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 = −4 − j and z₂ = −1 − j, find the real part of the quotient z₁/z₂.

z₁ = −4 − jz₂ = −1 − j
Re(z₁/z₂) =
2.Easy
ReIm22−2−244−4−4z₁z₂

For z₁ = 3 − j5 and z₂ = 1 + j, find the real part of w = −z₁ + 5z̄₂.

z₁ = 3 − j5z₂ = 1 + j
Re(w) =
3.Medium
ReIm11−1−122−2−233−3−3θz

Write z = 3 − j3 in exponential form r·ejθ. Give θ in radians, between −π and π.

z = 3 − j3
θ = rad
4.Medium
ReIm22−2−244−4−4z₁z₂w

Given z₁ = 5∠125° and z₂ = 4∠110°, find the argument θ of w = z₁/z₂ in degrees, between −180° and 180°.

z₁ = 5∠125°z₂ = 4∠110°
θ = °
5.Hard
ReIm11−1−122−2−2ABCD

Exactly one of the points A–D is a root of z⁶ = 64∠−90°. Click it.

w = 64∠−90°n = 6
  1. 1A (1∠−135°)
  2. 2B (2∠−135°)
  3. 3C (2∠135°)
  4. 4D (2∠−90°)
6.Hard
ReIm0.50.5−0.5−0.511−1−1z₁z₂

One root of z³ − 3z² + 4z − 2 = 0 is z₁ = 1 + j. The coefficients are real. Find the real root r.

z₁ = 1 + j
r =

Answer key

  1. 1. Re(z₁/z₂) = 2.5
  2. 2. Re(w) = 2
  3. 3. θ = −0.7854 rad
  4. 4. θ = 15 °
  5. 5. B (2∠−135°)
  6. 6. r = 1

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