MULTIPLICATION DIVISION AND PROPERTIES - PUNJAB-BOARD / 10TH-CLASS / MATHEMATICS / 1-10-MATHS-FULL-10TH-CLASS-NBF-2024 / COMPLEX-NUMBERS / MULTIPLICATION-DIVISION-AND-PROPERTIES

Example 6

Express $\frac{3 + 4 i}{5 - 7 i}$ in the form of $x + iy$ or $x + yi$ .

Solution

$\frac{3 + 4 i}{5 - 7 i} = \frac{3 + 4 i}{5 - 7 i} \times \frac{5 + 7 i}{5 + 7 i} = \frac{15 + 21 i + 20 i + 28 i^{2}}{(5)^{2} - (7 i)^{2}}$

$= \frac{15 - 28 + (21 + 20) i}{25 + 49}$

$= \frac{- 13 + 41 i}{74}$

$= \frac{- 13}{74} + \frac{41}{74} i$

1.2.6 Properties of Complex Numbers

📊 Data TableScroll ↔

Properties of Complex Numbers under Addition and Multiplication
Property	Addition	Multiplication
Closure Property	For any two complex numbers $z_1$ and $z_2$ , the sum $z_1 + z_2$ is also a complex number.	For any two complex numbers $z_1$ and $z_2$ , the product $z_1z_2$ is also a complex number.
The Commutative Property	For any two complex numbers $z_1$ and $z_2$ , $z_1 + z_2 = z_2 + z_1$	For any two complex numbers $z_1$ and $z_2$ , $z_1z_2 = z_2z_1$
The Associative Property	For any three complex numbers $z_1, z_2$ and $z_3$ , $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$	For any three complex numbers $z_1, z_2$ and $z_3$ , $(z_1z_2)z_3 = z_1(z_2z_3)$
The Additive / Multiplicative Identity	There exists a complex number $0 = 0 + 0i$ such that, for every complex number $z$ , $z + 0 = 0 + z = z$ . The complex number $0 = 0 + 0i$ is known as additive identity.	There exists a complex number $1 = 1 + 0i$ such that, for every complex number $z$ , $z \times 1 = 1 \times z = z$ . The complex number $1 = 1 + 0i$ is known as multiplicative identity.
The Additive / Multiplicative Inverse	For every complex number $z$ there exists a complex number $- z$ such that, $z + (-z) = (-z) + z = 0$ . $- z$ is called the additive inverse of $z$ .	For any non- zero complex number $z$ , there exists a complex number $w$ such that, $zw = wz = 1$ . $w$ is called the multiplicative inverse of $z$ and it is denoted by $z^{- 1}$ .
Distributive Property of Multiplication Over Addition	For any three complex numbers $z_{1},z_{2}$ , and $z_{3}$ : $z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}$ and $(z_{1} + z_{2})z_{3} = z_{1}z_{3} + z_{2}z_{3}$

Example 7

If $z = - 7 - 4i$ , then show that $z + 0 = z$ .

Solution

$z + 0 = (-7 - 4i) + (0 + 0i)$

$= -7 - 4i + 0 + 0i$

$= -7 - 4i = z$

Shows $z + 0 = z$ (0 is the additive identity)

1.2 Multiplication, Division, and Properties of Complex Numbers

Example 6

Solution

1.2.6 Properties of Complex Numbers

Example 7

Solution