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 = 4 + j5. Click the point that represents j·z₁.

z₁ = 4 + j5
  1. 1A (−5 + j4)
  2. 2B (4 − j5)
  3. 3C (5 − j4)
  4. 4D (5 + j4)
2.Easy
ReIm55−5−5z₁z₂

For z₁ = −6 − j7 and z₂ = −6 − j4, find the modulus of the quotient, |z₁/z₂|.

z₁ = −6 − j7z₂ = −6 − j4
|z₁/z₂| =
3.Medium
ReIm55−5−5θz

Find the argument θ of z = −5 − j7 in degrees, as the principal value between −180° and 180°.

z = −5 − j7
θ = °
4.Medium
ReIm11−1−122−2−233−3−344−4−455−5−566−6−677−7−788−8−8zABCD

The diagram shows z = 3 + j2. With w = 2∠−90°, click the point that represents z·w.

z = 3 + j2w = 2∠−90°
  1. 1A (6 + j4)
  2. 2B (2 − j3)
  3. 3C (4 − j6)
  4. 4D (−4 + j6)
5.Hard
ReIm22−2−244−4−4z₁z₂

Solve z² − 5z + 22.25 = 0. The roots are a ± jb; find the real part a.

p = −5q = 22.25
a =
6.Hard
ReIm11−1−122−2−233−3−3z₀z₁z₂z₃

The roots of zⁿ = w with n = 4 and w = 81∠90° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, 2, 3. Find the argument of z₀ in degrees, between −180° and 180°.

w = 81∠90°n = 4k = 0
θ₀ = °

Answer key

  1. 1. A (−5 + j4)
  2. 2. |z₁/z₂| = 1.279
  3. 3. θ = −125.5 °
  4. 4. C (4 − j6)
  5. 5. a = 2.5
  6. 6. θ₀ = 22.5 °

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