Question 1, Exercise 1.2

Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

If ${{z}_{1}}=2+i$and ${{z}_{2}}=1-i$, then verify commutative property w.r.t. addition and multiplication.

Given ${{z}_{1}}=2+i$, ${{z}_{2}}=1-i$. First, we prove commutative property under addition, that is, $$z_1+z_2=z_2+z_1.$$ We take $$\begin{align}z_1+z_2&=(2+i)+(1-i)\\ &=3 \ldots (i) \end{align}$$ Now $$\begin{align} z_2+z_1&=(1-i)+(2+i)\\ &=3 \ldots (ii)\end{align}$$ From (i) and (ii), we get required result.

Now, we prove commutative property under multiplication, that is, $$z_1 z_2=z_2 z_1.$$ $$\begin{align}z_1 z_2 &=(2+i)\cdot(1-i) \\ &=(2+1)+(1-2)i\\ &=3-i \ldots (1) \end{align}$$ Also $$\begin{align}z_2 z_1 &=(1-i)\cdot (2+i)\\ &=(2+1)+(1-2)i\\ &=3-i \ldots (2)\end{align}$$ From (1) and (2), we get required result.

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