Complex numbers worksheet with answers

Free printable Complex numbers worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
ReIm11−1−122−2−233−3−344−4−455−5−566−6−677−7−788−8−899−9−9z₁z₂ABCD

The Argand diagram shows z₁ = 4 + j3 and z₂ = 2 − j4. Click the point that represents z₁ + z₂.

z₁ = 4 + j3z₂ = 2 − j4
  1. 1A (2 + j7)
  2. 2B (−1 + j6)
  3. 3C (6 − j)
  4. 4D (6 + j)
2.Easy
ReIm55−5−5z₁z₂

For z₁ = −7 − j and z₂ = 5 − j3, find the real part of w = 2z₁ − 3z₂.

z₁ = −7 − jz₂ = 5 − j3
Re(w) =
3.Medium
ReIm11−1−122−2−2θz

The number z = 2·ej1.5 has its angle in radians. Find its real part a.

r = 2θ = 1.5 rad
a =
4.Medium
ReIm11−1−122−2−2z₁z₂w

Given z₁ = 1.5∠150° and z₂ = 2∠−95°, find the argument θ of w = z₁·z₂ in degrees, between −180° and 180°.

z₁ = 1.5∠150°z₂ = 2∠−95°
θ = °
5.Hard
ReIm11−1−122−2−233−3−3ABCD

Exactly one of the points A–D is a root of z³ = 27∠0°. Click it.

w = 27∠0°n = 3
  1. 1A (1.5∠0°)
  2. 2B (3∠0°)
  3. 3C (3∠60°)
  4. 4D (3∠30°)
6.Hard
ReIm11−1−122−2−2θz

Use De Moivre's theorem to find the real part of z⁸ for z = 1 − j√3.

z = 1 − j√3n = 8
Re(z⁸) =

Answer key

  1. 1. C (6 − j)
  2. 2. Re(w) = −29
  3. 3. a = 0.1415
  4. 4. θ = 55 °
  5. 5. B (3∠0°)
  6. 6. Re(z⁸) = −128

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