Mathematics 10 — Exercise Worked examples · Example 19

Full question

Simplify the following expression and find its real and imaginary parts.

Step-by-step solution

1. Start with the original quotient. For an inverse, the numerator is 1.

   $$\frac{3\times i+4}{2\times i+3}$$

2. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

   $$3-2\times i$$

3. Multiply by the conjugate over itself; this fraction equals 1.

   $$\left(\frac{3-2\times i}{3-2\times i}\right)\times \left(\frac{3\times i+4}{2\times i+3}\right)$$

4. Multiply numerators together and denominators together. Keep their brackets intact.

   $$\frac{\left(3-2\times i\right)\times \left(3\times i+4\right)}{\left(3-2\times i\right)\times \left(2\times i+3\right)}$$

5. Calculate the numerator first.

6. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

   $$3\times i\times \left(3-2\times i\right)+4\times \left(3-2\times i\right)$$

7. Distribute each of those products. Keep all four multiplications visible.

   $$4\times 3+4\times \left(-1\right)\times 2\times i+3\times 3\times i+3\times \left(-2\right)\times ii$$

8. Real × real: multiply the numerical coefficients.

   $$4\times 3=12$$

9. Real × imaginary: multiply the numerical coefficients.

   $$4\times \left(-1\right)\times 2\times i=-8\times i$$

10. Imaginary × real: multiply the numerical coefficients.

    $$3\times 3\times i=9\times i$$

11. Imaginary × imaginary: multiply the numerical coefficients.

    $$3\times \left(-2\right)\times ii=-6\times i^2$$

12. Substitute these four products into the expansion.

    $$12-8\times i+9\times i-6\times i^2$$

13. In the last term replace i² by −1.

    $$-6\times i^2=\left(-6\right)\times \left(-1\right)$$

14. Multiply its coefficient by −1.

    $$\left(-6\right)\times \left(-1\right)=6$$

15. Rewrite the expansion after this sign calculation.

    $$12-8\times i+9\times i+6$$

16. Group the real terms and factor i out of the imaginary terms.

    $$12+6+i\times \left(-8+9\right)$$

17. Calculate the sum of the real terms.

    $$12+6=18$$

18. Calculate the sum of the imaginary coefficients.

    $$-8+9=1$$

19. Insert both calculated sums in a + bi form.

    $$i+18$$

20. Now calculate the denominator independently.

21. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$2\times i\times \left(3-2\times i\right)+3\times \left(3-2\times i\right)$$

22. Distribute each of those products. Keep all four multiplications visible.

    $$3\times 3+3\times \left(-1\right)\times 2\times i+2\times 3\times i+2\times \left(-2\right)\times ii$$

