8.7 Complex Numbers Simplify numbers of the form where b > 0. Recognize subsets of the complex numbers. Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. Find powers of i.

Simplify numbers of the form . Objective 1 Simplify numbers of the form . Slide 8.7- 2

Simplify numbers of the form where b > 0. Imaginary Unit i The imaginary unit i is defined as That is, i is the principal square root of –1. Slide 8.7- 3

Simplify numbers of the form where b > 0. For any positive real number b, It is easy to mistake for with the i under the radical. For this reason, we usually write as as in the definition of Slide 8.7- 4

Write each number as a product of a real number and i. CLASSROOM EXAMPLE 1 Simplifying Square Roots of Negative Numbers Write each number as a product of a real number and i. Slide 8.7- 5

Multiply. Multiplying Square Roots of Negative Numbers CLASSROOM EXAMPLE 2 Multiplying Square Roots of Negative Numbers Multiply. Slide 8.7- 6

Divide. Dividing Square Roots of Negative Numbers CLASSROOM EXAMPLE 3 Slide 8.7- 7

Recognize subsets of the complex numbers. Objective 2 Recognize subsets of the complex numbers. Slide 8.7- 8

Complex Number Recognize subsets of the complex numbers. If a and b are real numbers, then any number of the form a + bi is called a complex number. In the complex number a + bi, the number a is called the real part and b is called the imaginary part. Slide 8.7- 9

Recognize subsets of the complex numbers. For a complex number a + bi, if b = 0, then a + bi = a, which is a real number. Thus, the set of real numbers is a subset of the set of complex numbers. If a = 0 and b ≠ 0, the complex number is said to be a pure imaginary number. For example, 3i is a pure imaginary number. A number such as 7 + 2i is a nonreal complex number. A complex number written in the form a + bi is in standard form. Slide 8.7- 10

Recognize subsets of the complex numbers. The relationships among the various sets of numbers. Slide 8.7- 11

Add and subtract complex numbers. Objective 3 Add and subtract complex numbers. Slide 8.7- 12

CLASSROOM EXAMPLE 4 Adding Complex Numbers Add. Slide 8.7- 13

Subtract. Subtracting Complex Numbers CLASSROOM EXAMPLE 5 Slide 8.7- 14

Multiply complex numbers. Objective 4 Multiply complex numbers. Slide 8.7- 15

Multiply. Multiplying Complex Numbers CLASSROOM EXAMPLE 6 Slide 8.7- 16

Multiply. Multiplying Complex Numbers (cont’d) CLASSROOM EXAMPLE 6 Slide 8.7- 17

Multiply. Multiplying Complex Numbers (cont’d) CLASSROOM EXAMPLE 6 Slide 8.7- 18

Multiply complex numbers. The product of a complex number and its conjugate is always a real number. (a + bi)(a – bi) = a2 – b2( –1) = a2 + b2 Slide 8.7- 19

Divide complex numbers. Objective 5 Divide complex numbers. Slide 8.7- 20

Find the quotient. Dividing Complex Numbers CLASSROOM EXAMPLE 7 Slide 8.7- 21

Find the quotient. Dividing Complex Numbers (cont’d) CLASSROOM EXAMPLE 7 Dividing Complex Numbers (cont’d) Find the quotient. Slide 8.7- 22

Objective 6 Find powers of i. Slide 8.7- 23

i3 = i6 = i4 = i7 = i5 = i8 = Find powers of i. Because i2 = –1, we can find greater powers of i, as shown below. i3 = i6 = i4 = i7 = i5 = i8 = Slide 8.7- 24

Find each power of i. Simplifying Powers of i CLASSROOM EXAMPLE 8 Slide 8.7- 25