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7-9: MORE About Complex Numbers Goal: Be able to solve equations with complex numbers, multiply complex numbers, find conjugates of complex numbers, and divide and find reciprocals of complex numbers. Equality for Complex Numbers If a + bi = c + di then a = c and b=d Ex 1: Solve for x and y. a.) 5 x 6i 10 2 yi 5 x 10 x2 6i 2 yi 2i 2i 3 y b.) 4 x 3 yi 8 6i 4x 8 x2 3 yi 6i 3i 3i y2 Ex 2: Multiply. Remember: i2 = -1 a.) 5i 2i 10i 2 10(1) 10 2 3 i 6 i 18 i 18(1) 18 b.) c.) (4i)2 16i 2 16(1) 16 d.) (7i) 2 49i 2 49(1) 49 2 (2 5 i )(3 4 i ) 6 8 i 15 i 20 i e.) 6 23i 20(1) 14 23i 2 ( 2 3 i )(6 5 i ) 12 10 i 18 i 15 i f.) 12 28i 15(1) 3 28i Complex Conjugates Conjugates: a + bi ; a - bi Ex 3: Find the conjugate of each number. a.) 2 5i ; 2 5i b.) 3 10i ; 3 10i c.) 8i ; 8i d.) 4 ; 4 (since 0 + 8i, 0 – 8i are conjugates) (since 4 + 0i, 4 – 0i are conjugates) Consider(a bi)(a bi) a 2 abi abi b2i 2 a 2 b2 (1) a 2 b2 (a bi)(a bi) a 2 b2 Ex 4: Multiply. a.) (4 2i)(4 2i) (4) 2 (2) 2 16 4 20 b.) (3 7i)(3 7i) (3) 2 (7) 2 9 49 58 c.) (5 i 3)(5 i 3) (5)2 ( 3) 2 25 (3) 28 Division and Reciprocals Multiply by “1;” Ex 5: Divide. use the conjugate 2 3 i 1 i a.) 1i 1 i 2 2i 3i 3i 2 (1) (1) 2 2 2 5i 3(1) 2 1 5i 2 2 Division and Reciprocals Ex 5: Divide. 1 2 i 3 4 i b.) 3 4i 3 4i 3 4i 6i 8i 2 (3)2 (4)2 Multiply by “1;” use the conjugate 3 2i 8(1) 9 16 11 2i 25 25 Ex 6: a.) Find the reciprocal of 2 7i. 1 2 7i 2 7i 2 7i 2 7i Multiply by “1;” use the conjugate 2 7i 2 2 4 49 (2) (7) 2 7i 53 53 Ex 6: b.) Find the reciprocal of 5 4i. 1 5 4i 5 4i 5 4i 5 4i Multiply by “1;” use the conjugate 5 4i 2 2 25 16 (5) (4) 5 4i 41 41