126 CHAPTER 1 Equations and Inequalities The product of two complex numbers is found by multiplying as though the numbers were binomials and using the fact that i2 = -1, as follows. 1a + bi21c + di2 = ac + adi + bic + bidi FOIL method = ac + adi + bci + bdi2 Commutative property; Multiply. = ac + 1ad + bc2i + bd1-12 Distributive property; i 2 = -1 = 1ac - bd2 + 1ad + bc2i Group like terms. With the definitions i2 = -1 and 2-a = i 2a for a 70, all properties of real numbers are extended to complex numbers. As a result, complex numbers are added, subtracted, multiplied, and divided using real number properties and the following definitions. Addition and Subtraction of Complex Numbers For complex numbers a + bi and c + di, 1 a +bi2 + 1c +di2 = 1a +c2 + 1b +d2i and 1a +bi2 − 1c +di2 = 1a −c2 + 1b −d2i. That is, to add or subtract complex numbers, add or subtract the real parts and add or subtract the imaginary parts. EXAMPLE 4 Adding and Subtracting Complex Numbers Find each sum or difference. Write answers in standard form. (a) 13 - 4i2 + 1-2 + 6i2 (b) 1-4 + 3i2 - 16 - 7i2 SOLUTION (a) 13 - 4i2 + 1-2 + 6i2 Add real Add imaginary parts. parts. (++)++* (+)+* = 33 + 1-224 + 3-4 + 64i Commutative, associative, distributive properties = 1 + 2i Standard form (b) 1-4 + 3i2 - 16 - 7i2 = 1-4 - 62 + 33 - 1-724i Subtract real parts. Subtract imaginary parts. = -10 + 10i Standard form S Now Try Exercises 47 and 49. Multiplication of Complex Numbers For complex numbers a + bi and c + di, 1 a +bi2 1c +di2 = 1ac −bd2 + 1ad +bc2i.
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