Mathematics 10 — Exercise Worked examples · Example 11

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.

   $$3\times i+4+2\times i+5$$

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

   $$4+3\times i+5+2\times i$$

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

   $$4+5+i\times \left(3+2\right)$$

4. Calculate the real part only.

   $$4+5=9$$

5. Calculate the imaginary coefficient only.

   $$3+2=5$$

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

   $$5\times i+9$$

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

   $$9-5\times i$$

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

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

   $$-(3\times i-4)-(2\times i-5)$$

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

    $$4-3\times i+5-2\times i$$

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

    $$4+5+i\times \left(-3-2\right)$$

12. Calculate the real part only.

    $$4+5=9$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 2=-2$$

14. Calculate the imaginary coefficient only.

    $$-3-2=-5$$

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

    $$9-5\times i$$

16. Both results are equal.

    $$9-5\times i=9-5\times i$$

Answer

The final answer is shown below.