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
ReIm11−1−122−2−233−3−344−4−4z₁ABCD

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

z₁ = −3 + j
  1. 1A (3 − j)
  2. 2B (1 − j3)
  3. 3C (3 + j)
  4. 4D (−3 − j)
2.Easy
ReIm55−5−5z₁z₂

For z₁ = −8 + j and z₂ = 2 − j6, find the imaginary part of the quotient z₁/z₂.

z₁ = −8 + jz₂ = 2 − j6
Im(z₁/z₂) =
3.Medium
ReIm22−2−244−4−4θz

The number z = 5·ej0.4 has its angle in radians. Find its imaginary part b.

r = 5θ = 0.4 rad
b =
4.Medium
ReIm55−5−51010−10−10zABCD

The diagram shows z = 1 + j4. With w = 0.5∠90°, click the point that represents z/w.

z = 1 + j4w = 0.5∠90°
  1. 1A (4 − j)
  2. 2B (−8 + j2)
  3. 3C (8 − j2)
  4. 4D (2 + j8)
5.Hard
ReIm11−1−122−2−2z₀z₁z₂z₃z₄z₅

The roots of zⁿ = w with n = 6 and w = 64∠−120° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, …, 5. Find the argument of z₀ in degrees, between −180° and 180°.

w = 64∠−120°n = 6k = 0
θ₀ = °
6.Hard
ReIm11−1−122−2−2θz

Use De Moivre's theorem to find the real part of z³ for z = −2 − j2.

z = −2 − j2n = 3
Re(z³) =

Answer key

  1. 1. A (3 − j)
  2. 2. Im(z₁/z₂) = −1.15
  3. 3. b = 1.947
  4. 4. C (8 − j2)
  5. 5. θ₀ = −20 °
  6. 6. Re(z³) = 16

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