z_2 z_3.$$
Logically, z_1 z_2 z_3 has no meaning as three number cannot be multiplies simultanously, but associate law tells us that the order in which we multiply three complex numbers doesn't matter; we will always end up with the same product. This property ensures consistency and helps simplify calculations involving complex numbers.

### Go to