Complex Numbers

MATH 018

Combined Algebra

S. Rook

2

Overview

• Section 10.7 in the textbook:– Introduction to imaginary numbers– Multiply and divide square roots with negative

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

3

Introduction to Imaginary Numbers

44

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– However it DOES exist in the complex number system

• Consider if we use the product rule to rewrite as is commonly encountered in the complex number

system and is called the imaginary unit or i

– Rewriting is called “poking out the i”

16

16116

1

16116

55

Introduction to Imaginary Numbers (Continued)

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

number– Also by definition:

11616 1i

12 i

66

Introduction to Imaginary Numbers (Continued)

• In summary:– A negative under an even root index

(e.g. ):• Does NOT exist in the real number system• DOES exist in the complex number system

– A negative under an odd root index (e.g. )

• ONLY exists in the real number system

4

5 32

Introduction to Imaginary Numbers (Example)

Ex 1: Simplify in either the real or complex number system:

a)

b)

c) 7

32

3 48

25

8

Multiply and Divide Square Roots with Negative Numbers

99

Multiply and Divide Square Roots with Negative Numbers

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

Product/Quotient rule for even roots ONLY applies to positive numbers under the radical!

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

Multiply and Divide Square Roots with Negative Numbers (Example)

Ex 2: Simplify in the complex number system:

a)

b)

c) 10

168

212

3

18

11

Adding and Subtracting Complex Numbers

1212

Definition of Complex Numbers

• We have discussed the complex number system in relation to the imaginary unit i

• Complex Number: a number written in the standard format a + bi where:a and b are real numbers

a is the real part

bi is the imaginary part

• Special cases:– If b = 0, the number is real (Chapters 1 – 10.6)– If a = 0, the number is imaginary (10.7)– What are some examples of each?

1313

Adding and Subtracting Complex Numbers

• Recall the definition of like terms:– e.g. (3a + 2b) + (5a – b) = ?

• What are the two components of a complex number?

• Think of these as “two different terms” and treat adding complex numbers as adding like terms– e.g. (3 + 2i) + (5 – i) = ?

• How would we simplify (6a – 8b) – (2a – 7b)?• Thus, how would we simplify (6 – 8i) – (2 – 7i)?• Write the result in standard form

Addition and Subtraction of Complex Numbers (Example)

Ex 3: Simplify and write in a + bi format:

a) c)

b) d)

14

ii 522

439 i

ii 68610

ii 32

15

Multiplication of Complex Numbers

1616

Multiplication of Complex Numbers

• Consider multiplying 3i · 2i :– Treat in two steps: multiply the real numbers

and then multiply the imaginary units (if they exist)

– Remember that it is only acceptable to leave i in the final answer

– What did we say about i2?

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

Multiplication of Complex Numbers (Example)

Ex 4: Simplify and write in a + bi format:

a)

b)

c)

17

i542

237 i

ii 652

18

Division of Complex Numbers

1919

Complex Conjugate

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

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

rationalizing in section 10.5– What would be the complex conjugate of (2 – i)?

2020

Division of Complex Numbers

• Goal is to write the quotient of complex numbers in the format a + bi

• Consider writing in standard form– How many terms are in the denominator?

• How would we write in standard form?• Finally, consider writing in standard form

– How many terms were in the denominator for the previous examples?

– What happens when we employ the complex conjugate?

• Goal is to get a single real number in the denominator

6

26 i

i

i3

i13

Division of Complex Numbers (Example)

Ex 5: Divide and then write in a + bi format:

a) c)

b)

21

i

i

25

34

i

i

3

6

i

i

41

23

22

Powers of i

2323

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 = -i

i4 = 1

• We can continue with subsequent powers until we notice a pattern

2424

Powers of i (Continued)

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

• How can we use this to derive ANY power of i?– e.g. Is i10 equivalent to i, -1, -i, or 1?– There are 4 powers of i in a cycle– How can we determine how many complete cycles we

must go through to arrive at 10?– Because we are dividing by 4, what are the possible

values for the remainder?– What does each possible value for the remainder mean

in terms of the next cycle?

Powers of i (Example)

Ex 6: Simplify the power of i to either i, -1, -i, or 1:

a)

b)

c)25

119i

105i

1212i

26

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

• Additional Practice– See the list of suggested problems for 10.7

• Next lesson– Solving Quadratic Equations by Completing the Square (Section

11.1)