2.5 Operations With Complex Numbers in Rectangular Form

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$2.5 Operations With Complex Numbers in Rectangular Form The computer-generated image shown is called a fractal.$

1 2.5 Operations With Complex Numbers in Rectangular Form The computer-generated image shown is called a fractal. Fractals are used in many ways, such as making realistic computer images for movies and squeezing high definition television (HDTV) signals into existing broadcast channels. Meteorologists use fractals to study cloud shapes, and seismologists use fractals to study earthquakes. To understand how fractals are generated, we need to extend our understanding of complex numbers. INVESTIGATE & INQUIRE 1. Describe each step in the following addition. (2 + 5x) + (4 3x) = 2 + 5x + 4 3x = x 3x = 6 + 2x 2. a) Use the steps from question 1 to simplify (4 + 3i) + (7 2i). b) Explain why the resulting complex number cannot be simplified further. 3. Describe each step in the following subtraction. (5 2x) (3 7x) = 5 2x 3 + 7x = 5 3 2x + 7x = 2 + 5x 4. Use the steps from question 3 to simplify (3 6i) (7 8i). 5. What are the results when the following operations are performed on the complex numbers a + bi and c + di? a) (a + bi) + (c + di) b) (a + bi) (c + di) 6. Write a rule for adding or subtracting complex numbers. 7. Simplify. a) (2 5i) + (5 4i) b) (3 + 2i) (7 i) 8. Describe each step in the following multiplication. (2 + 3x)(1 4x) = 2 8x + 3x 12x 2 = 2 5x 12x MHR Chapter 2

2 9. a) Use the steps from question 8 to simplify (3 + 4i)(2 5i). b) Explain how you can simplify the final term in the resulting expression. c) Write the expression in simplest form. d) Write a rule for multiplying complex numbers. 10. Simplify. a) (3 + i)(2 + 3i) b) (1 2i)(5 2i) Recall that a complex number is a number in the form a + bi, where a is the real part and bi is the imaginary part. Because there are two parts, any complex number can be represented as an ordered pair (a, b). The ordered pair can be graphed using rectangular axes on a plane called the complex plane. In the complex plane, the x-axis is referred to as the real axis, and the y-axis is referred to as the imaginary axis. A complex number in the form a + bi is said to be in rectangular form, because the ordered pair (a, b) includes the rectangular coordinates of the point a + bi in the complex plane. To add or subtract complex numbers in rectangular form, combine like terms, that is, combine the real parts and combine the imaginary parts. EXAMPLE 1 Adding and Subtracting Complex Numbers Simplify. a) (6 3i) + (5 + i) b) (2 3i) (4 5i) Imaginary y a b a + bi x Real a) (6 3i) + (5 + i) = 6 3i i = i + i = 11 2i b) (2 3i) (4 5i) = 2 3i 4 + 5i = 2 4 3i + 5i = 2 + 2i To perform the operations on a graphing calculator, change the mode settings to the a + bi (rectangular) mode. 2.5 Operations With Complex Numbers in Rectangular Form MHR 145

3 Complex numbers in rectangular form can be multiplied using the distributive property. EXAMPLE 2 Multiplying Complex Numbers Simplify. a) 2i(3 + 4i) b) (1 2i)(4 + 3i) c) (1 4i) 2 Use the distributive property. a) 2i(3 + 4i) = 2i(3 + 4i) = 6i + 8i 2 = 6i + 8( 1) = 8 + 6i Remember to change the mode settings to the a + bi (rectangular) mode. b) (1 2i)(4 + 3i) = (1 2i)(4 + 3i) = 4 + 3i 8i 6i 2 = 4 5i 6i 2 = 4 5i 6( 1) = 4 5i + 6 = 10 5i c) (1 4i) 2 = (1 4i)(1 4i) = 1 4i 4i + 16i 2 = 1 8i + 16( 1) = 1 8i 16 = 15 8i Since i is a radical, 1, any fraction with i in the denominator is not in simplest form. To simplify, rationalize the denominator. EXAMPLE 3 Rationalizing the Denominator 5 Simplify. 2 i 146 MHR Chapter 2

4 Multiply the numerator and the denominator by i. This is the same as multiplying the fraction by = i 2 i 2 i i 5i = 2 i 2 5i = 2( 1) 5i = 2 = 5 i 2 Note the use of brackets, because 5/2i means 5 2 i on the calculator. Recall that binomials of the form ab + cd and ab cd are known as conjugates. Since i represents the radical 1, complex numbers of the form a + bi and a bi are examples of conjugates and are known as complex conjugates. To simplify a fraction with a binomial complex number in the denominator, multiply the numerator and the denominator by the conjugate of the denominator. EXAMPLE 4 Rationalizing Binomial Denominators Simplify 2 + 3i. 1 2i Multiply the numerator and the denominator by the conjugate of 1 2i, which is 1 + 2i i = 2 + 3i 1 + 2i Use the Frac function to display the decimals as fractions. 1 2i 1 2i 1 + 2i Note that 7/5i means 7 i on the calculator. = ( i) ( 1 + 2i) ( 1 2i) ( 1 + 2i) = i + 3i + 6i 1 + 2i 2i 4i 2 = 2 + 7i + 6( 1) 1 4( 1) = 2 + 7i = 4 + 7i Operations With Complex Numbers in Rectangular Form MHR 147

