Mathematics 10 — Exercise 1.2 · Q4 (i)

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. Now independently compute the right side.

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

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

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

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

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

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

12. Calculate the real part only.

    $$1+2=3$$

13. Calculate the imaginary coefficient only.

    $$-3+5=2$$

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

    $$2\times i+3$$

15. The results are identical, proving the equality.

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

Answer

The final answer is shown below.