Complex Numbers : Properties of Complex Multiplication

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Properties of Complex Multiplication

what you'll learn...

Overview

» Complex Multiplication is closed.
    → $z_{1} \times z_{2} \in ℂ$ $z_1 xx z_2 in CC$

» Complex Multiplication is commutative.
    → $z_{1} \times z_{2} = z_{2} \times z_{1}$ $z_1 xx z_2 = z_2 xx z_1$

» Complex Multiplication is associative.
    → $z_{1} \times (z_{2} \times z_{3}) = (z_{1} \times z_{2}) \times z_{3}$ $z_1 xx (z_2 xx z_3) = (z_1 xx z_2) xx z_3$

» Complex Multiplicative identity is $1$ $1$ .
    → $z_{1} \times 1 = z_{1}$ $z_1 xx 1 = z_1$

» Complex Multiplicative inverse of $z$ $z$ is $\frac{1}{z}$ $1/z$ .
    → $z \times \frac{1}{z} = 1$ $z xx 1/(z) = 1$

» Complex Multiplication is distributive over complex addition.
    → $z_{1} \times (z_{2} + z_{3}) = z_{1} \times z_{2} + z_{1} \times z_{3}$ $z_1 xx (z_2 + z_3) = z_1 xx z_2 + z_1 xx z_3$

» Modulus of product is product of modulus.
    → $| z_{1} \times z_{2} | = | z_{1} | \times | z_{2} |$ $|z_1 xx z_2| = |z_1| xx |z_2|$

» argument of product is sum of arguments of multiplicand and multiplier.
    → $a r g (z_{1} \times z_{2}) = a r g (z_{1}) + a r g (z_{2})$ $arg (z_1 xx z_2) = arg(z_1) + arg(z_2)$

closure law

The word 'closure' means, closed and not open

Given complex numbers $z_{1} = a_{1} + b_{1} i$ $z_1=a_1+b_1i$ and $z_{2} = a_{2} + b_{2} i$ $z_2=a_2+b_2i$ , $a_{1}, a_{2}, b_{1}, b_{2} \in R$ $a_1,a_2,b_1,b_2 in R$ ,

$z_{1} \times z_{2} = a_{1} a_{2} - b_{1} b_{2} + i (a_{1} b_{2} + a_{2} b_{1})$ $z_1 xx z_2 = a_1a_2 -b_1b_2 + i (a_1b_2+a_2b_1)$ ,

It is noted that $a_{1} a_{2} - b_{1} b_{2}$ $a_1a_2 -b_1b_2$ and $a_{1} b_{2} + a_{2} b_{1}$ $a_1b_2+a_2b_1$ are real numbers (by closure law of real-number addition, subtraction, and multiplication) and so, $z_{1} \times z_{2}$ $z_1 xx z_2$ is a complex number.

Consider $z_{1} \div z_{2}$ $z_1 -: z_2$

The complex number division is equivalently multiplication by the conjugate. So, $z_{1} \div z_{2} = z_{1} \times \bar{z_{2}} / {| z_{2} |}^{2}$ $z_1 -: z_2 = z_1 xx bar(z_2) // |z_2|^2$ is a complex number

summary

Closure property of Multiplication and Division: For any given complex numbers $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$ , $z_{1} \frac{\times}{\div} z_{2} \in ℂ$ $z_1 xx / -: z_2 in CC$ .

Product or quotient of two complex number is another complex number - closed

Commutative Law

The word 'commute' means 'to go to and fro between two places on a regular basis'.

