Complex Numbers : Multiplication, Conjugate, & Division in Complex numbers

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Multiplication, Conjugate, & Division in Complex numbers

what you'll learn...

Overview

Complex number multiplication

» $(a + i b) \times (c + i d)$ $(a+ib) xx (c+id)$
by associative, commutative, distributive laws of real numbers, considering $i$ $i$ as a variable, and applying $i^{2} = - 1$ $i^2=-1$
    → $= (a c - b d) + i (c b + a d)$ $=(ac-bd) + i(cb+ad)$

Complex Number Conjugate

» Conjuage of $a + i b$ $a+ib$
Conjugate means "coupled or related". Conjugate of a complex number makes the number real by addition or multiplication.

    → $= \bar{a + i b}$ $=bar(a+ib)$
    → $= a - i b$ $=a-ib$

Complex Number Division

» $(a + i b) \div (c + i d)$ $(a+ib) -: (c+id)$
by multiplicative identity, multiplicative inverse, distributive properties.

    → $= \frac{a + i b}{c + i d}$ $= (a+ib)/(c+id)$ $\times \frac{c - i d}{c - i d}$ $xx (c-id)/(c-id)$

    → $= \frac{(a + i b) \times (c - i d)}{c^{2} + d^{2}}$ $= ((a+ib)xx(c-id)) / (c^2+d^2)$

multiplication

Consider two complex numbers $z_{1} = a_{1} + i b_{1}$ $z_1=a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . Then $z_{1} \times z_{2}$ $z_1 xx z_2$ is ' $(a_{1} a_{2} - b_{1} b_{2}) + i (a_{2} b_{1} + a_{1} b_{2})$ $(a_1a_2-b_1b_2) + i (a_2b_1+a_1b_2)$ '. This is from the associative and distributive laws of real numbers extended to numbers with $\sqrt{- 1}$ $sqrt(-1)$ .

$z_{1} = a_{1} + i b_{1}$ $z_1=a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . What is $z_{1} \times z_{2}$ $z_1 xx z_2$ ?
Solution :
$z_{3} = z_{1} \times z_{2}$ $z_3 = z_1 xx z_2$
$= (a_{1} + i b_{1}) \times (a_{2} + i b_{2})$ $quad quad = (a_1+ib_1)xx(a_2+ib_2)$
$= a_{1} \times (a_{2} + i b_{2})$ $quad quad = a_1 xx(a_2+ib_2)$
$+ i b_{1} \times (a_{2} + i b_{2})$ $quad quad quad quad +i b_1xx (a_2+ib_2)$
$= a_{1} a_{2} + i a_{1} b_{2} + i b_{1} a_{2}$ $quad quad = a_1a_2+ia_1b_2+ib_1a_2$
$+ i^{2} b_{1} b_{2}$ $quad quad quad quad +i^2b_1b_2$
$= (a_{1} a_{2} - b_{1} b_{2})$ $quad quad = (a_1a_2-b_1b_2)$
$+ i (b_{1} a_{2} + a_{1} b_{2})$ $quad quad quad quad + i(b_1a_2+a_1b_2)$

Given $z_{1} = 1 + 2 i$ $z_1=1+2i$ and $z_{2} = 3 - i$ $z_2=3-i$ what is $z_{1} \times z_{2}$ $z_1 xx z_2$ ?

The answer is ' $5 + 5 i$ $5+5i$ '

summary

Multiplication of two complex numbers : For any complex number $z_{1} = a_{1} + i b_{1} \in ℂ$ $z_1=a_1+ib_1 in CC$ and $z_{2} = a_{2} + i b_{2} \in ℂ$ $z_2 = a_2+ib_2 in CC$
$z_{1} \times z_{2} = (a_{1} a_{2} - b_{1} b_{2})$ $z_1 xx z_2 = (a_1a_2-b_1b_2)$
$+ i (a_{2} b_{1} + a_{1} b_{2})$ $quad quad quad quad quad +i(a_2b_1+a_1b_2)$

Multiplication of two complex numbers follows numerical expression laws and properties with $\sqrt{- 1}$ $sqrt(-1)$ handled as per the property ${(\sqrt{- 1})}^{2} = - 1$ $(sqrt(-1))^2 = -1$

conjugate

In numerical expressions or algebraic expressions, we can manipulate the expressions without modifying the value of the expression
• add and subtract : eg $3 = 3 + 1 - 1$ $3 = 3+1-1$
• multiply and divide : eg : $3 = 3 \times \frac{2}{2}$ $3 = 3 xx 2/2$
• etc.

. These manipulations help to arrive at a different form of expressions or help to solve.

