Learn – Algebra/Precalculus – Complex numbers

1 Geometric structure

1.1 Exercises

1.1.1 Problem set

For each of the following, mark the number on the complex (coordinate) plane.

  1. i
  2. 4i
  3. -3i
  4. 2+5i
  5. -5+2i
  6. -4-3i

2 Addition

2.1 Exercises

2.1.1 Problem set

For each of the following, simplify.

  1. 3i+4i
  2. 7i-2i
  3. -2i+6i
  4. -5i-4i

2.1.2 Problem set

For each of the following, simplify.

  1. (3+4i)+(5+2i)
  2. (-2+5i)+(5-6i)
  3. (-5-2i)-(-2+2i)
  4. (-4-7i)-(7-4i)

2.1.3 Problem set

In each of the following, complex numbers z_1 and z_2 are shown on the complex plane. Plot the complex number z_1 + z_2 on the plane.









3 Modulus and argument

3.1 Exercises

3.1.1 Problem set

For each of following complex numbers, mark the number on the complex plane and find the modulus of the number.

  1. -5
  2. 3i
  3. -7i
  4. 3+4i
  5. -4+3i
  6. -2-2i
  7. 5-5i
  8. -3+3\sqrt{3}i
  9. -\sqrt{3}-3i
  10. -7\sqrt{3}-21i

3.1.2 Problem set

For each of following complex numbers, mark the number on the complex plane and find the argument of the number.

  1. 4
  2. -100
  3. 123i
  4. -51.7i

3.1.3 Problem set

For each of following complex numbers, mark the number on the complex plane and find the argument of the number.

  1. 3+3i
  2. -5+5i
  3. -11-11i
  4. -2-2\sqrt{3}i
  5. 11-11\sqrt{3}i
  6. 6-2\sqrt{3}i
  7. -12+4\sqrt{3}i
  8. -5\sqrt{3}-15i

3.1.4 Problem set

In each of the following problems, find the complex number z.

  1. |z| = 4, \arg(z) = 0
  2. |z| = 3, \arg(z) = \frac{\pi}{2}
  3. |z| = 12.35, \arg(z) = \pi
  4. |z| = 134.5, \arg(z) = \frac{3\pi}{2}
  5. |z| = 4, \arg(z) = -\frac{\pi}{2}
  6. |z| = 4.7, \arg(z) = -\frac{3\pi}{2}

3.1.5 Problem set

In each of the following problems, find the complex number z.

  1. |z| = 2, \arg(z) = \frac{\pi}{4}
  2. |z| = 3, \arg(z) = \frac{\pi}{3}
  3. |z| = 5, \arg(z) = \frac{5\pi}{6}
  4. |z| = 10, \arg(z) = \frac{5\pi}{4}
  5. |z| = 6, \arg(z) = \frac{5\pi}{3}

3.1.6 Problem set

In each of the following problems, find the complex number z.

  1. |z| = 4, \arg(z) = -\frac{\pi}{4}
  2. |z| = 6, \arg(z) = -\frac{4\pi}{3}
  3. |z| = 10.3, \arg(z) = 19\pi
  4. |z| = 18, \arg(z) = \frac{20\pi}{3}
  5. |z| = 22, \arg(z) = -\frac{31\pi}{4}
  6. |z| = 3, \arg(z) = -\frac{194\pi}{3}

3.1.7 Problem set

For complex numbers z, z_1 and z_2, justify the following.

  1. |z_1+z_2| \le |z_1| + |z_2|
  2. |z_1z_2| = |z_1||z_2|

4 Polar form

4.1 Exercises

4.1.1 Problem set

Write each of the following complex numbers in polar form.

  1. 3+3i
  2. -4+4i
  3. -\sqrt{3}-i
  4. \sqrt{27}-3i
  5. -2+\sqrt{12}i

4.1.2 Problem set

Write each of the following complex numbers in polar form.

