Complex Numbers – Examples & Exercises

Complex Numbers - Powers of Z

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 \(\omega^2.\)

2 thoughts on “Complex Numbers – Examples & Exercises”

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