Mathematics 10 — Exercise 1.2 · Q4 (v)

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.

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

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

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

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

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

5. Calculate the real part only.

   $$2-2=0$$

6. Calculate the imaginary coefficient only.

   $$5-5=0$$

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

   $$0+0\times i$$

8. Now independently compute the right side.

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

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

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

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

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

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

12. Calculate the real part only.

    $$-2+2=0$$

13. Calculate the imaginary coefficient only.

    $$-5+5=0$$

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

    $$0+0\times i$$

15. The results are identical, proving the equality.

    $$0=0$$

Answer

The final answer is shown below.