Consider the multiplication and division of complex numbers. $z_{3} = z_{1} \times z_{2}$ $z_3 = z_1 xx z_2$ given as $a_{3} + i b_{3} = (a_{1} a_{2} - b_{1} b_{2}) + i (a_{2} b_{1} + a_{1} b_{2})$ $a_3+ib_3 = (a_1a_2-b_1b_2)+i(a_2b_1+a_1b_2)$ , where $a_{1}, a_{2}, b_{1}, b_{2} \in ℝ$ $a_1, a_2, b_1, b_2 in RR$ .
$z_{4} = z_{2} \times z_{1}$ $z_4 = z_2 xx z_1$ $= (a_{2} a_{1} - b_{2} b_{1}) + i (a_{1} b_{2} + a_{2} b_{1})$ $= (a_2a_1-b_2b_1)+i(a_1b_2+a_2b_1)$

It is noted that $z_{3} = z_{4}$ $z_3 = z_4$ as the real and imaginary parts of the two complex numbers are equal by commutative law of real number multiplication.

$z_{1} \times z_{2} = z_{2} \times z_{1}$ $z_1 xx z_2 = z_2 xx z_1$ will have the same

Commutative property is not defined for inverse operations. Division is the inverse of multiplication.

Division is the inverse of multiplication.

The commutative property has to be used in the following way.
$z_{1} \div z_{2}$ $z_1-:z_2$
$= z_{1} \times (\frac{1}{z_{2}})$ $quad quad = z_1 xx (1/z_2)$
$= (\frac{1}{z_{2}}) \times z_{1}$ $quad quad =(1/z_2) xx z_1$ (commutative property of multiplication)

Given for two complex numbers $z_{1}, z_{2}$ $z_1, z_2$ ; $z_{1} \times z_{2} = 3.1 + i 2.9$ $z_1 xx z_2 = 3.1+i2.9$ . Find the value of $z_{2} \times z_{1}$ $z_2 xx z_1$ .

The answer is ' $3.1 + i 2.9$ $3.1+i2.9$ '

summary

Commutative Property of Complex Multiplication: for any complex number $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$
$z_{1} \times z_{2} = z_{2} \times z_{1}$ $z_1 xx z_2 = z_2 xx z_1$

Complex numbers can be swapped in complex multiplication - commutative.

associative law

The word 'Associate' means 'to connect with; to join'.

When three complex numbers are multiplies,
• Person A associates the middle complex number with first complex number and, to the result of that, he multiplies the third complex number $(z_{1} \times z_{2}) \times z_{3}$ $(z_1 xx z_2) xx z_3$
• Person B associates the middle complex number with third complex number and to that she multiplies the first complex number $z_{1} \times (z_{2} \times z_{3})$ $z_1 xx (z_2 xx z_3)$ .

We can work out this in terms of real and imaginary parts of each number. It will prove that $(z_{1} \times z_{2}) \times z_{3}$ $(z_1 xx z_2) xx z_3$ $= z_{1} \times (z_{2} \times z_{3})$ $=z_1 xx (z_2 xx z_3)$

Associative property is not applicable to division. Instead, Division is handled as inverse of multiplication and assiciative property of multiplication is applicable. i.e. For $(z_{1} \div z_{2}) \div z_{3}$ $(z_1 -: z_2) -: z_3$ , it will be handled $(z_{1} \times \frac{1}{z_{2}}) \times \frac{1}{z_{3}}$ $(z_1 xx 1/z_2)xx 1/z_3$ , then associative property of multiplication applies.

Given $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$ , what will be the result of $z_{1} \times (z_{2} \times z_{1})$ $z_1 xx (z_2 xx z_1)$ ?

The answer is ' $z_{1}^{2} \times z_{2}$ $z_1^2 xx z_2$ '.

summary

Associative property of Complex Multiplication: For Any complex number $z_{1}, z_{2}, z_{3} \in ℂ$ $z_1,z_2,z_3 in CC$ ,
$z_{1} \times (z_{2} \times z_{3}) = (z_{1} \times z_{2}) \times z_{3}$ $z_1 xx (z_2 xx z_3) = (z_1 xx z_2) xx z_3$

Order of multiplication can be changed - associative

multiplicative identity

The word "identical" means "exactly alike; same in every detail"

Consider the complex number $1 + i 0$ $1+i0$ . The result of multiplying that with a complex number $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1 + i b_1$ is ' $z_{1}$ $z_1$ '.

