Mathematics 10 — Exercise 1.2 · Q4 (iii)

Full question

For z₁ = 2 + 5i, z₂ = 1 − 3i, z₃ = 2 + i, verify the stated property by computing both sides.

Step-by-step solution

1. Compute the left side first.

2. Write both complex numbers with their real and imaginary parts.

   $$5\times i+2-(3\times i-1)$$

3. Remove the brackets, preserving the signs of all four terms.

   $$2+5\times i+1-3\times i$$

4. Place the real terms together. Factor i out of the imaginary terms.

   $$2+1+i\times \left(5-3\right)$$

5. Calculate the real part only.

   $$2+1=3$$

6. Calculate the imaginary coefficient only.

   $$5-3=2$$

7. Put these two results into a + bi form.

   $$2\times i+3$$

8. Write both complex numbers with their real and imaginary parts.

   $$2\times i+3+i+2$$

9. Remove the brackets, preserving the signs of all four terms.

   $$3+2\times i+2+i$$

10. Place the real terms together. Factor i out of the imaginary terms.

    $$3+2+i\times \left(2+1\right)$$

11. Calculate the real part only.

    $$3+2=5$$

12. Calculate the imaginary coefficient only.

    $$2+1=3$$

13. Put these two results into a + bi form.

    $$3\times i+5$$

14. Now independently compute the right side.

15. Write both complex numbers with their real and imaginary parts.

    $$-(3\times i-1)+i+2$$

16. Remove the brackets, preserving the signs of all four terms.

    $$1-3\times i+2+i$$

17. Place the real terms together. Factor i out of the imaginary terms.

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

18. Calculate the real part only.

    $$1+2=3$$

19. Calculate the imaginary coefficient only.

    $$-3+1=-2$$

20. Put these two results into a + bi form.

    $$3-2\times i$$

21. Write both complex numbers with their real and imaginary parts.

    $$5\times i+2-(2\times i-3)$$

22. Remove the brackets, preserving the signs of all four terms.

    $$2+5\times i+3-2\times i$$

23. Place the real terms together. Factor i out of the imaginary terms.

    $$2+3+i\times \left(5-2\right)$$

24. Calculate the real part only.

    $$2+3=5$$

25. Calculate the imaginary coefficient only.

    $$5-2=3$$

26. Put these two results into a + bi form.

    $$3\times i+5$$

27. The results are identical, proving the equality.

    $$3\times i+5=3\times i+5$$

Answer

The final answer is shown below.