Mathematics 10 — Exercise Worked examples · Example 16

Full question

For the given z, verify the displayed identity.

Step-by-step solution

1. First write z and its conjugate. Only the imaginary sign changes.

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

2. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

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

3. Distribute each of those products. Keep all four multiplications visible.

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

4. Real × real: multiply the numerical coefficients.

   $$5\times 5=25$$

5. Real × imaginary: multiply the numerical coefficients.

   $$5\times \left(-1\right)\times 3\times i=-15\times i$$

6. Imaginary × real: multiply the numerical coefficients.

   $$3\times 5\times i=15\times i$$

7. Imaginary × imaginary: multiply the numerical coefficients.

   $$3\times \left(-3\right)\times ii=-9\times i^2$$

8. Substitute these four products into the expansion.

   $$25-15\times i+15\times i-9\times i^2$$

9. In the last term replace i² by −1.

   $$-9\times i^2=\left(-9\right)\times \left(-1\right)$$

10. Multiply its coefficient by −1.

    $$\left(-9\right)\times \left(-1\right)=9$$

11. Rewrite the expansion after this sign calculation.

    $$25-15\times i+15\times i+9$$

12. Group the real terms and factor i out of the imaginary terms.

    $$25+9+i\times \left(-15+15\right)$$

13. Calculate the sum of the real terms.

    $$25+9=34$$

14. Calculate the sum of the imaginary coefficients.

    $$-15+15=0$$

15. Insert both calculated sums in a + bi form.

    $$0\times i+34$$

16. Read a, the real coordinate.

    $$a=5$$

17. Read b, the coefficient of i, including its sign.

    $$b=3$$

18. Use |z| = √(a² + b²) and substitute both coordinates.

    $$\sqrt{5^2+3^2}$$

19. Square the real coordinate.

    $$5^2=25$$

20. Square the imaginary coordinate; a negative number squared is positive.

    $$3^2=9$$

21. Add the two squared values.

    $$25+9=34$$

22. Take the nonnegative square root of this sum.

    $$\sqrt{34}=\sqrt{34}$$

23. Square the modulus and compare it to the product.

    $$34=34$$

Answer

The final answer is shown below.