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Digital Lesson Complex Numbers The letter i represents the numbers whose square is –1. i2 = –1 i = 1 Imaginary unit If a is a positive real number, then the principal square root of negative a is the imaginary number i a . a = i a Examples: 4 = i 4 = 2i 36 = i 36 = 6i A complex number is a number of the form a + bi, where a and b are real numbers and i = 1 . The number a is the real part of a + bi, and b is the imaginary part. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Examples of complex numbers: Real Part a 2 20 Real Numbers: a + 0i + + – Imaginary Part bi 7i 3i Imaginary Numbers: 0 + bi Simplify: 1. 90 = i 90 = i 9 • 10 = 3i 10 2. 64 = i 64 = 8i 3. 16 + 50 = 16 + i 50 a + bi form = 16 + i 25 • 2 Simplify using the product property of = 4 + 5i 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. radicals. 3 To add or subtract complex numbers: 1. Write each complex number in the form a + bi. 2. Add or subtract the real parts of the complex numbers. 3. Add or subtract the imaginary parts of the complex numbers. (a + bi ) + (c + di ) = (a + c) + (b + d)i (a + bi ) – (c + di ) = (a – c) + (b – d )i Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Examples: Add (11 + 5i) + (8 – 2i ) = (11 + 8) + (5i – 2i ) Group real and imaginary terms. = 19 + 3i a + bi form Add (10 + 5 ) + (21 – 5) = (10 + i 5 ) + (21 – i 5 ) i = 1 = (10 + 21) + (i 5 – i 5) Group real and imaginary terms. = 31 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. a + bi form 5 Examples: Subtract: (– 21 + 3i ) – (7 – 9i) = (– 21 – 7) + [(3 – (– 9)]i = (– 21 – 7) + (3i + 9i) = –28 + 12i Group real and imaginary terms. a + bi form Subtract: (11 + 16 ) – (6 + 9 ) = (11 + i 16 ) – (6 + i 9 ) Group real and = (11 – 6) + [ 16 – 9 ]i = (11 – 6) + [ 4 – 3]i =5+i Copyright © by Houghton Mifflin Company, Inc. All rights reserved. imaginary terms. a + bi form 6 The product of two complex numbers is defined as: (a + bi)(c + di ) = (ac – bd ) + (ad + bc)i 1. Use the FOIL method to find the product. 2. Replace i2 by – 1. 3. Write the answer in the form a + bi. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Examples: 1. 25 • 5 = i 25 • i 5 = 5i • i 5 = 5i2 5 = 5 (–1) 5 = –5 5 2. 7i (11– 5i) = 77i – 35i2 = 77i – 35 (– 1) = 35 + 77i 3. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i2 = 12 + 4i – 21i2 = 12 + 4i – 21(–1) = 12 + 4i + 21 = 33 + 4i Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 The complex numbers a + bi and a - bi are called conjugates. The product of conjugates is the real number a2 + b2. (a + bi)(a – bi) = a2 – b2i2 = a2 – b2(– 1) = a2 + b2 Example: (5 + 2i)(5 – 2i) = (52 – 4i2) = 25 – 4 (–1) = 29 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Dividing Complex Numbers A rational expression, containing one or more complex numbers, is in simplest form when there are no imaginary numbers remaining in the denominator. 7 9i Example: 6i 7 9i • i i Multiply the expression by . 6i i i 2 7i 9i 6i 2 7i 3(–1) Replace i2 by –1 and simplify. 6(–1) 3 7i 1 7 i Write the answer in the form a + bi. 6 2 6 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Simplify: 5 3i 2 i 5 3i • 2 i 2i 2 i Multiply the numerator and denominator by the conjugate of 2 + i. 10 5 i 6 i 3i 2 In 2 + i, a = 2 and b = 1. a2 + b2 = 22 + 12 2 2 12 10 i 3(–1) 4 1 13 i 5 13 1 i 5 5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Replace i2 by –1 and simplify. Write the answer in the form a + bi. 11