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
ReIm22−2−244−4−4z₁z₂

For z₁ = −1 + j4 and z₂ = 2 − j4, find the imaginary part of the product z₁·z₂.

z₁ = −1 + j4z₂ = 2 − j4
Im(z₁z₂) =
2.Easy
ReIm55−5−5z₁z₂

For z₁ = 7 − j2 and z₂ = 6 − j3, find the real part of w = −z₁ − 2z₂.

z₁ = 7 − j2z₂ = 6 − j3
Re(w) =
3.Medium
ReIm22−2−244−4−4z₁z₂w

Given z₁ = 5∠150° and z₂ = 1.5∠65°, find the argument θ of w = z₁/z₂ in degrees, between −180° and 180°.

z₁ = 5∠150°z₂ = 1.5∠65°
θ = °
4.Medium
ReIm55−5−51010−10−10zABCD

The diagram shows z = 3 − j4. With w = 2∠−90°, click the point that represents z·w.

z = 3 − j4w = 2∠−90°
  1. 1A (−4 − j3)
  2. 2B (−8 − j6)
  3. 3C (6 − j8)
  4. 4D (8 + j6)
5.Hard
ReIm11−1−122−2−233−3−3ABCD

Exactly one of the points A–D is a root of z³ = 8∠−90°. Click it.

w = 8∠−90°n = 3
  1. 1A (3.6∠30°)
  2. 2B (2∠−30°)
  3. 3C (2∠150°)
  4. 4D (2∠−90°)
6.Hard
ReIm11−1−122−2−233−3−3z₀z₁z₂z₃z₄z₅

The roots of zⁿ = w with n = 6 and w = 729∠180° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, …, 5. Find the argument of z₄ in degrees, between −180° and 180°.

w = 729∠180°n = 6k = 4
θ₄ = °

Answer key

  1. 1. Im(z₁z₂) = 12
  2. 2. Re(w) = −19
  3. 3. θ = 85 °
  4. 4. B (−8 − j6)
  5. 5. B (2∠−30°)
  6. 6. θ₄ = −90 °

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