Operations with complex numbers - WordPress.com · Operations with complex numbers Book page 109 - 114 ©cgrahamphysics.com 2016. Equality •Two imaginary numbers are equal if and

Operations with complex numbersBook page 109 - 114

©cgrahamphysics.com 2016

Equality

• Two imaginary numbers are equal if and only if their real and imaginary parts are equal


Addition and subtraction• The sum of two (or more) complex numbers is made up of

the sum of their real parts plus the sum of their imaginary parts (multiplied by ‘i’).

• The difference of two (or more) complex numbers is made up of the difference of their real parts plus the difference of their imaginary parts (multiplied by ‘i’).


Multiplication• Multiplication by a real number:

• To multiply a complex number by a real number use the distributive property

• Multiplication by a complex number:

• To multiply a complex number by a complex number use the distributive property and the fact that 𝑖2 = −1


Example 1

A) z+w

B) 2z – 3w

C) Zw

D) 𝒘𝟐

Given that z = 3 + i and w = 1 – 2i, evaluate


Example 2• Find a complex number z that satisfies

4 − 2𝑖 ∙ 𝑧 = 3𝑧 + 2 − 5𝑖

Solution

• Use equality of two complex numbers

• Let z = a + ib

• 4 − 2𝑖 ∙ (𝑎 + 𝑖𝑏) = 3(𝑎 + 𝑖𝑏) + 2 − 5𝑖

• Expand and collect real and imaginary parts


continued


Example 3

• Find the real values x and y given3 − 2𝑦 𝑥 + 𝑖𝑦 = 12 − 5𝑖

Solution

• Equate terms


continued


Conjugate complex numbers

• Two complex numbers are a conjugate pair if they have equal real parts and opposite sign imaginary parts

• If 𝑧 = 𝑎 + 𝑖𝑏, 𝑡ℎ𝑒𝑛 𝑧∗ = 𝑎 − 𝑖𝑏,

• 𝑤ℎ𝑒𝑟𝑒 𝑧∗ is the conjugate of z


Example 4• Given 𝑧 = 𝑎 + 𝑖𝑏, find

• a) 𝑧 + 𝑧∗ b) 𝑧 − 𝑧∗ c) 𝑧 ∙ 𝑧∗

b)

c)


Conjugate complex numbers have these properties:

When a complex number is multiplied with its conjugate, the result is a real number


Division of complex numbers• Dividing a complex number by a complex number

using several properties studied so far

Multiply by the conjugate of the denominator


Division

In practice it is easier to derive the answer each time as the formula is not easy to beremembered


Example 5• Given that 𝑧1 = 5 + 5𝑖

𝑧2 = 1 + 2𝑖𝑧3 = 3 − 2𝑖

• Find a) 𝑧1

𝑧2and b)

𝑧12

𝑧2∙𝑧3∗


Example 6• Find the complex number z that satisfies

𝑧 + 1

3 + 𝑖=𝑧 − 5𝑖

2𝑖 − 1


continued


• Book page 111 exercise 3E all

• Book page 114 exercise 3F all (1 – 5 as many as you need)


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Operations with complex numbers - WordPress.com · Operations with complex numbers Book page 109 - 114 ©cgrahamphysics.com 2016. Equality •Two imaginary numbers are equal if and