Mathematics 10 — Exercise Worked examples · Example 20

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.

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

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

   $$i+1$$

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

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

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

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

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.

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

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

   $$1\times 1+1\times i+1\times i+1\times ii$$

8. Real × real: multiply the numerical coefficients.

   $$1\times 1=1$$

9. Real × imaginary: multiply the numerical coefficients.

   $$1\times i=i$$

10. Imaginary × real: multiply the numerical coefficients.

    $$1\times i=i$$

11. Imaginary × imaginary: multiply the numerical coefficients.

    $$1\times ii=i^2$$

12. Substitute these four products into the expansion.

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

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

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

14. Multiply its coefficient by −1.

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

15. Rewrite the expansion after this sign calculation.

    $$1+i+i-1$$

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

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

17. Calculate the sum of the real terms.

    $$1-1=0$$

18. Calculate the sum of the imaginary coefficients.

    $$1+1=2$$

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

    $$0+2\times i$$

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.

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

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

    $$1\times 1+1\times i-1\times i-1\times ii$$

23. Real × real: multiply the numerical coefficients.

    $$1\times 1=1$$

24. Real × imaginary: multiply the numerical coefficients.

    $$1\times i=i$$

25. Imaginary × real: multiply the numerical coefficients.

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

26. Imaginary × imaginary: multiply the numerical coefficients.

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

27. Substitute these four products into the expansion.

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

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

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

29. Multiply its coefficient by −1.

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

30. Rewrite the expansion after this sign calculation.

    $$1+i-i+1$$

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

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

32. Calculate the sum of the real terms.

    $$1+1=2$$

33. Calculate the sum of the imaginary coefficients.

    $$1-1=0$$

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

    $$0\times i+2$$

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

    $$2=2$$

36. Put the calculated numerator above the calculated denominator.

    $$\frac{0+2\times i}{2}$$

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

    $$\frac{0}{2}+\frac{2\times i}{2}$$

38. Move i outside its numerical fraction.

    $$\frac{0}{2}+\left(\frac{2}{2}\right)\times i$$

39. Reduce the real coefficient by dividing numerator and denominator by 2.

    $$\frac{0}{2}=\frac{\frac{0}{2}}{\frac{2}{2}}$$

40. Simplify the real coefficient fraction.

    $$\frac{0}{2}=0$$

41. Reduce the imaginary coefficient by dividing numerator and denominator by 2.

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

42. Simplify the imaginary coefficient fraction.

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

43. Combine the reduced real and imaginary terms.

    $$0+i$$

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

    $$\frac{1}{i}=\frac{1}{i}$$

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

    $$\frac{1}{i}$$

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

    $$-i$$

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

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

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

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

49. Calculate the numerator first.

50. Multiplication by 1 leaves the conjugate unchanged.

    $$-i=-i$$

51. Now calculate the denominator independently.

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

    $$0\times \left(-1\right)\times i+\left(-i\right)\times i$$

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

    $$0\times 0+0\times \left(-1\right)\times i+0\times i+\left(-1\right)\times ii$$

54. Real × real: multiply the numerical coefficients.

    $$0\times 0=0$$

55. Real × imaginary: multiply the numerical coefficients.

    $$0\times \left(-1\right)\times i=0\times i$$

56. Imaginary × real: multiply the numerical coefficients.

    $$0\times i=0\times i$$

57. Imaginary × imaginary: multiply the numerical coefficients.

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

58. Substitute these four products into the expansion.

    $$0+0\times i+0\times i-i^2$$

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

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

60. Multiply its coefficient by −1.

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

61. Rewrite the expansion after this sign calculation.

    $$0+0\times i+0\times i+1$$

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

    $$0+1+i\times \left(0+0\right)$$

63. Calculate the sum of the real terms.

    $$0+1=1$$

64. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

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

    $$0\times i+1$$

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

    $$1=1$$

67. Put the calculated numerator above the calculated denominator.

    $$\frac{1}{1}\times \left(0-i\right)$$

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

    $$0\times \frac{1}{1}+\frac{1}{1}\times \left(-i\right)$$

69. Move i outside its numerical fraction.

    $$0\times \frac{1}{1}+\left(-\frac{1}{1}\right)\times i$$

70. Simplify the real coefficient fraction.

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

71. Calculate this operation before combining the remaining terms.

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

72. Calculate this operation before combining the remaining terms.

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

73. Simplify the imaginary coefficient fraction.

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

74. Combine the reduced real and imaginary terms.

    $$0-i$$

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

    $$0\times \left(-1\right)\times i-\left(-i\right)\times i$$

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

    $$0\times 0+0\times \left(-1\right)\times i-0\times i-\left(-1\right)\times ii$$

77. Real × real: multiply the numerical coefficients.

    $$0\times 0=0$$

78. Real × imaginary: multiply the numerical coefficients.

    $$0\times \left(-1\right)\times i=0\times i$$

79. Imaginary × real: multiply the numerical coefficients.

    $$-0\times i=0\times i$$

80. Imaginary × imaginary: multiply the numerical coefficients.

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

81. Substitute these four products into the expansion.

    $$0+0\times i+0\times i+i^2$$

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

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

83. Multiply its coefficient by −1.

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

84. Rewrite the expansion after this sign calculation.

    $$0+0\times i+0\times i-1$$

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

    $$0-1+i\times \left(0+0\right)$$

86. Calculate the sum of the real terms.

    $$0-1=-1$$

87. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

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

    $$0\times i-1$$

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

    $$(\mathrm{Re}=-1,\mathrm{Im}=0)$$

Answer

The final answer is shown below.