====== 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. 

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