124 CHAPTER 1 Equations and Inequalities Some graphing calculators, such as the TI-84 Plus, are capable of working with complex numbers, as seen in Figure 3. 7 The following important concepts apply to a complex number a + bi. 1. If b = 0, then a + bi = a, which is a real number. (This means that the set of real numbers is a subset of the set of complex numbers. See Figure 4.) 2. If b≠0, then a + bi is a nonreal complex number. Examples: 7 + 2i, -1 - i 3. If a = 0 and b≠0, then the nonreal complex number is a pure imaginary number. Examples: 3i, -16i The form a + bi (or a + ib) is standard form. (The form a + ib is used to write expressions such as i 25, because 25i could be mistaken for 25i.) The relationships among the subsets of the complex numbers are shown in Figure 4. Figure 4 Complex Numbers a + bi, for a and b Real Real numbers a + bi, b = 0 Irrational numbers Integers –11, –6, –3, –2, –1 Nonreal complex numbers a + bi, b ≠ 0 Whole numbers 0 Natural numbers 1, 2, 3, 4, 5, 37, 40 Ï2 Ï15 2Ï8 p 4 p 7 + 2i, 5 – iÏ3, – i 1 2 3 2 + Pure imaginary numbers a + bi, a = 0 and b ≠ 0 3i, – i, – i, iÏ5 2 3 Rational numbers , – , 4 9 5 8 11 7 For a positive real number a, the expression 2-a is defined as follows. Meaning of !−a For a 70, !−a =i !a. Figure 3 The calculator is in complex number mode. This screen supports the definition of i. It also shows how the calculator returns the real and imaginary parts of the complex number 7 + 2i. EXAMPLE 1 Writing !−a as i !a Write each number as the product of a real number and i. (a) 2-16 (b) 2-70 (c) 2-48 SOLUTION (a) 2-16 = i216 = 4i (b) 2-70 = i270 (c) 2-48 = i 248 = i 216 # 3 = 4i 23 Product rule for radicals S Now Try Exercises 21, 23, and 25.
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