 PERFORM COMPUTATIONS INVOLVING COMPLEX NUMBERS. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 3.1 The Complex Numbers

Complex Numbers Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. The symbol i represents. Imaginary Number a + bi, a ≠ 0, b ≠ 0 Pure Imaginary Number a + bi, a = 0, b ≠ 0

The Complex Number System Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

The Complex-Number System Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Express each number in terms of i.

Addition and Subtraction Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Add or subtract and simplify each of the following. a. (8 + 6i) + (3 + 2i)b. (4 + 5i) – (6 – 3i)

Multiplication Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley When and are real numbers, This is not true when and are not real numbers. Note: Remember i 2 = –1

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Multiply and simplify each of the following.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Simplify each of the following

Conjugates Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 i 8i and  8i The product of a complex number and its conjugate is a real number.

Multiplying Conjugates - Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Multiply each of the following. a. (5 + 7i)(5 – 7i) b. (8i)(–8i)

Dividing Using Conjugates - Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Divide 2  5i by 1  6i.