In case of complex numbers, $a + i b$ $a+ib$ , one modification stands out "convert the complex number to real number". This can be achieved either by addition or multiplication with the number $a - i b$ $a-ib$ .
• addition $(a + i b) + (a - i b) = 2 a$ $(a+ib) + (a-ib) = 2a$
• multiplication $(a + i b) \times (a - i b) = a^{2} + b^{2}$ $(a+ib) xx (a-ib) = a^2+b^2$

For a given complex number $z = a + i b$ $z= a+i b$ , the connected number that give a real number on multiplication is $a - i b$ $a-ib$ . It is named as conjugate of z and represented as $\bar{z}$ $bar(z)$ or $\bar{a + i b} = a - i b$ $bar(a+ib) = a-ib$

The word 'conjugate' means 'coupled; joined ; related in reciprocal or complementary'

Find $\bar{1 - 3 i}$ $bar(1-3i)$ .

summary

Conjugate of a Complex NumberFor a complex number $z = a + i b \in ℂ$ $z=a+ib in CC$ the conjugate of $z$ $z$ is given as $\bar{z} = a - i b$ $bar(z) = a-ib$ .

Conjugate of a complex number is the number with the same real part and negative of imaginary part.

division

Consider two complex numbers $z_{1} = a_{1} + i b_{1}$ $z_1=a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . Then, $\frac{z_{1}}{z_{2}}$ $z_1 / z_2$ in $a + b i$ $a+bi$ form is ' $\frac{(a_{1} a_{2} + b_{1} \times b_{2}) + i (a_{2} b_{1} - a_{1} b_{2})}{a_{2}^{2} + b_{2}^{2}}$ $((a_1a_2+b_1xxb_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)$ '

$z_{1} = a_{1} + i b_{1}$ $z_1=a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . What is $\frac{z_{1}}{z_{2}}$ $z_1 / z_2$ ?
Solution :
$z_{3} = \frac{z_{1}}{z_{2}}$ $z_3 = z_1 / z_2$
$= \frac{a_{1} + i b_{1}}{a_{2} + i b_{2}}$ $quad quad = (a_1+ib_1)/(a_2+ib_2)$
$= \frac{a_{1} + i b_{1}}{a_{2} + i b_{2}} \times \frac{a_{2} - i b_{2}}{a_{2} - i b_{2}}$ $quad quad = (a_1+ib_1)/(a_2+ib_2) xx (a_2-ib_2)/(a_2-ib_2)$
$= \frac{(a_{1} + i b_{1}) \times (a_{2} - i b_{2})}{(a_{2} + i b_{2}) \times (a_{2} - i b_{2})}$ $quad quad = ((a_1+ib_1)xx (a_2-ib_2))/((a_2+ib_2) xx (a_2-ib_2))$
$= \frac{(a_{1} a_{2} + b_{1} b_{2}) + i (a_{2} b_{1} - a_{1} b_{2})}{a_{2}^{2} + b_{2}^{2}}$ $quad quad = ((a_1a_2+b_1b_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)$

Given $z_{1} = 10 + 5 i$ $z_1 = 10+5i$ and $z_{2} = 3 - 4 i$ $z_2=3-4i$ then $\frac{z_{1}}{z_{2}}$ $z_1/z_2$ is ' $\frac{10 + 11 i}{5}$ $(10+11i)/5$ '.

summary

Division of two complex numbers : For any complex number $z_{1} = a_{1} + i b_{1} \in ℂ$ $z_1=a_1+ib_1 in CC$ and $z_{2} = a_{2} + i b_{2} \in ℂ$ $z_2 = a_2+ib_2 in CC$
$\frac{z_{1}}{z_{2}} =$ $z_1 / z_2 =$ $\frac{(a_{1} a_{2} + b_{1} b_{2}) + i (a_{2} b_{1} - a_{1} b_{2})}{a_{2}^{2} + b_{2}^{2}}$ $((a_1a_2+b_1b_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)$

Division of two complex numbers follows numerical expression laws and properties with $i$ $i$ handled as ${(\sqrt{- 1})}^{2} = - 1$ $(sqrt(-1))^2 = -1$ to arrive at $a + b i$ $a+bi$ form.

Outline

The outline of material to learn "complex numbers" is as follows.

Note : Click here for detailed overview of Complex-Numbers

→ Complex Numbers in Number System

→ Representation of Complex Number (incomplete)

→ Euler's Formula

→ Generic Form of Complex Numbers

→ Argand Plane & Polar form

→ Complex Number Arithmetic Applications

→ Understanding Complex Artithmetics

→ Addition & Subtraction

→ Multiplication, Conjugate, & Division

→ Exponents & Roots

→ Properties of Addition

→ Properties of Multiplication

→ Properties of Conjugate

→ Algebraic Identities