127 1.3 Complex Numbers To find a given product in routine calculations, it is often easier just to multiply as with binomials and use the fact that i2 = -1. Example 5(c) showed that 16 + 5i216 - 5i2 = 61. The numbers 6 + 5i and 6 - 5i differ only in the sign of their imaginary parts and are complex conjugates. The product of a complex number and its conjugate is always a real number. This product is the sum of the squares of the real and imaginary parts. Property of Complex Conjugates For real numbers a and b, 1 a +bi2 1a −bi2 =a2 +b2. To find the quotient of two complex numbers in standard form, we multiply both the numerator and the denominator by the complex conjugate of the denominator. This screen shows how the TI-84 Plus displays the results found in Example 5. EXAMPLE 5 Multiplying Complex Numbers Find each product. Write answers in standard form. (a) 12 - 3i213 + 4i2 (b) 14 + 3i22 (c) 16 + 5i216 - 5i2 SOLUTION (a) 12 - 3i213 + 4i2 = 2132 + 214i2 - 3i132 - 3i14i2 FOIL method = 6 + 8i - 9i - 12i2 Multiply. = 6 - i - 121-12 Combine like terms; i2 = -1 = 18 - i Standard form (b) 14 + 3i22 = 42 + 214213i2 + 13i22 Square of a binomial = 16 + 24i + 9i2 Multiply. = 16 + 24i + 91-12 i2 = -1 = 7 + 24i Standard form (c) 16 + 5i216 - 5i2 = 62 - 15i22 Product of the sum and difference of two terms = 36 - 251-12 Square 6; 15i22 = 52i2 = 251-12 = 36 + 25 Multiply. = 61, or 61 + 0i Standard form S Now Try Exercises 55, 59, and 63. Remember to add twice the product of the two terms.
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