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
ReIm55−5−5z₁z₂

For z₁ = 5 − j7 and z₂ = 1 + j, find the real part of the product z₁·z₂.

z₁ = 5 − j7z₂ = 1 + j
Re(z₁z₂) =
2.Easy
ReIm55−5−5z₁z₂

For z₁ = 8 − j4 and z₂ = −6 − j3, find the real part of the quotient z₁/z₂.

z₁ = 8 − j4z₂ = −6 − j3
Re(z₁/z₂) =
3.Medium
ReIm22−2−244−4−466−6−6θz

Write z = 6∠−75° in rectangular form a + jb and give the real part a.

r = 6θ = −75°
a =
4.Medium
ReIm11−1−122−2−233−3−344−4−455−5−5zABCD

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

z = 2 + jw = 2∠90°
  1. 1A (2 − j4)
  2. 2B (−2 + j4)
  3. 3C (−1 + j2)
  4. 4D (4 + j2)
5.Hard
ReIm22−2−244−4−4ABCD

Exactly one of the points A–D is a root of z³ = 27∠90°. Click it.

w = 27∠90°n = 3
  1. 1A (5.4∠−30°)
  2. 2B (3∠−90°)
  3. 3C (3∠−60°)
  4. 4D (3∠90°)
6.Hard
ReIm11−1−122−2−233−3−3z₀z₁z₂z₃z₄

The roots of zⁿ = w with n = 5 and w = 243∠180° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, …, 4. Find the imaginary part of z₄.

w = 243∠180°n = 5k = 4
Im(z₄) =

Answer key

  1. 1. Re(z₁z₂) = 12
  2. 2. Re(z₁/z₂) = −0.8
  3. 3. a = 1.553
  4. 4. B (−2 + j4)
  5. 5. B (3∠−90°)
  6. 6. Im(z₄) = −1.763

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