The product is identical to the number being multiplied.

Multiplicative identity is 'an element, when multiplied, will result in product identical to the multiplicand'.

Consider $z_{1} = \sqrt{2} + \sqrt{3} i$ $z_1=sqrt(2)+sqrt(3)i$ What will be the product $z_{1} \times (1 + i 0)$ $z_1 xx (1+i0)$ ?

The answer is ' $z_{1}$ $z_1$ '

summary

Multiplicative Identity in Complex Numbers: For any complex number $z_{1} \in ℂ$ $z_1 in CC$ , there exists $1 = 1 + i 0, \in ℂ$ $1 = 1+i0, in CC$ such that
$z_{1} \times 1 = z_{1}$ $z_1 xx 1 = z_1$

$1 = 1 + i 0$ $1=1+i0$ is the multiplicative identity.

multiplicative inverse

The word 'inverse' means 'opposite; converse'.

Consider two complex numbers $z_{1} = a + i b$ $z_1 = a+ i b$ and $z_{2} = \frac{1}{a + i b}$ $z_2 = 1 /(a+ib)$

$z_{1} \times z_{2}$ $z_1 xx z_2$ is $1$ $1$ multiplicative identity.

$z_{2} = \frac{1}{a + i b}$ $z_2 = 1/(a+ib)$
$= \frac{1}{a + i b} \times \frac{a - i b}{a - i b}$ $quad quad quad = 1/(a+ib) xx (a-ib)/(a-ib)$
$= \frac{a - i b}{a^{2} + b^{2}}$ $quad quad quad = (a-ib)/(a^2+b^2)$

$\frac{1}{z}$ $1/z$ or (a-ib)/(a^2+b^2) is the multiplicative inverse.

What is the multiplicative inverse of $2 - i$ $2-i$ ?

The answer is ' $\frac{2 + i}{5}$ $(2+i)/5$ '

summary

Multiplicative Inverse: For any complex number $z = a + i b \in ℂ$ $z = a+ib in CC$ , there exists $\frac{1}{z} = \frac{a - i b}{a^{2} + b^{2}} \in ℂ$ $1/z = (a - i b)/(a^2+b^2) in CC$ such that
$z \times \frac{1}{z} = 1$ $z xx 1/z = 1$

For any complex number $z$ $z$ there exists $\frac{1}{z}$ $1/z$ : multiplicative inverse

distributive property

The word 'distribute' means 'to share; to spread'

Consider three complex numbers $z_{1}, z_{2}, z_{3} \in ℂ$ $z_1, z_2, z_3 in CC$ , and the following

add first $z_{2} + z_{3}$ $z_2+z_3$ and multiply that by $z_{1}$ $z_1$ . That is, $z_{1} \times (z_{2} + z_{3})$ $z_1 xx (z_2 + z_3)$
multiply $z_{1} \times z_{2}$ $z_1 xx z_2$ and $z_{1} \times z_{3}$ $z_1 xx z_3$ and added the results. That is $z_{1} \times z_{2} + z_{1} \times z_{3}$ $z_1 xx z_2 + z_1 xx z_3$
These two results will be the same".

The complex numbers addition and multiplication is considered to be operations on numerical expressions of real numbers.

$z_{1} \times (z_{2} + z_{3})$ $z_1 xx (z_2 + z_3)$
$= z_{1} \times z_{2} + z_{1} \times z_{3}$ $quad quad = z_1 xx z_2 + z_1 xx z_3$

proof:
$z_{1} \times (z_{2} + z_{3})$ $z_1 xx (z_2 + z_3)$
$= (a_{1} + i b_{1}) \times (a_{2} + a_{3} + i (b_{2} + b_{3}))$ $quad quad = (a_1+ib_1) xx (a_2+a_3 + i (b_2+b_3))$
$= a_{1} a_{2} - b_{1} b_{2} + i (a_{1} b_{2} + a_{2} b_{1})$ $quad quad = a_1a_2-b_1b_2 + i(a_1b_2+a_2b_1)$
$+ a_{1} a_{3} - b_{1} b_{3} + i (a_{1} b_{3} + a_{3} b_{1})$ $quad quad quad quad +a_1a_3-b_1b_3 + i(a_1b_3+a_3b_1)$
$= z_{1} \times z_{2} + z_{1} \times z_{3}$ $quad quad = z_1 xx z_2 + z_1 xx z_3$

