4.8 C OMPLEX N UMBERS Part 1: Introduction to Complex and Imaginary Numbers

4.8 COMPLEX NUMBERSPart 1: Introduction to Complex and Imaginary Numbers

REAL NUMBERS

See Page 12 in Textbook

COMPLEX NUMBERS

The set of Real Numbers is a subset of a larger set of numbers called Complex NumbersThe complex numbers are based on

a number whose square root is –1 The imaginary unit i is the complex

number whose square root is –1 .

SQUARE ROOT OF A NEGATIVE REAL NUMBER

For any real number a,

EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER

EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER

EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER

REAL AND IMAGINARY NUMBERS

An imaginary number is any number of the form a + bi, where a and b are real numbers and b ≠ 0.

If b = 0, then the number is a real number. If a = 0 and b ≠ 0, then the number is a

pure imaginary number

a + bi↑ ↑

Real Part

ImaginaryPart

COMPLEX NUMBERS

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

POWERS OF IMAGINARY NUMBERS

2

3

4

5

6

7

8

1

1

1

1

1

1

i

i

i i

i

i

i

i i

i

EVALUATING POWERS

Divide the exponent by 4 and determine the remainder.

Equivalent power depends on the remainder

2

3

4

i

i

i

i

Remainder of 1

Remainder of 2

Remainder of 3

Remainder of 0

TRY THESE

15

20

201

26

i

i

i

i

GRAPHING COMPLEX NUMBERS In the complex number plane,

The x – axis represents the real part The y – axis represents the imaginary part

The point (a, b) represents the complex number a + bi

The absolute value of a complex number is its distance from the origin in the complex plane.

EXAMPLE: WHAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER?

EXAMPLE: WHAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER?

ADDING AND SUBTRACTING COMPLEX NUMBERS

To add or subtract, combine like terms

3 4 2 5 5 9

3 4 2 5 3 4 2 5 1

i i i

i i i i i

ADDING AND SUBTRACTING 2

Add or subtract

5 9 25

3 12 2 75

MULTIPLYING COMPLEX NUMBERS

THE QUADRATIC FORMULA

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

We can use and the quadratic formula to solve all quadratic equations.

FIND ALL SOLUTIONS TO EACH QUADRATIC EQUATION

FIND ALL SOLUTIONS TO EACH QUADRATIC EQUATION

HOMEWORK

P253 #1, 2, 8 – 17 all, 39 – 44 all