5 EXAMPLE 5 Checking Imaginary Roots Solve and check x 2 4x + 6 = 0. Use the quadratic formula. For x 2 4x + 6 = 0, a = 1, b = 4, and c = 6. x = b ± bc 2 4a 2a ( 4) ± ( 4) 2 ) 4(1)(6 = 2 4 ± = 2 4 ± 8 = 2 = 4 ± 2i2 2 = 2 ± i2 x = 2 + i2 or x = 2 i2 Check. For x = 2 + i2, L.S. = x 2 4x + 6 R.S. = 0 = (2 + i2) 2 4(2 + i2) + 6 = (2 + i2)(2 + i2) 4(2 + i2) + 6 = 4 + 4i i2 + 6 = 0 L.S. = R.S. For x = 2 i2, L.S. = x 2 4x + 6 R.S. = 0 = (2 i2) 2 4(2 i2) + 6 = (2 i2)(2 i2) 4(2 i2) + 6 = 4 4i i2 + 6 = 0 L.S. = R.S. The roots are 2 + i2 148 MHR Chapter 2 and 2 i2.

6 Functions that generate some fractals are in the form F = z 2 + c, where c is a complex number. Fractals are created by iteration, which means that the function F is evaluated for some input value of z, and then the result is used as the next input value, and so on. EXAMPLE 6 Fractals Find the first three output values for F = z 2 + 2i. Use z = 0 as the first input value: Use z = 2i as the second input value: F = z 2 + 2i F = i = 2i F = (2i) 2 + 2i = 4i 2 + 2i = 4 + 2i Use z = 4 + 2i as the third input value: F = ( 4 + 2i) 2 + 2i = 16 16i + 4i 2 + 2i = 16 16i 4 + 2i = 12 14i The first three output values are 2i, 4 + 2i, and 12 14i. Key Concepts To add or subtract complex numbers, combine like terms. To multiply complex numbers, use the distributive property. To simplify a fraction with a pure imaginary number in the denominator, multiply the numerator and the denominator by i. To simplify a fraction with a binomial complex number in the denominator, multiply the numerator and the denominator by the conjugate of the denominator. Communicate Your Understanding 1. Explain why the complex number 5 3i cannot be simplified. 2. Describe how you would simplify each of the following. a) (3 2i) (4 7i) b) (5 + 3i)(1 4i) 3 4 c) d) 4 i 2 + 3i Web Connection To learn more about fractals, visit the above web site. Go to Math Resources, then to MATHEMATICS 11, to find out where to go next. Summarize the various types of fractals. Then, make your own fractal and write the rule that generates it. 2.5 Operations With Complex Numbers in Rectangular Form MHR 149

7 Practise A 1. Simplify. a) (4 + 2i) + (3 4i) b) (2 5i) + (1 6i) c) (3 2i) (1 + 3i) d) (6 i) (5 7i) e) (4 + 6i) + (7i 6) f) (i 8) + (4i 3) g) (9i 6) (10i 3) h) (3i + 11) (6i 13) i) 2(1 7i) + 3(4 i) j) 3(2i 4) (5 + 6i) 2. Simplify. a) 2(4 3i) b) 3i(1 + 2i) c) 4i(3 5i) d) 2i(3i 2 4i + 2) e) (2 4i)(1 + 3i) f) (3 + 4i)(3 5i) g) (3i 1)(4i 5) h) (1 5i)(1 + 5i) i) (1 + 2i) 2 j) (4i 3) 2 k) (i 1) 2 l) (i 2 1) 2 3. Simplify a) b) c) i 3i 4i d) e) f) 5i 2i 7i 4. Simplify. 3 + i 2 2i 5 + 2i a) b) c) i i 2i 3 4i 4 + 3i d) e) 3i 2i 5. Write the conjugate of each complex number. a) 3 + 2i b) 7 3i c) 5 4i d) 6 + 7i 6. Simplify. 3 5 a) b) c) 2 i 1 + 2i i 4 + i d) e) f) 4 + 3i 3 i 2 + 3i 4 3i g) h) 2 3i 2 + 2i 2i 3 2i 2 2i 3 + i 7. Solve and check. a) x 2 + 2x + 2 = 0 b) y 2 4y + 8 = 0 c) x 2 6x + 10 = 0 d) n 2 + 4n + 6 = 0 e) z 2 2z = 6 f) x 2 = 8x 19 Apply, Solve, Communicate 8. Fractals Find the first four output values of F = z 2 if the first input value is (1 i). B 9. Application Imaginary numbers are used in electricity. Three of the basic quantities that can be measured or calculated for an electrical circuit are as follows. the electric current, I, measured in amperes (symbol A) the resistance or impedance, Z, measured in ohms (symbol Ω) 150 MHR Chapter 2

