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3 Objectives 1.Simplify powers of i. 2.Simplify square roots containing negative radicands. 3.Perform operations with complex numbers. 4.Rationalize a denominator, expressing the answer in a + bi form. 1 1 2 2 3 3 4 4

4 Objectives Solve a quadratic equation that has complex- number solutions. 5 5

5 Complex Numbers The solutions of some quadratic equations are not real numbers. For example, we solve the equation x 2 + 2x + 5 = 0 and obtain the following solutions or Each of these solutions involves, which is not a real number. Thus, the solutions of this equation are not real numbers. As we will see, the solutions of this equation are from a set called the set of complex numbers.

6 Simplify powers of i 1.

7 Simplify powers of i The imaginary number is usually denoted by the letter i. Since it follows that i 2 = –1 The powers of i produce an interesting pattern: i = = i i 5 = i 4  i = 1  i = i i 2 = = –1 i 6 = i 4  i 2 = 1(–1) = –1 i 3 = i 2  i = –1  i = –i i 7 = i 4  i 3 = 1(–i) = –i i 4 = i 2  i 2 = (–1)(–1) = 1 i 8 = i 4  i 4 = (1)(1) = 1

8 Simplify powers of i The pattern continues: i, –1, –i, 1,....

9 Simplify square roots containing negative radicands 2.

10 Simplify square roots containing negative radicands If we assume that multiplication of imaginary numbers is commutative and associative, then (2i ) 2 = 2 2 i 2 = 4(–1) = –4 Since (2i ) 2 = –4, it follows that 2i is a square root of –4, and we write

11 Simplify square roots containing negative radicands This result could have been obtained by the following process: Likewise, we have

12 Simplify square roots containing negative radicands In general, we have the following rules. Properties of Radicals If at least one of a and b is a nonnegative real number, then and

13 Perform operations with complex numbers 3.

14 Perform operations with complex numbers Imaginary numbers such as,, and form a subset of a broader set of numbers called complex numbers. Complex Numbers A complex number is any number that can be written in the form a + bi where a and b are real numbers, and. The number a is called the real part, and the number b is called the imaginary part of the complex number a + bi.

15 Perform operations with complex numbers If b = 0, the complex number a + bi is the real number a. If b  0 and a = 0, the complex number 0 + bi (or just bi ) is an imaginary number. Figure 9-13 shows the relationship of the real numbers to the imaginary and complex numbers. Figure 9-13

16 Perform operations with complex numbers Equality of Complex Numbers The complex numbers a + bi and c + di are equal if and only if a = c and b = d Here are several examples of equal complex numbers., because and. x + yi = 4 + 7i if and only if x = 4 and y = 7.

17 Perform operations with complex numbers Addition and Subtraction of Complex Numbers Complex numbers are added and subtracted as if they were binomials: (a + bi ) + (c + di ) = (a + c) + (b + d )i (a + bi ) – (c + di ) = (a – c) + (b – d )i

18 Example Perform each operation: a. (8 + 4i ) + (12 + 8i ) b. (7 – 4i ) + (9 + 2i ) c. (–6 + i ) – (3 – 4i ) d. (2 – 4i ) – (–4 + 3i )

19 Example – Solution We will use the rules for addition and subtraction of complex numbers. a. (8 + 4i ) + (12 + 8i ) = 8 + 4i + 12 + 8i = 20 + 12i b. (7 – 4i ) + (9 + 2i ) = 7 – 4i + 9 + 2i = 16 – 2i c. (–6 + i ) – (3 – 4i ) = –6 + i – 3 + 4i = –9 + 5i

20 Example – Solution d. (2 – 4i ) – (–4 + 3i ) = 2 – 4i + 4 – 3i = 6 – 7i cont’d

21 Perform operations with complex numbers To multiply a complex number by an imaginary number, we use the distributive property to remove parentheses and then simplify. Multiplication of Complex Numbers Complex numbers are multiplied as if they were binomials, with i 2 = –1: (a + bi ) (c + di ) = ac + adi + bci + bdi 2 = (ac – bd) + (ad + bc)i

22 Rationalize a denominator, expressing the answer in a + bi form 4.

23 Rationalize a denominator, expressing the answer in a + bi form The fraction is not in simplest form because the denominator contains a radical: To simplify the fraction, we must rationalize the denominator just as we did when we simplified rational expressions.

24 Example Rationalize each denominator and write the result in a + bi form. a. b. Solution: We can multiply each numerator and denominator by i and simplify. a.

25 Example – Solution b. cont’d

26 Rationalize a denominator, expressing the answer in a + bi form To rationalize the denominators of fractions such as,, and, we must multiply the numerator and denominator by the complex conjugate of the denominator. Complex Conjugates The complex numbers a + bi and a – bi are called complex conjugates of each other.

27 Rationalize a denominator, expressing the answer in a + bi form For example, 3 + 4i and 3 – 4i are complex conjugates. 5 – 7i and 5 + 7i are complex conjugates. 8 + 17i and 8 – 17i are complex conjugates. In general, the product of the complex number a + bi and its complex conjugate a – bi is the real number a 2 + b 2. (a + bi )(a – bi ) = a 2 – abi + abi – b 2 i 2 = a 2 – b 2 (–1) = a 2 + b 2

28 Solve a quadratic equation that has complex-number solutions 5.

29 Solve a quadratic equation that has complex number solutions The solutions of many quadratic equations are not real numbers. For example, the solutions of the equation x 2 + x + 1 = 0 are not real numbers. We will show that this is true in the next example.

30 Example Solve: x 2 + x + 1 = 0 Solution: We will solve the equation by using the quadratic formula with a = 1, b = 1, and c = 1:

31 Example – Solution Expressed in a + bi form, the solutions are cont’d