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
ReIm55−5−51010−10−10z₁z₂ABCD

The Argand diagram shows z₁ = 3 − j2 and z₂ = −5 − j. Click the point that represents z₁ + z₂.

z₁ = 3 − j2z₂ = −5 − j
  1. 1A (−2 + j3)
  2. 2B (−3 − j2)
  3. 3C (8 − j)
  4. 4D (−2 − j3)
2.Easy
ReIm22−2−244−4−466−6−6z₁z₂

For z₁ = −4 − j6 and z₂ = −2 − j6, find the imaginary part of w = 3z₁ + 2z₂.

z₁ = −4 − j6z₂ = −2 − j6
Im(w) =
3.Medium
ReIm11−1−122−2−233−3−3θz

Write z = 4∠140° in rectangular form a + jb and give the real part a.

r = 4θ = 140°
a =
4.Medium
ReIm11−1−122−2−233−3−344−4−455−5−566−6−677−7−788−8−8zABCD

The diagram shows z = −1 − j3. With w = 0.5∠−90°, click the point that represents z/w.

z = −1 − j3w = 0.5∠−90°
  1. 1A (6 − j2)
  2. 2B (−2 − j6)
  3. 3C (3 − j)
  4. 4D (−6 + j2)
5.Hard
ReIm0.50.5−0.5−0.511−1−1θz

Use De Moivre's theorem to find the imaginary part of z³ for z = −1 + j.

z = −1 + jn = 3
Im(z³) =
6.Hard
ReIm11−1−122−2−233−3−3z₀z₁z₂z₃z₄z₅

The roots of zⁿ = w with n = 6 and w = 729∠135° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, …, 5. Find the imaginary part of z₀.

w = 729∠135°n = 6k = 0
Im(z₀) =

Answer key

  1. 1. D (−2 − j3)
  2. 2. Im(w) = −30
  3. 3. a = −3.064
  4. 4. A (6 − j2)
  5. 5. Im(z³) = 2
  6. 6. Im(z₀) = 1.148

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