Mathematics 10 — Exercise Review · Q4

Full question

Find the additive and multiplicative inverses of z = 8 + 9i.

Step-by-step solution

1. For the additive inverse, change both signs.

   $$-9\times i-8$$

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

   $$\frac{1}{9\times i+8}$$

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

   $$8-9\times i$$

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

   $$\left(\frac{8-9\times i}{8-9\times i}\right)\times \left(\frac{1}{9\times i+8}\right)$$

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

   $$\frac{8-9\times i}{\left(8-9\times i\right)\times \left(9\times i+8\right)}$$

6. Calculate the numerator first.

7. Multiplication by 1 leaves the conjugate unchanged.

   $$8-9\times i=8-9\times i$$

8. Now calculate the denominator independently.

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

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

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

    $$8\times 8+8\times \left(-1\right)\times 9\times i+9\times 8\times i+9\times \left(-9\right)\times ii$$

11. Real × real: multiply the numerical coefficients.

    $$8\times 8=64$$

12. Real × imaginary: multiply the numerical coefficients.

    $$8\times \left(-1\right)\times 9\times i=-72\times i$$

13. Imaginary × real: multiply the numerical coefficients.

    $$9\times 8\times i=72\times i$$

14. Imaginary × imaginary: multiply the numerical coefficients.

    $$9\times \left(-9\right)\times ii=-81\times i^2$$

15. Substitute these four products into the expansion.

    $$64-72\times i+72\times i-81\times i^2$$

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

    $$-81\times i^2=\left(-81\right)\times \left(-1\right)$$

17. Multiply its coefficient by −1.

    $$\left(-81\right)\times \left(-1\right)=81$$

18. Rewrite the expansion after this sign calculation.

    $$64-72\times i+72\times i+81$$

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

    $$64+81+i\times \left(-72+72\right)$$

20. Calculate the sum of the real terms.

    $$64+81=145$$

21. Calculate the sum of the imaginary coefficients.

    $$-72+72=0$$

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

    $$0\times i+145$$

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

    $$145=145$$

24. Put the calculated numerator above the calculated denominator.

    $$\frac{8-9\times i}{145}$$

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

    $$\frac{8}{145}+\frac{-9\times i}{145}$$

26. Move i outside its numerical fraction.

    $$\frac{8}{145}+\left(-\frac{9}{145}\right)\times i$$

27. Simplify the real coefficient fraction.

    $$\frac{8}{145}=\frac{8}{145}$$

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

    $$\frac{\left(-9\right)\times 1}{1\times 145}$$

29. Calculate these two products separately.

    $$-\frac{9}{145}$$

30. Calculate this operation before combining the remaining terms.

    $$\left(-9\right)\times \frac{1}{145}=-\frac{9}{145}$$

31. Simplify the imaginary coefficient fraction.

    $$-\frac{9}{145}=-\frac{9}{145}$$

32. Combine the reduced real and imaginary terms.

    $$\frac{8}{145}-\frac{9\times i}{145}$$

Answer

The final answer is shown below.