  1. 3
  2. -6.9
  3. -8i
  4. -10.3i
  5. -\pi
  6. -1.2\overline{34}i

5 Multiplication

5.1 Exercises

5.1.1 Problem set

Evaluate the following.

  1. -5\times(-4-7i)
  2. (3+4i)\times(2+3i)
  3. (3+0i)\times(-10+0i)
  4. (-2+5i)\times(3-4i)
  5. \left(3\,\cos \frac{19\pi}{6}+i\,3\,\sin \frac{19\pi}{6}\right)\times\left(2\,\cos \left(-\frac{19\pi}{6}\right)+i\,2\,\sin \left(-\frac{19\pi}{6}\right)\right)
  6. \left(\sqrt{2}\,\cos \frac{13\pi}{3}+i\,\sqrt{2}\,\sin\frac{13\pi}{3}\right)\times\left(\sqrt{8}\,\cos\frac{5\pi}{3}+i\,\sqrt{8}\,\sin\frac{5\pi}{3}\right)
  7. \left(\sqrt{6}\,\cos \left(-\frac{\pi}{3}\right)+i\,\sqrt{6}\,\sin\left(-\frac{\pi}{3}\right)\right)\times\left(\sqrt{3}\,\cos\left(-\frac{\pi}{6}\right)+i\,\sqrt{3}\,\sin\left(-\frac{\pi}{6}\right)\right)
  8. \left(\sqrt{4}\,\cos \left(-\frac{\pi}{4}\right)+i\,\sqrt{4}\,\sin\left(-\frac{\pi}{4}\right)\right)\times\left(\sqrt{6}\,\cos\frac{5\pi}{4}+i\,\sqrt{6}\,\sin\frac{5\pi}{4}\right)

5.1.2 Problem set

In each of the following, complex numbers z_1 and z_2 are shown on the complex plane. Plot the complex number z_1z_2 on the plane.







5.1.3 Problem set

In each of the following, complex numbers z_1 and z_2 are shown on the complex plane. Plot the complex number z_1z_2 on the plane.






6 Conjugate

6.1 Exercises

6.1.1 Problem set

For each of the following problems, find the conjugate of the given complex number.

  1. -\sqrt{2}+4.3i
  2. -\sqrt{5}-\sqrt{3}i
  3. -\pi i
  4. -5

6.1.2 Problem set

For each of the following problems, find the conjugate of the given complex number, and express your answer in polar form.

  1. 3\cos \frac{\pi}{6}+i\,3\sin \frac{\pi}{6}
  2. 2\cos \frac{\pi}{7}+i\,2\sin \frac{\pi}{7}
  3. 5\cos \frac{31\pi}{9}+i\,5\sin \frac{31\pi}{9}
  4. 31.256\cos \left(\frac{-93\pi}{5}\right)+i\,31.256\sin \left(\frac{-93\pi}{5}\right)

6.1.3 Problem set

For each of the following problems, plot the conjugate of the given complex number.







6.1.4 Problem set

For complex numbers z, z_1 and z_2, justify the following.

  1. \overline{z_1+z_2} = \overline{z_1}+\overline{z_2}
  2. \overline{z_1z_2} = \overline{z_1}\,\,\overline{z_2}
  3. \overline{\frac{z_1}{z_2}} = \frac{\overline{z_1}}{\overline{z_2}}
  4. \overline{z^{10}} = \left(\overline{z}\right)^{10}

6.1.5 Problem set

In each of the following problems, solve for z.

  1. z\overline{z} = 11, \arg(z) = -\frac{\pi}{2}
  2. z+\overline{z} = -6, \arg{z} = -\frac{3\pi}{4}
  3. z+\overline{z} = -5, |z| = 5
  4. z+\overline{z} = -10, z-\overline{z} = 8i
  5. z+\overline{z} = -40, z-\overline{z} = 0
  6. z\times \left(r\cos\frac{4\pi}{11}+i\,r\sin\frac{4\pi}{11}\right) = 4r
  7. w\times z \times \overline{w} = \left(\frac{|w|^2}{2}-i\,\frac{|w|^2}{2}\right)

7 Division

7.1 Exercises

7.1.1 Problem set

In each of the following problems, solve for z.

