Mathematics 10 — Exercise 1.3 · Q5 (ii)

Full question

For z₁ = 2 + 3i, z₂ = −1 + i, find the imaginary part of z₁z₂.

Step-by-step solution

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

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

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

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

3. Real × real: multiply the numerical coefficients.

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

4. Real × imaginary: multiply the numerical coefficients.

   $$2\times 1\times i=2\times i$$

5. Imaginary × real: multiply the numerical coefficients.

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

6. Imaginary × imaginary: multiply the numerical coefficients.

   $$3\times 1\times ii=3\times i^2$$

7. Substitute these four products into the expansion.

   $$-2+2\times i-3\times i+3\times i^2$$

8. In the last term replace i² by −1.

   $$3\times i^2=3\times \left(-1\right)$$

9. Multiply its coefficient by −1.

   $$3\times \left(-1\right)=-3$$

10. Rewrite the expansion after this sign calculation.

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

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

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

12. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 3=-3$$

13. Calculate the sum of the real terms.

    $$-2-3=-5$$

14. Calculate the sum of the imaginary coefficients.

    $$2-3=-1$$

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

    $$-i-5$$

16. In a + bi the real part is a, and the imaginary part is b, not bi.

    $$\mathrm{Im}=-1$$

Answer

The final answer is shown below.