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§ 7.7 Complex Numbers Complex Numbers The Imaginary Unit i The imaginary unit i is defined as i 1, where i 2 1. The Square Root of a Negative Number If b is a positive real number, then b (1)b 1 b i b or Blitzer, Intermediate Algebra, 4e – Slide #94 bi. Complex Numbers EXAMPLE Write as a multiple of i: (a) 300 (b) 20 5. SOLUTION (a) 300 3001 300 1 100 3 1 10 3i (b) 20 5 20 51 20 5 1 20 5i Blitzer, Intermediate Algebra, 4e – Slide #95 Complex Numbers Complex Numbers & Imaginary Numbers The set of all numbers in the form a bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part of the complex number a bi. If b 0, then the complex number is called an imaginary number. Blitzer, Intermediate Algebra, 4e – Slide #96 Complex Numbers Adding & Subtracting Complex Numbers 1) a bi c di a c b d i In words, this says that you add complex numbers by adding their real parts, adding their imaginary parts, and expressing the sum as a complex number. 2) a bi c di a c b d i In words, this says that you subtract complex numbers by subtracting their real parts, subtracting their imaginary parts, and expressing the difference as a complex number. Blitzer, Intermediate Algebra, 4e – Slide #97 Complex Numbers EXAMPLE Perform the indicated operations, writing the result in the form a + bi: (a) (-9 + 2i) – (-17 – 6i) (b) (-2 + 6i) + (4 - i). SOLUTION (a) (-9 + 2i) – (-17 – 6i) = -9 + 2i + 17 + 6i Remove the parentheses. Change signs of the real and imaginary parts being subtracted. = -9 + 17 + 2i + 6i Group real and imaginary terms. = (-9 + 17) + (2 + 6)i = 8 + 8i Add real parts and imaginary parts. Simplify. Blitzer, Intermediate Algebra, 4e – Slide #98 Complex Numbers CONTINUED (b) (-2 + 6i) + (4 - i) = -2 + 6i + 4 - i = -2 + 4 + 6i - i = (-2 + 4) + (6 - 1)i = 2 + 5i Remove the parentheses. Group real and imaginary terms. Add real parts and imaginary parts. Simplify. Blitzer, Intermediate Algebra, 4e – Slide #99 Complex Numbers EXAMPLE Find the products: (a) -6i(3 – 5i) (b) (-4 + 2i)(-4 - 2i). SOLUTION (a) -6i(3 – 5i) 6i 3 6i 5i 18i 30i 2 18i 301 30 18i Distribute -6i through the parentheses. Multiply. 2 Replace i with -1. Simplify and write in a + bi form. (b) (-4 + 2i)(-4 – 2i) 16 8i 8i 4i 2 Use the FOIL method. Blitzer, Intermediate Algebra, 4e – Slide #100 Complex Numbers CONTINUED 16 8i 8i 41 i 2 1 16 4 8i 8i Group real and imaginary terms. 20 Combine real and imaginary terms. Blitzer, Intermediate Algebra, 4e – Slide #101 Complex Numbers Multiplying Complex Numbers Because the product rule for radicals only applies to real numbers, multiplying radicands is incorrect. When performing operations with square roots of negative numbers, begin by expressing all square roots in terms of i. Then perform the indicated operation. Blitzer, Intermediate Algebra, 4e – Slide #102 Complex Numbers EXAMPLE Multiply: 16 4. SOLUTION 16 4 16 1 4 1 16i 4i 64i 2 Express square roots in terms of i. 16 4 64 and i i i 2 . 64 1 i 2 1 8 The square root of 64 is 8. Blitzer, Intermediate Algebra, 4e – Slide #103 Complex Numbers EXAMPLE 6 3i . Divide and simplify to the form a + bi: 4 2i SOLUTION The conjugate of the denominator is 4 – 2i. Multiplication of both the numerator and the denominator by 4 – 2i will eliminate i from the denominator. 6 3i 6 3i 4 2i 4 2i 4 2i 4 2i Multiply by 1. 24 12i 12i 6i 2 2 4 2 2i Use FOIL in the numerator and A B A B A2 B 2 in the denominator. Blitzer, Intermediate Algebra, 4e – Slide #104 Complex Numbers CONTINUED 24 24i 6i 2 16 4i 2 24 24i 6 1 16 4 1 24 24i 6 16 4 18 24i 20 18 24 i 20 20 9 6 i 10 5 Simplify. i 2 1 Perform the multiplications involving -1. Combine like terms in the numerator and denominator. Express answer in the form a + bi. Simplify. Blitzer, Intermediate Algebra, 4e – Slide #105 Complex Numbers EXAMPLE 5i . Divide and simplify to the form a + bi: 4i SOLUTION The conjugate of the denominator, 0 - 4i, is 0 + 4i. Multiplication of both the numerator and the denominator by 4i will eliminate i from the denominator. 5 i 5 i 4i 4i 4i 4i 20i 4i 2 16i 2 Multiply by 1. Multiply. Use the distributive property in the numerator. Blitzer, Intermediate Algebra, 4e – Slide #106 Complex Numbers CONTINUED 20i 4 1 16 1 i 2 1 20i 4 16 4 20 i 16 16 Perform the multiplications involving -1. 1 5 i 4 4 Simplify real and imaginary parts. Express the division in the form a + bi. Blitzer, Intermediate Algebra, 4e – Slide #107 Complex Numbers Simplifying Powers of i 2 1) Express the given power of i in terms of i . 2) Replace i 2 with -1 and simplify. Use the fact that -1 to an even power is 1 and -1 to an odd power is -1. Blitzer, Intermediate Algebra, 4e – Slide #108 Complex Numbers EXAMPLE Simplify: a i 46 b i 400 c i 13. SOLUTION i a i 46 b i 400 i c i 13 1 2 23 23 2 200 i 12 1 1 200 1 i i i 16 i 1 i i 2 6 Blitzer, Intermediate Algebra, 4e – Slide #109