4-8 COMPLEX NUMBERSChapter 4 Quadratic Functions and Equations

©Tentinger

ESSENTIAL UNDERSTANDING AND OBJECTIVES

Essential Understanding: the complex numbers are based on a number whose square is -1

Objectives: Students will be able to:

Identify complex numbers Graph and perform operations using complex

numbers Find complex number solutions to quadratic

equations

IOWA CORE CURRICULUM

Number and Quantity N.CN.1. Know there is a complex number I

such that i2 = -1, and every complex number has the form a + bi with a and b real.

N.CN.2. Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

N.CN.7. Solve quadratic equations with real coefficients that have complex solutions

N.CN.8 (+) Expand polynomial identities to the complex numbers

IMAGINARY NUMBERS

What is an imaginary number?

Represented as i i = √-1 i2 = -1 

Rewrite using the imaginary unit i

√-18 √-12 √-7 √-25 √-24

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−a = −1⋅ a = i a

IMAGINARY NUMBERS

Imaginary number: any number in the form a + bi, where a and b are real numbers and b does not equal zero.

Imaginary numbers and real numbers make up the set of complex numbers

ABSOLUTE VALUE

Absolute value of a complex number: the distance from the origin in the complex plane

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a + bi = a2 + b2

GRAPHING IN THE COMPLEX NUMBER PLANE

The y-axis is the imaginary numbers, the x-axis is the real numbers

What are the graph and absolute values of each number?

 5 – i

-3 – 2i

1 + 4i

ADDING AND SUBTRACTING

You can define operations on the set of complex numbers so that when you restrict the operations to the subset of real numbers, you get the familiar operations on the real numbers.

Adding and Subtracting Complex Numbers

Combine the real numbers and the imaginary numbers

(7-2i)+(-3+i) (8 + 6i)-(8-6i) (1+5i)-(3-2i) (-3+9i)+(3+9i)

MULTIPLYING

Multiplying Complex Numbers Multiply as you would binomials (3i)(-5+2i)

(7i)(3i)

(2-3i)(4+5i)

(-4+5i)(-4-5i)

DIVIDING

Complex Conjugates: when (a+bi)(a-bi) = a real number

Use Complex Conjugates to simplify quotients

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9 +12i

3i

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2 + 3i

1 − 4i

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5 −2i

3+ 4i

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4 − i

6i

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8 − 7i

8 + 7i

FACTORING

Factoring Using Complex Conjugates 2x2 + 32

5x2 + 20

x2 +81

IMAGINARY SOLUTIONS

Every quadratic equation has complex number solutions (that sometimes are real numbers)

Finding Imaginary Solutions 2x2 – 3x + 5 = 0

3x2 – x + 2 = 0

x2 – 4x + 5 = 0

HOMEWORK

Pg. 253 – 254 #9 – 69 (3s)