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Chapter 2 Polynomial and Rational Functions Section 2.4 Complex Numbers Course/Section Lesson Number Date Section Objectives: Students will know how to perform operations with complex numbers and plot complex numbers in the complex plane. I. The Imaginary Unit i (p. 127) Pace: 5 minutes • We need more numbers because simple equations such as x2 + 1 = 0 do not have a real solution. We need a number whose square is –1. So define i2 = -1. i is called the imaginary unit. By adding real numbers to multiples of the imaginary unit, we get the set of complex numbers, defined as {a + bi | a is real, b is real, and i2 = -1}. a + bi is the standard form of a complex number. a is called the real part and bi is the imaginary part. • Two complex numbers a + bi and c + di are equal to each other if and only if a = c and b = d. II. Operations with Complex Numbers (pp.128 – 129) Pace: 10 minutes • To add two complex numbers, we add the two real parts then add the two imaginary parts. That is, (a + bi) + (c + di) = (a + c) + (b + d )i • Also note that the additive identity of the complex numbers is zero. Furthermore, the additive inverse of a + bi is –( a + bi) = -a – bi. Example 1. Add or subtract the following complex numbers. a) (6 − 3i) + (1+ 2i) = (6 + 1) + (−3 + 2 )i = 7 − i b) (5 − i) − (2 − 4i) = 5− i − 2 + 4i = 3 + 3i • Many of the properties of the real numbers are valid for the complex numbers also, such as both of the Associative and Commutative Properties, and both Distributive Properties. • Using these properties, much as we did with polynomials, is the best method for multiplying two complex numbers. • Now is a good time to examine the Exploration on page 129 of the text. Example 2. Multiply the following complex numbers. a) 2(17 − 5i) = 2(17) − 2(5i) = 34 − 10i b) (3 − i )(5 + 4i) = 15 +12i − 5i − 4i2 = 15 +12i − 5i + 4 = 19 + 7i c) (1+ 7i)2 = 12 + 2(1)(7i ) + (7i)2 = 1+ 14i − 49 = −48 +14i 2 d) (4 + 5i)(4 − 5i ) = 4 2 − (5i) = 16 + 25 = 41 Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, 4e Precalculus, Functions and Graphs: A Graphing Approach 4e Instructor Success Organizer Copyright © Houghton Mifflin Company. All rights reserved. 2.4-1 Tip: Note that in the last example, we took the product of two complex numbers and got a real number. This leads to our next topic. III. Complex Conjugates (p. 130) Pace: 5 minutes • Two complex numbers of the forms a + bi and a – bi are called complex conjugates. Note how their product is a real number: (a + bi)( a – bi) = a2 – (bi)2 = a2 + b2 To find the quotient of two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. Example 3. Find the quotient of the following complex numbers. 2 1 + i 2(1 + i) 2(1+ i) 2 a) = ⋅ = = = 1+ i 1 − i 1− i 1 + i 12 + 12 2 b) 2 − i 4 − 3i 2 −i = ⋅ 4 + 3i 4 + 3i 4 − 3i 8 − 6i − 4i + 3i 42 + 32 8 − 3 − 6i − 4i = 16 + 9 5 − 10i = 25 1 2 = − i 5 5 = 2 IV. Fractals and the Mandelbrot Set (pp. 131 - 132) Pace: 5 minutes • State that each complex number corresponds to a point in the complex plane. • Transform the rectangular coordinate system into the complex plane by stating that the y-axis becomes the imaginary axis and the x-axis becomes the real axis. The complex number a +bi is plotted just as the point (a, b). Example 4. Plot each of the complex numbers in the complex plane. a) 3 + 2i b) 3i c) -4 d) -1 –3i 4 Imaginary Axis 3 0 2 3 1 -4 -6 -4 0 -2 -1 0 2 4 -2 -1-3 -4 Real Axis 2.4-2 Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, 4e Precalculus, Functions and Graphs: A Graphing Approach 4e Instructor Success Organizer Copyright © Houghton Mifflin Company. All rights reserved.