Mathematics 10 — Exercise 1.3 · Q2 (i)

Full question

For the given z₁,z₂, verify the displayed conjugate property.

Step-by-step solution

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

   $$7\times i+2-(3\times i-4)$$

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

   $$2+7\times i+4-3\times i$$

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

   $$2+4+i\times \left(7-3\right)$$

4. Calculate the real part only.

   $$2+4=6$$

5. Calculate the imaginary coefficient only.

   $$7-3=4$$

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

   $$4\times i+6$$

7. Take the conjugate of that result by changing the sign of its imaginary coefficient.

   $$6-4\times i$$

8. Now combine the conjugates of the original numbers independently.

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

   $$-(7\times i-2)+3\times i+4$$

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

    $$2-7\times i+4+3\times i$$

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

    $$2+4+i\times \left(-7+3\right)$$

12. Calculate the real part only.

    $$2+4=6$$

13. Calculate the imaginary coefficient only.

    $$-7+3=-4$$

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

    $$6-4\times i$$

15. Both results are equal.

    $$6-4\times i=6-4\times i$$

Answer

The final answer is shown below.