Complex Numbers

MATH 017

Intermediate Algebra

S. Rook

2

Overview

• Section 7.7 in the textbook– Introduction to imaginary numbers– Multiply and divide square roots with

imaginary numbers– Addition and subtraction of complex numbers– Multiplication of complex numbers– Division of complex numbers– Powers of i

Introduction to Imaginary Numbers

4

Introduction to Imaginary Numbers

• Thus far we have discussed numbers exclusively in the real number system

• Consider – Does not exist in the real number system

16

5

Introduction to Imaginary Numbers (Continued)

• Consider if we use the product rule to rewrite as– This step is called “poking out the i”– We know how to evaluate

• Imaginary unit: – Thus, – Any number with an i is called an imaginary

number– Also by definition:

16116

16

i416 1i

12 i

6

Introduction to Imaginary Numbers (Continued)

• A negative under an even root:– Does NOT exist in the real number system– DOES exist in the complex number system

• A negative under an odd root:– ONLY exists in the real number system

7

Introduction to Imaginary Numbers (Example)

Ex 1: Evaluate each root in the complex number system

50,28,25,9

Multiply and Divide Square Roots with Imaginary Numbers

9

Multiply and Divide Square Roots with Imaginary Numbers

• First step is to ALWAYS “poke out the i”WRONG

CORRECT

• After “poking out the i” use the product or quotient rule on the REAL roots– After checking whether the REAL roots can be

simplified of course– Only acceptable to have i in the final answer –

• i2 can be simplified to -1

1111

11111 2 iii

10

Multiply and Divide Square Roots with Imaginary Numbers (Example) Ex 2: Multiply or divide:

10

40,

5

20,245,149,32

Addition and Subtraction of Complex Numbers

12

Definition of Complex Numbers

• Complex Number: a number written in the format a + bi where:– a and b are real numbers– a is the real part– bi is the imaginary part

13

Addition and Subtraction of Complex Numbers

• To add complex numbers– Add the real parts– Add the imaginary parts– The real and imaginary parts cannot be

combined any further

• To subtract 2 complex numbers– Distribute the negative to the second complex

number– Treat as adding complex numbers

14

Addition and Subtraction of Complex Numbers (Example)

Ex 3: Add or subtract the complex numbers – leave the final answer in a + bi format:

iiii 726),39(24

15

Addition and Subtraction of Complex Numbers (Example)

Ex 4: Add or subtract the complex numbers – leave the final answer in a + bi format:

iii 11798),1(4

Multiplication of Complex Numbers

17

Multiplication of Complex Numbers

• To multiply 3i · 2i– Multiply the real numbers first: 6– Multiply the i s: i · i = i2

3i · 2i = 6i2 = -6• Remember that it is only acceptable to leave i in

the final answer

• To multiply complex numbers in general– Use the distributive property or FOIL

18

Multiplication of Complex Numbers (Example)

Ex 5: Multiply the complex numbers – leave the final answer in a + bi format:

iiii 37,52

19

Multiplication of Complex Numbers (Example)

Ex 6: Multiply the complex numbers – leave the final answer in a + bi format:

iiii 52,34

20

Multiplication of Complex Numbers (Example)

Ex 7: Multiply the complex numbers – leave the final answer in a + bi format:

26,37210 iii

Division of Complex Numbers

22

Complex Conjugate

• Consider (3 + i)(3 – i)– What do you notice?

• Complex conjugate: the same complex number with real parts a and imaginary part bi except with the opposite sign– Very similar to conjugates when we discussed

rationalizing– What would be the complex conjugate of (2 – i)?

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Division of Complex Numbers

• Goal is to write the quotient of complex numbers in the format a + bi– Multiply the numerator and denominator by the

complex conjugate of the denominator (dealing with an expression)

– The numerator simplifies to a complex number– The denominator simplifies to a single real number– Divide the denominator into each part of the

numerator and write the result in a + bi format

24

Division of Complex Numbers

Ex 8: Divide the complex numbers:

i33

5

25

Division of Complex Numbers

Ex 9: Divide the complex numbers:

i

i

74

62

26

Division of Complex Numbers

Ex 10: Divide the complex numbers:

i

i

1

5

27

Division of Complex Numbers

Ex 11: Divide the complex numbers:

i

i

2

47

Powers of i

29

Powers of i

• So far we have discussed two powers of i:i

i2 = -1

• We can use these to obtain subsequent powers of i:

i3 = i2 · i = -i

i4 = i3 · i = -i · i = -i2 = 1

30

Powers of i (Continued)

• Of course we don’t have to stop there:i5 = i4 · i = 1 · i = i

i6 = i5 · i = i · i = i2 = -1

i7 = i6 · i = -1 · i = -i

i8 = i7 · i = -i · i = -i2 = 1

:– Notice the pattern?

31

Powers of i (Continued)

• Thus, there are only 4 distinct powers of i i, i2 = -1, i3 = -I, i4 = 1

• Knowing this, we can derive ANY power of i– Divide the power by 4

• The quotient represents how many complete cycles we would need to go through

• The remainder represents the power of i in the next cycle0 i4

1 i

2 i2

3 i3

32

Powers of i (Example)

Ex 12: Evaluate the following powers of i – the only acceptable answers are i, -1, -i, or 1

13572543117 ,,,, iiiii

33

Summary

• After studying these slides, you should know how to do the following:– Understand the concept of imaginary numbers– Multiply and divide square roots with

imaginary numbers– Add, subtract, multiply, and divide complex

numbers• Understand the form a + bi

– Calculate any power of i