Mathematics 10 — Exercise 1.4 · Q1 (vii)

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.

   $$\frac{1-2\times i}{i+1}$$

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

   $$1-i$$

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

   $$\left(\frac{1-2\times i}{i+1}\right)\times \left(\frac{1-i}{1-i}\right)$$

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

   $$\frac{\left(1-2\times i\right)\times \left(1-i\right)}{\left(1-i\right)\times \left(i+1\right)}$$

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.

   $$1-i-2\times i\times \left(1-i\right)$$

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

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

8. Real × real: multiply the numerical coefficients.

   $$1\times 1=1$$

9. Real × imaginary: multiply the numerical coefficients.

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

10. Imaginary × real: multiply the numerical coefficients.

    $$-2\times 1\times i=-2\times i$$

11. Imaginary × imaginary: multiply the numerical coefficients.

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

12. Substitute these four products into the expansion.

    $$1-i-2\times i+2\times i^2$$

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

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

14. Multiply its coefficient by −1.

    $$2\times \left(-1\right)=-2$$

15. Rewrite the expansion after this sign calculation.

    $$1-i-2\times i-2$$

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

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

17. Calculate the sum of the real terms.

    $$1-2=-1$$

18. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 2=-2$$

19. Calculate the sum of the imaginary coefficients.

    $$-1-2=-3$$

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

    $$-3\times i-1$$

21. Now calculate the denominator independently.

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

    $$1-i+i\times \left(1-i\right)$$

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

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

24. Real × real: multiply the numerical coefficients.

    $$1\times 1=1$$

25. Real × imaginary: multiply the numerical coefficients.

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

26. Imaginary × real: multiply the numerical coefficients.

    $$1\times i=i$$

27. Imaginary × imaginary: multiply the numerical coefficients.

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

28. Substitute these four products into the expansion.

    $$1-i+i-i^2$$

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

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

30. Multiply its coefficient by −1.

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

31. Rewrite the expansion after this sign calculation.

    $$1-i+i+1$$

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

    $$1+1+i\times \left(-1+1\right)$$

33. Calculate the sum of the real terms.

    $$1+1=2$$

34. Calculate the sum of the imaginary coefficients.

    $$-1+1=0$$

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

    $$0\times i+2$$

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

    $$2=2$$

37. Put the calculated numerator above the calculated denominator.

    $$\frac{-3\times i-1}{2}$$

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

    $$-\frac{1}{2}+\frac{-3\times i}{2}$$

39. Move i outside its numerical fraction.

    $$-\frac{1}{2}+\left(-\frac{3}{2}\right)\times i$$

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

    $$\frac{\left(-1\right)\times 1}{1\times 2}$$

41. Calculate these two products separately.

    $$-\frac{1}{2}$$

42. Calculate this operation before combining the remaining terms.

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

43. Simplify the real coefficient fraction.

    $$-\frac{1}{2}=-\frac{1}{2}$$

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

    $$\frac{\left(-3\right)\times 1}{1\times 2}$$

45. Calculate these two products separately.

    $$-\frac{3}{2}$$

46. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{2}=-\frac{3}{2}$$

47. Simplify the imaginary coefficient fraction.

    $$-\frac{3}{2}=-\frac{3}{2}$$

48. Combine the reduced real and imaginary terms.

    $$-\frac{3\times i}{2}-\frac{1}{2}$$

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

    $$-\frac{3\times i\times \left(-\frac{3\times i}{2}-\frac{1}{2}\right)}{2}-\frac{-\frac{3\times i}{2}-\frac{1}{2}}{2}$$

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

    $$-(-\frac{1}{4})-(-\frac{3\times i}{4})-\frac{-3\times i}{4}-\frac{-9\times i^2}{4}$$

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

    $$\frac{\left(-1\right)\times \left(-1\right)}{2\times 2}$$

52. Calculate these two products separately.

    $$\frac{1}{4}$$

53. Real × real: multiply the numerical coefficients.

    $$\left(-\frac{1}{2}\right)\times \left(-\frac{1}{2}\right)=\frac{1}{4}$$

54. Real × imaginary: multiply the numerical coefficients.

    $$-\frac{-3\times i}{4}=\frac{3\times i}{4}$$

55. Imaginary × real: multiply the numerical coefficients.

    $$-\frac{-3\times i}{4}=\frac{3\times i}{4}$$

56. Imaginary × imaginary: multiply the numerical coefficients.

    $$-\frac{-9\times i^2}{4}=\frac{9\times i^2}{4}$$

57. Substitute these four products into the expansion.

    $$\frac{1}{4}+\frac{3\times i}{4}+\frac{3\times i}{4}+\frac{9\times i^2}{4}$$

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

    $$\frac{9\times i^2}{4}=\frac{9}{4}\times \left(-1\right)$$

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

    $$\frac{9\times 1}{1\times 4}$$

60. Calculate these two products separately.

    $$\frac{9}{4}$$

61. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{4}=\frac{9}{4}$$

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

    $$\frac{9\times \left(-1\right)}{4\times 1}$$

63. Calculate these two products separately.

    $$-\frac{9}{4}$$

64. Multiply its coefficient by −1.

    $$\frac{9}{4}\times \left(-1\right)=-\frac{9}{4}$$

65. Rewrite the expansion after this sign calculation.

    $$\frac{1}{4}+\frac{3\times i}{4}+\frac{3\times i}{4}-\frac{9}{4}$$

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

    $$\frac{1}{4}-\frac{9}{4}+i\times \left(\frac{3}{4}+\frac{3}{4}\right)$$

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

    $$\frac{1\times 1}{1\times 4}$$

68. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

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

    $$\frac{\left(-1\right)\times 9}{1\times 4}$$

70. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{9}{4}=-\frac{9}{4}$$

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

    $$\mathrm{LCD}=4$$

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

    $$\frac{1}{4}=\frac{1\times 1}{4\times 1}$$

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

    $$-\frac{9}{4}=\frac{\left(-9\right)\times 1}{4\times 1}$$

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

    $$\frac{1-9}{4}=-\frac{8}{4}$$

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

    $$\mathrm{GCD}=4$$

76. Show the two divisions explicitly.

    $$\frac{-\frac{8}{4}}{\frac{4}{4}}$$

77. Evaluate the numerator division.

    $$-\frac{8}{4}=-2$$

78. Evaluate the denominator division.

    $$\frac{4}{4}=1$$

79. Write the reduced fraction.

    $$-2\times \frac{1}{1}$$

80. Calculate the sum of the real terms.

    $$\frac{1}{4}-\frac{9}{4}=-2$$

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

    $$\frac{3\times 1}{1\times 4}$$

82. Calculate these two products separately.

    $$\frac{3}{4}$$

83. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{4}=\frac{3}{4}$$

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

    $$\frac{3}{4}=\frac{3\times 1}{4\times 1}$$

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

    $$\frac{3+3}{4}=\frac{6}{4}$$

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

    $$\mathrm{GCD}=2$$

87. Show the two divisions explicitly.

    $$\frac{\frac{6}{2}}{\frac{4}{2}}$$

88. Evaluate the numerator division.

    $$\frac{6}{2}=3$$

89. Evaluate the denominator division.

    $$\frac{4}{2}=2$$

90. Write the reduced fraction.

    $$\frac{3}{2}$$

91. Calculate the sum of the imaginary coefficients.

    $$\frac{3}{4}+\frac{3}{4}=\frac{3}{2}$$

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

    $$\frac{3\times i}{2}-2$$

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

    $$(\mathrm{Re}=-2,\mathrm{Im}=\frac{3}{2})$$

Answer

The final answer is shown below.