  1. 3\times z = -9+15i
  2. (7+4i)\times z = 65
  3. z \times (-4-i) = 68

7.1.2 Problem set

Evaluate the following.

  1. \frac{9-12i}{3}
  2. \frac{35-42i}{-7}

7.1.3 Problem set

Find the values of each of the following.

  1. \frac{7+4i}{2-3i}

  2. \frac{-3-4i}{-3+4i}

  3. \frac{-2+5i}{2-5i}

  4. \frac{9\cos\frac{3\pi}{4}+i\,9\sin\frac{3\pi}{4}}{3\cos\frac{\pi}{4}+i\,3\sin\frac{\pi}{4}}

  5. \frac{8\cos\left(-\frac{2\pi}{3}\right)+i\,8\sin\left(-\frac{2\pi}{3}\right)}{2\cos\left(-\frac{\pi}{6}\right)+i\,2\sin\left(-\frac{\pi}{6}\right)}

  6. \frac{12\cos\left(-\frac{2019\pi}{3}\right)+i\,12\sin\left(-\frac{2019\pi}{3}\right)}{6\cos\left(-\frac{2019\pi}{3}\right)+i\,6\sin\left(-\frac{2019\pi}{3}\right)}

  7. \frac{\frac{23}{8}\cos\frac{23\pi}{8}+i\,\frac{23}{8}\sin\frac{23\pi}{8}}{\frac{25}{8}\cos\frac{25\pi}{8}+i\,\frac{25}{8}\sin\frac{25\pi}{8}}

7.1.4 Problem set

Evaluate the following.

  1. \frac{-3i}{i}
  2. \frac{-1}{i}
  3. \frac{-2}{i}
  4. \frac{5}{i}
  5. \frac{-2+3i}{i}
  6. \frac{1}{i}\times (8+i)

8 Exponentiation

8.1 Exercises

8.1.1 Problem set

Evaluate the following.

  1. \left(2\cos \frac{\pi}{4}+i\,2\sin \frac{\pi}{4}\right)^5
  2. \left(3\cos \frac{7\pi}{3}+i\,3\sin \frac{7\pi}{3}\right)^4
  3. \left(4\cos \left(-\frac{5\pi}{6}\right)+i\,4\sin \left(-\frac{5\pi}{6}\right)\right)^{18}
  4. \left(5\cos \left(-\frac{\pi}{3}\right)+i\,5\sin \left(-\frac{\pi}{3}\right)\right)^{191}
  5. \left(-\frac{\sqrt{3}}{2}-i\,\frac{1}{2}\right)^{105}
  6. \left(\frac{1}{\sqrt{2}}-i\,\frac{1}{\sqrt{2}}\right)^{2014}
  7. \left(1-i\,\sqrt{3}\right)^{3993}
  8. \left(-i\right)^{535}

8.1.2 Problem set

In the each of the following, the complex number w is defined in terms of the complex number z, and the number z is shown on the complex plane. Plot the number w on the plane.

  1. w = z^4


  2. w = z^7


  3. w = z^7


  4. w = z^3


  5. w = z^4


  6. w = z^{1003}









8.1.3 Problem set

Evaluate the following.

  1. i^4
  2. (-i)^3
  3. {(-i)}^2
  4. -i^2
  5. i^{103}
  6. i^{899}
  7. (-2i)^6
  8. i^{-899}

9 Roots

9.1 Exercises

9.1.1 Problem set

Find all the possible values of the following.

  1. \sqrt[3]{27}
  2. \sqrt[3]{1}
  3. \sqrt[3]{-8}
  4. \sqrt[3]{-27}

9.1.2 Problem set

Find all the possible values of the following.

