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
ReIm11−1−122−2−233−3−344−4−455−5−566−6−6z₁ABCD

The Argand diagram shows z₁ = 1 + j5. Click the point that represents −z₁.

z₁ = 1 + j5
  1. 1A (−1 − j5)
  2. 2B (1 − j5)
  3. 3C (−1 + j5)
  4. 4D (−5 − j)
2.Easy
ReIm55−5−51010−10−101515−15−15z₁z₂ABCD

The Argand diagram shows z₁ = −5 − j4 and z₂ = −2 + j5. Click the point that represents z₁ + z₂.

z₁ = −5 − j4z₂ = −2 + j5
  1. 1A (−7 + j)
  2. 2B (1 − j7)
  3. 3C (−7 − j)
  4. 4D (−3 − j9)
3.Medium
ReIm22−2−244−4−4θz

Write z = 6∠140° in rectangular form a + jb and give the real part a.

r = 6θ = 140°
a =
4.Medium
ReIm22−2−244−4−4θz

Find the argument θ of z = −5 + j in degrees, as the principal value between −180° and 180°.

z = −5 + j
θ = °
5.Hard
ReIm11−1−122−2−233−3−3z₁z₂

One root of z³ + 9z² + 36z + 54 = 0 is z₁ = −3 + j3. The coefficients are real. Find the real root r.

z₁ = −3 + j3
r =
6.Hard
ReIm11−1−122−2−233−3−3θz

Find the real part of (−1 + j3)³.

z = −1 + j3n = 3
Re(z³) =

Answer key

  1. 1. A (−1 − j5)
  2. 2. A (−7 + j)
  3. 3. a = −4.596
  4. 4. θ = 168.7 °
  5. 5. r = −3
  6. 6. Re(z³) = 26

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