8 the electromotive force, E, sometimes called the voltage and measured in volts (symbol V) These quantities are related by the formula E = IZ. To avoid confusion with the symbol for electric current, I, engineers use j instead of i to represent the imaginary unit. a) In a circuit, the electric current is (8 + 3j) A and the impedance is (4 j) Ω. What is the voltage? b) In a 110-V circuit, the electric current is (5 + 3j) A. What is the impedance? c) In a 110-V circuit, the impedance is (6 2j) Ω. What is the electric current? 10. Communication a) Find the first four output values of F = iz if the first input value is (1 + i). b) Predict the next four output values of F = iz. Explain your reasoning. 11. If y = x 2 + 4x + 5, determine the value of y for each of the following values of x. a) 1 + i b) 2 + i c) 1 i 12. Simplify. a) (4 + i) 2 + (1 3i) 2 b) (3 2i) 2 (4 + 3i) 2 c) 2i(6 + 3i) i(3 2i) d) 3i( 2 + 3i) + 4i( 3 + 2i) e) (3 + i)(2 + i)(1 i) f) (4 2i)( 1 + 3i)(3 i) 13. Factoring The binomial a 2 + b 2 cannot be factored over the real numbers. It can be factored over the complex numbers. Factor a 2 + b Reciprocal Write the reciprocal of a + bi in simplest form. 15. Quadratic equations Write a quadratic equation that has each pair of roots i 3 2i a) 1 + i and 1 i b) and Communication Suppose that the quadratic equation ax 2 + bx + c = 0 has real coefficients and complex roots. Explain why the roots must be complex conjugates of each other. C 17. Determine the values of x and y for which each equation is true. a) 3x + 4yi = 15 16i b) 2x 5yi = 6(1 + 5i) c) (x + y) + (x y)i = i d) (x 2y) (3x + 4y)i = 4 2i 2.5 Operations With Complex Numbers in Rectangular Form MHR 151

9 18. Complex plane If the graph of a complex number a + bi is a point on the imaginary axis in the complex plane, what can you conclude about each of the following? Explain. a) the value of a b) the value of b 19. Transformation Name the transformation that maps the graph of a complex number onto the graph of its conjugate in the complex plane. 20. Quartic equations A quartic equation is a fourth-degree polynomial equation. Quartic equations in the form ax 4 + bx 2 + c = 0 can be solved using the same techniques used to solve quadratic equations. To solve x 4 x 2 12 = 0, first factor the left side. Then, equate each factor to 0 and solve for x. x 4 x 2 12 = 0 (x 2 4)(x 2 + 3) = 0 x 2 4 = 0 or x = 0 x 2 = 4 x 2 = 3 x =±2 x =± 3 x =±i3 The solutions are 2, 2, i3, and i3. Solve the following quartic equations. a) x 4 8x = 0 b) x 4 + 2x = 0 c) x 4 + 3x 2 4 = 0 d) x 4 5x = 0 e) y 4 y 2 6 = 0 f) 3r 4 5r = 0 g) 2x 4 + 5x = 0 h) 2x 4 + x 2 = 6 i) 4a 4 1 = 0 j) 9x 4 4x 2 = Quartic equations Is it possible for a quartic equation to have three real roots and one imaginary root? Explain. 22. Fourth roots a) What are the two square roots of 1 and the two square roots of 1 in the complex number system? b) What are the fourth roots of 1 in the complex number system? c) What are the fourth roots of 1 in the complex number system? A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application a) Express the number 25 as a product of two complex conjugates, a + bi and a bi, in two different ways, with a and b both natural numbers. b) Find another perfect square that can be expressed as a product of two complex conjugates, a + bi and a bi, in two different ways, with a and b both natural numbers. c) Describe the most efficient method for finding numbers that satisfy the above relationship. 152 MHR Chapter 2

3-3 Complex Numbers. Simplify. SOLUTION: 2. SOLUTION: 3. (4i)( 3i) SOLUTION: 4. SOLUTION: 5. SOLUTION: esolutions Manual - Powered by Cognero Page 1

3-3 Complex Numbers. Simplify. SOLUTION: 2. SOLUTION: 3. (4i)( 3i) SOLUTION: 4. SOLUTION: 5. SOLUTION: esolutions Manual - Powered by Cognero Page 1 1. Simplify. 2. 3. (4i)( 3i) 4. 5. esolutions Manual - Powered by Cognero Page 1 6. 7. Solve each equation. 8. Find the values of a and b that make each equation true. 9. 3a + (4b + 2)i = 9 6i Set the