Complex Numbers

MATH 109 - PrecalculusS. Rook

Overview

• Section 2.4 in the textbook:– Imaginary numbers & complex numbers– Adding & subtracting complex numbers– Multiplying complex numbers– Dividing complex numbers

2

Imaginary Numbers & Complex Numbers

4

Imaginary Numbers

• In the real number system, recall that a value under a radical must be greater than or equal to 0– Otherwise the value is non-real

• Consider if we decomposed and rewrote it 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

16

5

Multiplying and Dividing Imaginary Numbers

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

CORRECT

• After “poking out the i” use the product or quotient rule for radicals 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

6

Complex Numbers

• 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

• The set of real numbers along with the set of imaginary numbers comprises the set of complex numbers– i.e. a complex number is exclusively real when b = 0

and exclusively imaginary when a = 0

Imaginary Numbers (Example)

Ex 1: Perform the following operations and write the answer in standard form:

a)

b)

7

272

26

Adding & Subtracting Complex Numbers

9

Adding & Subtracting 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 two complex numbers:– Distribute the negative to the second complex

number– Treat as adding complex numbers

Adding & Subtracting Complex Numbers (Example)

Ex 2: Add or subtract and write the result in standard form:

a) (5 + i) + (6 – 2i)

b) 13i – (14 – 7i)

10

Multiplication of Complex Numbers

12

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

Multiplication of Complex Numbers (Example)

Ex 3: Multiply and write in standard form:

a) (4 + 5i)2

b) (1 + i)(3 – 2i)

13

Division of Complex Numbers

15

Complex Conjugate

• Consider (3 + i)(3 – i)– What is noticeable?

• 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)?

16

Division of Complex Numbers

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

• To divide complex numbers:– 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

Division of Complex Numbers (Example)

Ex 4: Divide and write in standard form:

a)

b)

17

i

i

3

3

i

i56

Summary

• After studying these slides, you should be able to:– Understand the principles of imaginary and complex

numbers– Understand the standard form for a complex number– Add, subtract, multiply, and divide complex numbers

• Additional Practice– See the list of suggested problems for 2.4

• Next lesson– Zeros of Polynomial Functions (Section 2.5)

18