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
ReIm22−2−244−4−466−6−6z₁z₂

For z₁ = 4 + j6 and z₂ = 6 − j5, find the real part of w = 2z₁ − 3z̄₂.

z₁ = 4 + j6z₂ = 6 − j5
Re(w) =
2.Easy
ReIm22−2−244−4−466−6−6z₁z₂

For z₁ = −6 − j and z₂ = −4 − j5, find the real part of the quotient z₁/z₂.

z₁ = −6 − jz₂ = −4 − j5
Re(z₁/z₂) =
3.Medium
ReIm22−2−244−4−4z₁z₂w

Given z₁ = 5∠100° and z₂ = 4∠110°, find the argument θ of w = z₁/z₂ in degrees, between −180° and 180°.

z₁ = 5∠100°z₂ = 4∠110°
θ = °
4.Medium
ReIm55−5−5θz

Write z = 7 + j5 in exponential form r·ejθ. Give θ in radians, between −π and π.

z = 7 + j5
θ = rad
5.Hard
ReIm11−1−122−2−2θz

Use De Moivre's theorem to find the real part of z⁴ for z = −√3 + j.

z = −√3 + jn = 4
Re(z⁴) =
6.Hard
ReIm11−1−122−2−2z₀z₁z₂z₃z₄

The roots of zⁿ = w with n = 5 and w = 32∠135° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, …, 4. Find the imaginary part of z₀.

w = 32∠135°n = 5k = 0
Im(z₀) =

Answer key

  1. 1. Re(w) = −10
  2. 2. Re(z₁/z₂) = 0.7073
  3. 3. θ = −10 °
  4. 4. θ = 0.6202 rad
  5. 5. Re(z⁴) = −8
  6. 6. Im(z₀) = 0.908

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