# Complex Numbers – Examples & Exercises

This Post contains Worked Out Examples and Unsolved Exercises on Complex Numbers. Answers and hints are contained as per locations. To view or un-hide them, tap or move the mouse over them. For brevity, some steps in the solutions may not have been indicated in as many words. A note of advice though: try to do the examples as well – first, on your own.

Complex Numbers – Beyond Real Numbers

### Examples

1. Solve for x and y : $$(x + iy)(2 – 3i) = 4 + i.$$

Solution 1:

$$x + iy = \frac{4 + i}{2 – 3i} = \frac{(4 + i)(2 + 3i)}{(2 – 3i)(2 + 3i)} = \frac{5 + 14i}{2^2 + 3^2}$$

$$= \frac{5}{13} + i\frac{14}{13}\Rightarrow x = \frac{5}{13}, y = \frac{14}{13}$$

2. Find the value of $$x^3 + 7x^2 – x + 16,$$ when $$x = 1 + 2\sqrt{-1}.$$

Solution 2:

Given, $$x = 1 + 2\sqrt{-1} = 1 + 2i$$

$$\Rightarrow x – 1 = 2i$$

$$\Rightarrow (x – 1)^2 = (2i)^2$$

$$\Rightarrow x^2 – 2x + 5 = 0$$

Dividing $$x^3 + 7x^2 – x + 16$$ by $$x^2 – 2x + 5$$, we get

$$x^3 + 7x^2 – x + 16 = (x^2 – 2x + 5)(x + 9) + (12x – 29)$$

$$= (12x – 29) = 12(1 + 2i) – 29 = 24i – 17.$$

3. Find the square root of $$9 + 40i.$$

Solution 3:

$$9 + 40i = 9 + 2 \times 20i = 9 + 2 \times 5 \times 4i$$

$$= 5^2 + (4i)^2 + 2 \times 5 \times 4i = (5 + 4i)^2$$

$$\therefore$$ Square root $$= (5 + 4i)$$ or $$– (5 + 4i).$$

4. What will be the conjugate of $$\frac{1}{2 + 3i}?$$

Solution 4:

$$\frac{1}{2 + 3i} = \frac{2 – 3i}{(2 + 3i)(2 – 3i)} = \frac{2 – 3i}{13} = \frac{2}{13} – \frac{3}{13}i$$

5. Evaluate: $$\sqrt[3]{-i}$$

Solution 5:

$$\sqrt[3]{-i} = \sqrt[3]{i^3} = i \times \sqrt[3]{1} = i \times 1, i\omega , i\omega^2$$

6. Express $$\left( -\sqrt{3} + \sqrt{-2} \right)\left( 2\sqrt{3} – i \right)$$ in the form of a+ib.

Solution 6:

$$\left( -\sqrt{3} + \sqrt{-2} \right)\left( 2\sqrt{3} – i \right) = \left( -\sqrt{3} + \sqrt{2}i \right)\left( 2\sqrt{3} – i \right)$$

$$= -6 + \sqrt{3}i + 2\sqrt{6}i – \sqrt{2}i^2 = -\left( 6 + \sqrt{2} \right) + \sqrt{3}\left( 1 + 2\sqrt{2} \right) i$$

7. Express $$i^{-35}$$ in the a + ib form.

Solution 7:

$$i^{-35} = \frac{1}{i^{35}} = \frac{1}{(i^2)^{17}i} = \frac{1}{-i} \times \frac{i}{i} = \frac{i}{-i^2} = i$$

8. Evaluate: $$a^6 + a^4 + a^2 + 1$$, when $$a = \frac{1 + i}{\sqrt{2}}.$$

Solution 8:

$$a = \frac{1 + i}{\sqrt{2}} \Rightarrow \sqrt{2}a – 1 = i \Rightarrow 2a^2 + 1 – 2\sqrt{2}a = i^2 \Rightarrow a^2 – \sqrt{2}a + 1 = 0$$

So, $$a^6 + a^4 + a^2 + 1$$

$$= \left( a^2 – \sqrt{2}a + 1 \right)\left( a^4 + \sqrt{2}a^3 + 2a^2 + \sqrt{2}a + 1 \right) + 0 = 0 + 0 = 0.$$

Dave’s Short Course on Complex Numbers

### Exercises

1. Find the least positive value of $$n$$ if $$\left( \frac{1 + i}{1 – i} \right)^n = 1.$$

2. If $$\sqrt[3]{a + ib} = x + iy,$$ prove that $$\sqrt[3]{a – ib} = x – iy.$$

3. Express $$(1 + a^2)(1 + b^2)$$ as the sum of two squares.

4. Find the square root of $$8 – 15i.$$

5. What is the square root of $$(\sqrt{i} + \sqrt{-i})?$$

6. Find the product of $$(1 – i)$$ and its conjugate.

7. Find the modulus and argument of $$\frac{1}{\cos \theta – i\sin \theta}.$$

8. Convert into the trigonometrical form: $$\frac{1 + i}{2 – i}.$$

9. Find the value of the complex number $$z$$ when $$z^2 = (i\overline{z})^2.$$

10. If $$\frac{z – 1}{z + 1}$$ be purely imaginary, then prove that $$|z| = 1.$$

11. Evaluate : $$\sqrt{-2 + 2\sqrt{-2 + 2\sqrt{-2 + ⋯ : \infty}}}$$

12. Prove that : $$\left( \frac{i + \sqrt{3}}{2} \right)^{100} + \left( \frac{i – \sqrt{3}}{2} \right)^{100} = -1$$

13. Find the cube root of $$i$$.

14. Resolve into linear factors : $$a^3 – b^3$$

15. Prove that : $$\sqrt{-1 – \sqrt{-1 – \sqrt{-1 – ⋯ : \infty}}} = \omega$$ or $$\omega^2.$$