  1. \sqrt[4]{16\cos\frac{2\pi}{3}+i\,16\sin\frac{2\pi}{3}}
  2. \sqrt[3]{8\cos\frac{\pi}{2}+i\,8\sin\frac{\pi}{2}}
  3. \sqrt{9\cos\left(-\frac{\pi}{2}\right)+i\,9\sin\left(-\frac{\pi}{2}\right)}
  4. \sqrt[3]{-27i}
  5. \sqrt[4]{-\frac{1}{2}-i\,\frac{\sqrt{3}}{2}}

9.1.3 Problem set

Find all the possible values of the following.

  1. \sqrt[4]{16}
  2. \sqrt[4]{-16}
  3. \sqrt{-1}
  4. \sqrt[3]{i}
  5. \sqrt[3]{-i}

9.1.4 Problem set

In the each of the following, the complex number w is defined in terms of the complex number z, and the number z is shown on the complex plane. Plot the complex numbers that w could be equal to, on the plane.

  1. w = \sqrt[5]{z}


  2. w = \sqrt[4]{z}


  3. w = \sqrt[3]{z}


  4. w = \sqrt[3]{z}


  5. w = \sqrt[3]{z}


  6. w = \sqrt[5]{z}


  7. w = \sqrt[4]{z}


10 Miscellaneous

10.1 Exercises

10.1.1 Problem set

Following are the notations used for this problem set:

  • \mathbb{N} – Set of Natural numbers
  • \mathbb{Q} – Set of Rational numbers
  • \mathbb{Z} – Set of Integers
  • \mathbb{W} – Set of Whole numbers
  • \mathbb{R} – Set of Real numbers
  • \mathbb{C} – Set of Complex numbers
  1. If \mathbb{G} denotes the set of purely imaginary numbers, which of the following is/are true: \mathbb{G}\subseteq\mathbb{R}, \mathbb{R}\subseteq\mathbb{G}, \mathbb{G}\subseteq\mathbb{C}?
  2. If \mathbb{I} denotes the set of irrational numbers, which of the following is/are true: \mathbb{Q}\subseteq\mathbb{I}, \mathbb{I}\subseteq\mathbb{R}, \mathbb{I}\subseteq\mathbb{C}, \mathbb{R}\subseteq \mathbb{Q}\bigcup \mathbb{I}?
  3. If \mathbb{G} denotes the set of purely imaginary numbers, which of the following is true: \mathbb{R}\bigcup\mathbb{G}\subseteq\mathbb{C}, \mathbb{C}\subseteq\mathbb{R}\bigcup\mathbb{G}?
  4. Is this true or false: \mathbb{Q}\subseteq(\mathbb{C}-\mathbb{R})?
  5. Fill in the missing sets in the set inclusion hierarchy: \mathbb{N}\subseteq \mathbb{W} \subseteq ? \subseteq \mathbb{Q} \subseteq ? \subseteq \mathbb{C}.

10.1.2 Problem set

  1. Plot the function x^2 - 4. What are the zeroes of this function?
  2. Plot the function x^2 + 1. What are the zeroes of this function?
  3. Plot the function -x^2 - 1. What are the zeroes of this function?

10.1.3 Problem set

Express the following in the form of a+bi, where a and b are real numbers.

  1. For a complex number z, |z| = 5. What is the value of |-2z|?
  2. In general for two complex numbers z_1 and z_2, triangle inequality |z_1+z_2| \le |z_1| + |z_2| holds.
    1. What happens when z_2 = 2z_1?
    2. What happens when z_2 = -2z_1?
  3. z_1 = 3+4i and z_2 = 9+bi. Evaluate b such that |z_1+z_2| = |z_1|+|z_2|.
  4. z_1 = a+bi. Determine z_2 such that z_1z_2 = |z_1|^2.