Section 3.1 The Complex Numbers Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives Perform computations involving complex numbers.

The Complex-Number System Some functions have zeros that are not real numbers. The complex-number system is used to find zeros of functions that are not real numbers. When looking at a graph of a function, if the graph does not cross the x-axis, then it has no x-intercepts, and thus it has no real-number zeros.

Example Express each number in terms of i.

Example (continued)

Complex Numbers A complex number is a number of the form a + bi, where a and b are real numbers. The number a is said to be the real part of a + bi and the number b is said to be the imaginary part of a + bi. Imaginary Number a + bi, a ≠ 0, b ≠ 0 Pure Imaginary Number a + bi, a = 0, b ≠ 0

Addition and Subtraction Complex numbers obey the commutative, associative, and distributive laws. We add or subtract them as we do binomials. We collect the real parts and the imaginary parts of complex numbers just as we collect like terms in binomials.

Example Add or subtract and simplify each of the following. a. (8 + 6i) + (3 + 2i) b. (4 + 5i) – (6 – 3i) a. (8 + 6i) + (3 + 2i) = (8 + 3) + (6i + 2i) = 11 + (6 + 2)i = 11 + 8i b. (4 + 5i) – (6 – 3i) = (4 – 6) + [5i  (–3i)] =  2 + 8i

Multiplication When and are real numbers, This is not true when and are not real numbers. Note: Remember i 2 = –1

Example Multiply and simplify each of the following.

Example (continued) Solution continued

Simplifying Powers of i Recall that 1 raised to an even power is 1, and 1 raised to an odd power is 1. Simplifying powers of i can then be done by using the fact that i 2 = –1 and expressing the given power of i in terms of i 2. Note that powers of i cycle through i, –1, –i, and 1.

Example Simplify the following. a. i 37 b. i 58 a. b.

Conjugates The conjugate of a complex number a + bi is a  bi. The numbers a + bi and a  bi are complex conjugates. Examples: 3 + 7i and 3  7i 14  5i and 14 + 5i 8i and 8i The product of a complex number and its conjugate is a real number.

Example Multiply each of the following. a. (5 + 7i)(5 – 7i) b. (8i)(–8i) a. (5 + 7i)(5  7i) = 52  (7i)2 = 25  49 i 2 = 25  49(1) = 25 + 49 = 74 b. (8i)(–8i) = 64 i 2 = 64(1) = 64

Example Divide 2  5i by 1  6i. Write fraction notation. Multiply by 1, using the conjugate of the denominator to form the symbol for 1.