Given $z_{1} z_{2} = 3 + 4 i$ $z_1z_2= 3+4i$ and $z_{2} z_{3} = 1 - 3 i$ $z_2z_3=1-3i$ what is $(z_{1} + z_{3}) z_{2}$ $(z_1+z_3)z_2$ ?

The answer is ' $4 + i$ $4+i$ '

summary

Distributive Property of Complex Numbers: For any given complex numbers $z_{1}, z_{2}, z_{3} \in ℂ$ $z_1, z_2, z_3 in CC$
$z_{1} \times (z_{2} + z_{3})$ $z_1 xx (z_2 + z_3)$
$= z_{1} \times z_{2} + z_{1} \times z_{3}$ $quad quad = z_1 xx z_2 + z_1 xx z_3$

Product with sum of complex numbers is sum of, products with the complex numbers - distributive.

modulus in multiplication

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . The modulus of product $| z_{1} \times z_{2} |$ $|z_1 xx z_2|$ is ' $| z_{1} | \times | z_{2} |$ $|z_1| xx |z_2|$ '.

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ , the modulus of product
$| z_{1} \times z_{2} |$ $|z_1 xx z_2|$
$= | a_{1} a_{2} - b_{1} b_{2} + i (b_{1} a_{2} + a_{1} b_{2}) |$ $quad quad = |a_1a_2-b_1b_2 + i(b_1a_2+a_1b_2)|$
$= \sqrt{a_{1}^{2} + b_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2}}$ $quad quad = sqrt(a_1^2+b_1^2)sqrt(a_2^2+b_2^2)$
$= | z_{1} | \times | z_{2} |$ $quad quad = |z_1| xx |z_2|$

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ what is the modulus of quotient $| z_{1} \div z_{2} |$ $|z_1 -: z_2|$ ?

The answer is ' $| z_{1} | \div | z_{2} |$ $|z_1| -: |z_2|$ '.

summary

Modulus of Product: For complex numbers $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$

$| z_{1} \times z_{2} | = | z_{1} | \times | z_{2} |$ $|z_1 xx z_2| = |z_1| xx |z_2|$

Modulus of product is product of modulus.

argument in multiplication

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ , the argument of product $arg (z_{1} \times z_{2})$ $text(arg)(z_1 xx z_2)$ is 'text(arg)z_1 + text(arg)z_2'.

$arg (z_{1} \times z_{2})$ $text(arg)(z_1 xx z_2)$
$= arg (r_{1} r_{2} e^{i θ_{1} + θ_{2}})$ $quad quad = text(arg)(r_1r_2 e^(i theta_1 + theta_2))$
$= θ_{1} + θ_{2}$ $quad quad = theta_1 + theta_2$
$= arg z_{1} + arg z_{2}$ $quad quad = text(arg)z_1 + text(arg)z_2$

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ , what is the argument of quotient $arg (z_{1} \div z_{2})$ $text(arg)(z_1 -: z_2)$ ?

The answer is ' $arg z_{1} - arg z_{2}$ $text(arg)z_1 - text(arg)z_2$ '.

summary

Argument of Product: For complex numbers $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$
$arg (z_{1} \times z_{2}) = arg z_{1} + arg z_{2}$ $text(arg)(z_1 xx z_2) = text(arg)z_1 + text(arg)z_2$

argument of product is sum of arguments.

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