Mathematics 10 — Exercise 1.2 · Q3 (ii)

Full question

Find the multiplicative inverse of the following complex number.

Step-by-step solution

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

   $$\frac{1}{2\times i+6}$$

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

   $$6-2\times i$$

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

   $$\left(\frac{6-2\times i}{6-2\times i}\right)\times \left(\frac{1}{2\times i+6}\right)$$

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

   $$\frac{6-2\times i}{\left(6-2\times i\right)\times \left(2\times i+6\right)}$$

5. Calculate the numerator first.

6. Multiplication by 1 leaves the conjugate unchanged.

   $$6-2\times i=6-2\times i$$

7. Now calculate the denominator independently.

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

   $$2\times i\times \left(6-2\times i\right)+6\times \left(6-2\times i\right)$$

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

   $$6\times 6+6\times \left(-1\right)\times 2\times i+2\times 6\times i+2\times \left(-2\right)\times ii$$

10. Real × real: multiply the numerical coefficients.

    $$6\times 6=36$$

11. Real × imaginary: multiply the numerical coefficients.

    $$6\times \left(-1\right)\times 2\times i=-12\times i$$

12. Imaginary × real: multiply the numerical coefficients.

    $$2\times 6\times i=12\times i$$

13. Imaginary × imaginary: multiply the numerical coefficients.

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

14. Substitute these four products into the expansion.

    $$36-12\times i+12\times i-4\times i^2$$

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

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

16. Multiply its coefficient by −1.

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

17. Rewrite the expansion after this sign calculation.

    $$36-12\times i+12\times i+4$$

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

    $$36+4+i\times \left(-12+12\right)$$

19. Calculate the sum of the real terms.

    $$36+4=40$$

20. Calculate the sum of the imaginary coefficients.

    $$-12+12=0$$

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

    $$0\times i+40$$

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

    $$40=40$$

23. Put the calculated numerator above the calculated denominator.

    $$\frac{6-2\times i}{40}$$

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

    $$\frac{6}{40}+\frac{-2\times i}{40}$$

25. Move i outside its numerical fraction.

    $$\frac{6}{40}+\left(-\frac{2}{40}\right)\times i$$

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

    $$\frac{6}{40}=\frac{\frac{6}{2}}{\frac{40}{2}}$$

27. Simplify the real coefficient fraction.

    $$\frac{6}{40}=\frac{3}{20}$$

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

    $$\frac{\left(-2\right)\times 1}{1\times 40}$$

29. Calculate these two products separately.

    $$-\frac{2}{40}$$

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

    $$\mathrm{GCD}=2$$

31. Show the two divisions explicitly.

    $$\frac{-\frac{2}{2}}{\frac{40}{2}}$$

32. Evaluate the numerator division.

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

33. Evaluate the denominator division.

    $$\frac{40}{2}=20$$

34. Write the reduced fraction.

    $$-\frac{1}{20}$$

35. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times \frac{1}{40}=-\frac{1}{20}$$

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

    $$-\frac{2}{40}=\frac{-\frac{2}{2}}{\frac{40}{2}}$$

37. Simplify the imaginary coefficient fraction.

    $$-\frac{2}{40}=-\frac{1}{20}$$

38. Combine the reduced real and imaginary terms.

    $$\frac{3}{20}-\frac{i}{20}$$

39. A multiplicative inverse must give 1 when multiplied by the original.

    $$1=1$$

Answer

The final answer is shown below.