Study Guide and Intervention Pure Imaginary Numbers A square root of a number n is a number whose square is n. For nonnegative real numbers a and b, ab = a b and a b = a, b 0. b The imaginary unit i is defined to have the property that i = -1. Simplified square root expressions do not have radicals in the denominator, and any number remaining under the square root has no perfect square factor other than 1. Example 1 Example a. Simplify -48. -48 = 16 (-3) = 16 3-1 = 4i 3 b. Simplify -63. -63 = -1 7 9 = -1 7. 9 = 3i 7 a. Simplify -3i 4i. -3i 4i = -1i = -1(-1) = 1 b. Simplify -3-15. -3-15 = i 3 i 15 = i 45 = -1 9 5 = -3 5 Example 3 Solve x + 5 = 0. x + 5 = 0 Original equation. x = -5 x = ± 5 i Square Root Property. Exercises Subtract 5 from each side. 1. -7. -4 3. -84 4. ( + i) ( - i) Solve each equation. 5. 5x + 45 = 0 6. 4x + 4 = 0 7. -9x = 9 8. 7x + 84 = 0 Chapter 4 4 Glencoe Algebra
Study Guide and Intervention Operations with (continued) Complex Number Addition and Subtraction of Multiplication of Complex Conjugate A complex number is any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (i = -1). a is called the real part, and b is called the imaginary part. Combine like terms. (a + bi) + (c + di) = (a + c) + (b + d )i (a + bi) - (c + di) = (a - c) + (b - d )i Use the definition of i and the FOIL method: (a + bi)(c + di) = (ac - bd ) + (ad + bc)i a + bi and a - bi are complex conjugates. The product of complex conjugates is always a real number. To divide by a complex number, first multiply the dividend and divisor by the complex conjugate of the divisor. Example 1 Simplify (6 + i) + (4-5i). Example (6 + i) + (4-5i) = (6 + 4) + (1-5)i = 10-4i (8 + 3i) - (6 - i) = (8-6) + [3 - (-)]i = + 5i Simplify (8 + 3i) - (6 - i). Example 3 Simplify ( - 5i) (-4 + i). Example 4 ( - 5i) (-4 + i) = (-4) + (i) + (-5i)(-4) + (-5i)(i) = -8 + 4i + 0i - 10i = -8 + 4i - 10(-1) = + 4i Exercises 3 - i + 3i = 3 - i + 3i Simplify 3 - i + 3i. 1. (-4 + i) + (6-3i). (5 - i) - (3 - i) 3. (6-3i) + (4 - i) 4. (-11 + 4i) - (1-5i) 5. (8 + 4i) + (8-4i) 6. (5 + i) - (-6-3i) 7. ( + i)(3 - i) 8. (5 - i)(4 - i) 9. (4 - i)(1 - i) = - 3i - 3i 6-9i - i + 3 i 4-9 i = 3-11i 13 = 3 13-11 13 i Lesson 5 10. 3 + i 11. 7-13i i 1. 6-5i 3i Chapter 4 5 Glencoe Algebra
Skills Practice 1. 99. 7 49 3. 5x 3 y 5 4. -108x 7 5. -81x 6 6. -3-46 7. (3i)(-i)(5i) 8. i 11 9. i 65 10. (7-8i) + (-1-4i) 11. (-3 + 5i) + (18-7i) 1. (10-4i) - (7 + 3i) 13. (7-6i)( - 3i) 14. (3 + 4i)(3-4i) 15. 8-6i 3i Solve each equation. 16. 3i 4 + i 17. 3x + 3 = 0 18. 5x + 15 = 0 19. 4x + 0 = 0 0. -x - 16 = 0 1. x + 18 = 0. 8x + 96 = 0 Find the values of l and m that make each equation true. 3. 0-1i = 5l + (4m)i 4. l - 16i = 3 - (m)i 5. (4 + l) + (m)i = 9 + 14i 6. (3 - m) + (7l - 14)i = 1 + 7i Chapter 4 6 Glencoe Algebra
