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 = −7 − j and z₂ = −4 + j2, find the modulus of the quotient, |z₁/z₂|.

z₁ = −7 − jz₂ = −4 + j2
|z₁/z₂| =
2.Easy
ReIm22−2−244−4−4z₁z₂

For z₁ = −4 − j2 and z₂ = −1 − j5, find the real part of w = 3z₁ + 5z̄₂.

z₁ = −4 − j2z₂ = −1 − j5
Re(w) =
3.Medium
ReIm55−5−5θz

Find the argument θ of z = −5 − j7 in degrees, as the principal value between −180° and 180°.

z = −5 − j7
θ = °
4.Medium
ReIm55−5−51010−10−10zABCD

The diagram shows z = −2 − j4. With w = 2∠−90°, click the point that represents z·w.

z = −2 − j4w = 2∠−90°
  1. 1A (−8 + j4)
  2. 2B (−4 − j8)
  3. 3C (−4 + j2)
  4. 4D (8 − j4)
5.Hard
ReIm0.50.5−0.5−0.511−1−1θz

Use De Moivre's theorem to find the real part of z⁴ for z = 1 + j.

z = 1 + jn = 4
Re(z⁴) =
6.Hard
ReIm11−1−122−2−2z₁z₂

One root of z³ + 3z² − 5z + 25 = 0 is z₁ = 1 + j2. The coefficients are real. Find the real root r.

z₁ = 1 + j2
r =

Answer key

  1. 1. |z₁/z₂| = 1.581
  2. 2. Re(w) = −17
  3. 3. θ = −125.5 °
  4. 4. A (−8 + j4)
  5. 5. Re(z⁴) = −4
  6. 6. r = −5

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