23. Real × real: multiply the numerical coefficients.

    $$3\times 3=9$$

24. Real × imaginary: multiply the numerical coefficients.

    $$3\times \left(-1\right)\times 2\times i=-6\times i$$

25. Imaginary × real: multiply the numerical coefficients.

    $$2\times 3\times i=6\times i$$

26. Imaginary × imaginary: multiply the numerical coefficients.

    $$2\times \left(-2\right)\times ii=-4\times i^2$$

27. Substitute these four products into the expansion.

    $$9-6\times i+6\times i-4\times i^2$$

28. In the last term replace i² by −1.

    $$-4\times i^2=\left(-4\right)\times \left(-1\right)$$

29. Multiply its coefficient by −1.

    $$\left(-4\right)\times \left(-1\right)=4$$

30. Rewrite the expansion after this sign calculation.

    $$9-6\times i+6\times i+4$$

31. Group the real terms and factor i out of the imaginary terms.

    $$9+4+i\times \left(-6+6\right)$$

32. Calculate the sum of the real terms.

    $$9+4=13$$

33. Calculate the sum of the imaginary coefficients.

    $$-6+6=0$$

34. Insert both calculated sums in a + bi form.

    $$0\times i+13$$

35. The imaginary coefficient in the denominator is zero, so the denominator is real.

    $$13=13$$

36. Put the calculated numerator above the calculated denominator.

    $$\frac{i+18}{13}$$

37. Separate the real term and imaginary term into two fractions with the same denominator.

    $$\frac{18}{13}+\frac{i}{13}$$

38. Move i outside its numerical fraction.

    $$\frac{18}{13}+\left(\frac{1}{13}\right)\times i$$

39. Simplify the real coefficient fraction.

    $$\frac{18}{13}=\frac{18}{13}$$

40. Simplify the imaginary coefficient fraction.

    $$\frac{1}{13}=\frac{1}{13}$$

41. Combine the reduced real and imaginary terms.

    $$\frac{i}{13}+\frac{18}{13}$$

42. A negative exponent asks for a reciprocal. Find the reciprocal first.

    $$\frac{1}{\frac{i}{13}+\frac{18}{13}}=\frac{1}{\frac{i}{13}+\frac{18}{13}}$$

43. Start with the original quotient. For an inverse, the numerator is 1.

    $$\frac{1}{\frac{i}{13}+\frac{18}{13}}$$

44. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

    $$\frac{18}{13}-\frac{i}{13}$$

45. Multiply by the conjugate over itself; this fraction equals 1.

    $$\left(\frac{\frac{18}{13}-\frac{i}{13}}{\frac{18}{13}-\frac{i}{13}}\right)\times \left(\frac{1}{\frac{i}{13}+\frac{18}{13}}\right)$$

46. Multiply numerators together and denominators together. Keep their brackets intact.

    $$\frac{\frac{18}{13}-\frac{i}{13}}{\left(\frac{18}{13}-\frac{i}{13}\right)\times \left(\frac{i}{13}+\frac{18}{13}\right)}$$

47. Calculate the numerator first.

48. Multiplication by 1 leaves the conjugate unchanged.

    $$\frac{18}{13}-\frac{i}{13}=\frac{18}{13}-\frac{i}{13}$$

49. Now calculate the denominator independently.

50. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$\frac{i\times \left(\frac{18}{13}-\frac{i}{13}\right)}{13}+\frac{\frac{324}{13}-\frac{18\times i}{13}}{13}$$

51. Distribute each of those products. Keep all four multiplications visible.

    $$\frac{18}{13}\times \frac{18}{13}+\frac{-18\times i}{169}+\frac{18\times i}{169}+\frac{-i^2}{169}$$

52. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{18\times 1}{1\times 13}$$

53. Calculate these two products separately.

    $$\frac{18}{13}$$

54. Calculate this operation before combining the remaining terms.

    $$18\times \frac{1}{13}=\frac{18}{13}$$

55. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{18\times 18}{13\times 13}$$

56. Calculate these two products separately.

    $$\frac{324}{169}$$

57. Real × real: multiply the numerical coefficients.

    $$\frac{18}{13}\times \frac{18}{13}=\frac{324}{169}$$

58. Real × imaginary: multiply the numerical coefficients.

    $$\frac{-18\times i}{169}=-\frac{18\times i}{169}$$

59. Imaginary × real: multiply the numerical coefficients.

    $$\frac{18\times i}{169}=\frac{18\times i}{169}$$

60. Imaginary × imaginary: multiply the numerical coefficients.

    $$\frac{-i^2}{169}=-\frac{i^2}{169}$$

61. Substitute these four products into the expansion.

    $$\frac{324}{169}-\frac{18\times i}{169}+\frac{18\times i}{169}-\frac{i^2}{169}$$

62. In the last term replace i² by −1.

    $$-\frac{i^2}{169}=\left(-\frac{1}{169}\right)\times \left(-1\right)$$

63. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 169}$$

64. Calculate these two products separately.

    $$-\frac{1}{169}$$

65. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{169}=-\frac{1}{169}$$

66. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times \left(-1\right)}{169\times 1}$$

67. Calculate these two products separately.

    $$\frac{1}{169}$$

68. Multiply its coefficient by −1.

    $$\left(-\frac{1}{169}\right)\times \left(-1\right)=\frac{1}{169}$$

69. Rewrite the expansion after this sign calculation.

    $$\frac{324}{169}-\frac{18\times i}{169}+\frac{18\times i}{169}+\frac{1}{169}$$

70. Group the real terms and factor i out of the imaginary terms.

    $$\frac{324}{169}+\frac{1}{169}+i\times \left(-\frac{18}{169}+\frac{18}{169}\right)$$

71. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{324\times 1}{1\times 169}$$

72. Calculate this operation before combining the remaining terms.

    $$324\times \frac{1}{169}=\frac{324}{169}$$

73. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 169}$$

74. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{169}=\frac{1}{169}$$

75. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=169$$

76. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{324}{169}=\frac{324\times 1}{169\times 1}$$

77. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1}{169}=\frac{1\times 1}{169\times 1}$$

78. Keep the common denominator and add the signed numerators.

    $$\frac{324+1}{169}=\frac{325}{169}$$

79. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=13$$

80. Show the two divisions explicitly.

    $$\frac{\frac{325}{13}}{\frac{169}{13}}$$

81. Evaluate the numerator division.

    $$\frac{325}{13}=25$$

82. Evaluate the denominator division.

    $$\frac{169}{13}=13$$

83. Write the reduced fraction.

    $$\frac{25}{13}$$

84. Calculate the sum of the real terms.

    $$\frac{324}{169}+\frac{1}{169}=\frac{25}{13}$$

85. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-18\right)\times 1}{1\times 169}$$

86. Calculate these two products separately.

    $$-\frac{18}{169}$$

87. Calculate this operation before combining the remaining terms.

    $$\left(-18\right)\times \frac{1}{169}=-\frac{18}{169}$$

88. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{18\times 1}{1\times 169}$$

89. Calculate these two products separately.

    $$\frac{18}{169}$$

90. Calculate this operation before combining the remaining terms.

    $$18\times \frac{1}{169}=\frac{18}{169}$$

91. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{18}{169}=\frac{\left(-18\right)\times 1}{169\times 1}$$

92. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{18}{169}=\frac{18\times 1}{169\times 1}$$

93. Keep the common denominator and add the signed numerators.

    $$\frac{-18+18}{169}=\frac{0}{169}$$

94. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=169$$

95. Show the two divisions explicitly.

    $$\frac{\frac{0}{169}}{\frac{169}{169}}$$

96. Evaluate the numerator division.

    $$\frac{0}{169}=0$$

97. Evaluate the denominator division.

    $$\frac{169}{169}=1$$

98. Write the reduced fraction.

    $$0\times \frac{1}{1}$$

99. Calculate the sum of the imaginary coefficients.

    $$-\frac{18}{169}+\frac{18}{169}=0$$

100. Insert both calculated sums in a + bi form.

     $$0\times i+\frac{25}{13}$$

101. The imaginary coefficient in the denominator is zero, so the denominator is real.

     $$\frac{25}{13}=\frac{25}{13}$$

102. Put the calculated numerator above the calculated denominator.

     $$\frac{\frac{18}{13}-\frac{i}{13}}{\frac{25}{13}}$$

103. Separate the real term and imaginary term into two fractions with the same denominator.

     $$\frac{18}{25}+\frac{-\frac{i}{13}}{\frac{25}{13}}$$

104. Move i outside its numerical fraction.

     $$\frac{18}{25}+\left(-\frac{1}{25}\right)\times i$$

105. Reduce the real coefficient by dividing numerator and denominator by 13.

     $$\frac{234}{325}=\frac{\frac{234}{13}}{\frac{325}{13}}$$

106. Simplify the real coefficient fraction.

     $$\frac{18}{25}=\frac{18}{25}$$

107. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-13\right)\times 1}{1\times 325}$$

108. Calculate these two products separately.

     $$-\frac{13}{325}$$

109. Show the two divisions explicitly.

     $$\frac{-\frac{13}{13}}{\frac{325}{13}}$$

110. Evaluate the numerator division.

     $$-\frac{13}{13}=-1$$

111. Write the reduced fraction.

     $$-\frac{1}{25}$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-13\right)\times \frac{1}{325}=-\frac{1}{25}$$

113. Reduce the imaginary coefficient by dividing numerator and denominator by 13.

     $$-\frac{13}{325}=\frac{-\frac{13}{13}}{\frac{325}{13}}$$

114. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-1\right)\times 1}{1\times 25}$$

115. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{25}=-\frac{1}{25}$$

116. Simplify the imaginary coefficient fraction.

     $$-\frac{1}{25}=-\frac{1}{25}$$

117. Combine the reduced real and imaginary terms.

     $$\frac{18}{25}-\frac{i}{25}$$

118. Now read the constant term as Re(z) and the coefficient of i as Im(z).

     $$(\mathrm{Re}=\frac{18}{25},\mathrm{Im}=-\frac{1}{25})$$

Answer

The final answer is shown below.