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 = −5 − j4 and z₂ = −3 − j, find the imaginary part of the product z₁·z₂.

z₁ = −5 − j4z₂ = −3 − j
Im(z₁z₂) =
2.Easy
ReIm11−1−122−2−233−3−344−4−4z₁ABCD

The Argand diagram shows z₁ = 3 + j. Click the point that represents z̄₁.

z₁ = 3 + j
  1. 1A (1 + j3)
  2. 2B (3 − j)
  3. 3C (−3 − j)
  4. 4D (−1 − j3)
3.Medium
ReIm22−2−244−4−466−6−6θz

Write z = −6 − j4 in exponential form r·ejθ. Give θ in radians, between −π and π.

z = −6 − j4
θ = rad
4.Medium
ReIm55−5−51010−10−10θz

Find the argument θ of z = −9 + j6 in degrees, as the principal value between −180° and 180°.

z = −9 + j6
θ = °
5.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 imaginary part of z₃.

w = 81∠90°n = 4k = 3
Im(z₃) =
6.Hard
ReIm11−1−122−2−233−3−3z₁z₂

One root of z³ − 2z² − 11z + 52 = 0 is z₁ = 3 + j2. The coefficients are real. Find the real root r.

z₁ = 3 + j2
r =

Answer key

  1. 1. Im(z₁z₂) = 17
  2. 2. B (3 − j)
  3. 3. θ = −2.554 rad
  4. 4. θ = 146.3 °
  5. 5. Im(z₃) = −2.772
  6. 6. r = −4

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