Learn – Algebra/Precalculus – Complex numbers

Geometric structure

Exercises

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

Addition

Exercises

Problem set

For each of the following, simplify.

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

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)

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.









Modulus and argument

Exercises

Problem set

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

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

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

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. -2-2\sqrt{3}i
  4. 6-2\sqrt{3}i
  5. -12+4\sqrt{3}i
  6. -5\sqrt{3}-15i

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}

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}

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}

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|

Polar form

Exercises

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

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

Multiplication

Exercises

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)

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.







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.






Conjugate

Exercises

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

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)

Problem set

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







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}

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)

Division

Exercises

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

Problem set

Evaluate the following.

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

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}}

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)

Exponentiation

Exercises

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}

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}


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}

Roots

Exercises

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}


Problem set

Find all the possible values of the following.

  1. \sqrt[3]{8\cos\frac{\pi}{2}+i\,8\sin\frac{\pi}{2}}
  2. \sqrt[4]{16\cos\frac{2\pi}{3}+i\,16\sin\frac{2\pi}{3}}
  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}}

Problem set

Find all the possible values of the following.

  1. \sqrt[3]{1}
  2. \sqrt[4]{16}
  3. \sqrt[3]{-27}
  4. \sqrt[4]{-16}
  5. \sqrt{-1}
  6. \sqrt[3]{i}
  7. \sqrt[3]{-i}

Miscellaneous

Exercises

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}.

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?

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.