Mathematics 10 — Exercise 1.2 · Q3 (iv)

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.

   $$\frac{1}{-4\times i+\sqrt{5}}$$

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

   $$4\times i+\sqrt{5}$$

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

   $$\left(\frac{4\times i+\sqrt{5}}{4\times i+\sqrt{5}}\right)\times \left(\frac{1}{-4\times i+\sqrt{5}}\right)$$

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

   $$\frac{4\times i+\sqrt{5}}{\left(-4\times i+\sqrt{5}\right)\times \left(4\times i+\sqrt{5}\right)}$$

5. Calculate the numerator first.

6. Multiplication by 1 leaves the conjugate unchanged.

   $$4\times i+\sqrt{5}=4\times i+\sqrt{5}$$

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.

   $$-4\times i\times \left(4\times i+\sqrt{5}\right)+\sqrt{5}\times \left(4\times i+\sqrt{5}\right)$$

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

   $$\sqrt{5}\times \sqrt{5}+4\times \sqrt{5}\times i-4\times \sqrt{5}\times i-4\times 4\times ii$$

10. Real × real: multiply the numerical coefficients.

    $$\sqrt{5}\times \sqrt{5}=5$$

11. Real × imaginary: multiply the numerical coefficients.

    $$4\times \sqrt{5}\times i=\left(4\times \sqrt{5}\right)\times i$$

12. Imaginary × real: multiply the numerical coefficients.

    $$-4\times \sqrt{5}\times i=\left(\left(-4\right)\times \sqrt{5}\right)\times i$$

13. Imaginary × imaginary: multiply the numerical coefficients.

    $$-4\times 4\times ii=-16\times i^2$$

14. Substitute these four products into the expansion.

    $$5+\left(4\times \sqrt{5}\right)\times i+\left(\left(-4\right)\times \sqrt{5}\right)\times i-16\times i^2$$

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

    $$-16\times i^2=\left(-16\right)\times \left(-1\right)$$

16. Multiply its coefficient by −1.

    $$\left(-16\right)\times \left(-1\right)=16$$

17. Rewrite the expansion after this sign calculation.

    $$5+\left(4\times \sqrt{5}\right)\times i+\left(\left(-4\right)\times \sqrt{5}\right)\times i+16$$

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

    $$5+16+i\times \left(4\times \sqrt{5}-4\times \sqrt{5}\right)$$

19. Calculate the sum of the real terms.

    $$5+16=21$$

20. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 4\times \sqrt{5}=\left(-4\right)\times \sqrt{5}$$

21. Calculate the sum of the imaginary coefficients.

    $$4\times \sqrt{5}-4\times \sqrt{5}=0$$

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

    $$0\times i+21$$

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

    $$21=21$$

24. Put the calculated numerator above the calculated denominator.

    $$\frac{4\times i+\sqrt{5}}{21}$$

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

    $$\frac{4\times i}{21}+\frac{\sqrt{5}}{21}$$

26. Move i outside its numerical fraction.

    $$\left(\frac{4}{21}\right)\times i+\frac{\sqrt{5}}{21}$$

27. Calculate this operation before combining the remaining terms.

    $$\sqrt{5}\times \frac{1}{21}=\frac{1}{21}\times \sqrt{5}$$

28. Simplify the real coefficient fraction.

    $$\frac{\sqrt{5}}{21}=\frac{1}{21}\times \sqrt{5}$$

29. Simplify the imaginary coefficient fraction.

    $$\frac{4}{21}=\frac{4}{21}$$

30. Combine the reduced real and imaginary terms.

    $$\frac{4\times i}{21}+\frac{1}{21}\times \sqrt{5}$$

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

    $$1=1$$

Answer

The final answer is shown below.