Practice 1. -36. -8-3 3. -15-5 4. (-3i) (4i)(-5i) 5. (7i) (6i) 6. i 4 7. i 55 8. i 89 9. (5 - i) + (-13-8i) 10. (7-6i) + (9 + 11i) 11. (-1 + 48i) + (15 + 1i) 1. (10 + 15i) - (48-30i) 13. (8-4i) - (10-30i) 14. (6-4i) (6 + 4i) 15. (8-11i) (8-11i) 16. (4 + 3i) ( - 5i) 17. (7 + i) (9-6i) 18. 6 + 5i -i 19. 7-8i 0. 3 - i - i 1. - 4i 1 + 3i Solve each equation.. 5n + 35 = 0 3. m + 10 = 0 4. 4m + 76 = 0 5. -m - 6 = 0 6. -5m - 65 = 0 7. 3 4 x + 1 = 0 Find the values of l and m that make each equation true. 8. 15-8i = 3l + (4m)i 9. (6 - l) + (3m)i = -1 + 7i 30. (3l + 4) + (3 - m)i = 16-3i 31. (7 + m) + (4l - 10)i = 3-6i 3. ELECTRICITY The impedance in one part of a series circuit is 1 + 3j ohms and the impedance in another part of the circuit is 7-5j ohms. Add these complex numbers to find the total impedance in the circuit. 33. ELECTRICITY Using the formula E = IZ, find the voltage E in a circuit when the current I is 3 - j amps and the impedance Z is 3 + j ohms. Lesson Chapter 4 7 Glencoe Algebra
Word Problem Practice 1. SIGN ERRORS Jennifer and Jessica come up with different answers to the same problem. They had to multiply (4 + i)(4 - i) and give their answer as a complex number. Jennifer claims that the answer is 15 and Jessica claims that the answer is 17. Who is correct? Explain. 4. ROTATIONS Complex numbers can be used to perform rotations in the plane. For example, if (x, y) are the coordinates of a point in the plane, then the real and imaginary parts of i(x + yi) are the horizontal and vertical coordinates of the 90 counterclockwise rotation of (x, y) about the origin. What are the real and imaginary parts of i(x + yi)?. COMPLEX CONJUGATES You have seen that the product of complex conjugates is always a real number. Show that the sum of complex conjugates is also always a real number. 3. PYTHAGOREAN TRIPLES If three integers a, b, and c satisfy a + b = c, then they are called a Pythagorean triple. Suppose that a, b, and c are a Pythagorean triple. Show that the real and imaginary parts of (a + bi), together with the number c, form another Pythagorean triple. 5. ELECTRICAL ENGINEERING Alternating current (AC) in an electrical circuit can be described by complex numbers. In any electrical circuit, Z, the impedance in the circuit, is related to the voltage V and the current I by the formula Z = V. The standard electrical I voltage in Europe is 0 volts, so in these problems use V = 0. a. Find the impedance in a standard European circuit if the current is 11i amps. b. Find the current in a standard European circuit if the impedance is 10 5i watts. c. Find the impedance in a standard European circuit if the current is 0i amps. Chapter 4 8 Glencoe Algebra
Enrichment Conjugates and Absolute Value When studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z = x + yi. We denote the conjugate of z by z. Thus, z = x - yi. We can define the absolute value of a complex number as follows. z = x + yi = x + y There are many important relationships involving conjugates and absolute values of complex numbers. Example 1 Let z = x + yi. Then, zz z = (x + yi)(x - yi) = x + y = (x + y ) = z Show z = z z for any complex number z. Example Show complex number z. z z is the multiplicative inverse for any nonzero We know z = z z. If z 0, then we have z ( z z ) = 1. z Thus, is the multiplicative inverse of z. z Exercises For each of the following complex numbers, find the absolute value and multiplicative inverse. 1. i. -4-3i 3. 1-5i 4. 5-1i 5. 1 + i 6. 3 - i Lesson 7. 3 3 + 3 i 8. 3 - i 9. 1-3 i Chapter 4 9 Glencoe Algebra