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
ReIm11−1−122−2−233−3−3z₁z₂

For z₁ = 3 + j and z₂ = 3 + j, find the imaginary part of w = 4z₁ − 3z₂.

z₁ = 3 + jz₂ = 3 + j
Im(w) =
2.Easy
ReIm22−2−244−4−466−6−6z₁z₂

For z₁ = −6 − j6 and z₂ = −4 − j4, find the real part of the quotient z₁/z₂.

z₁ = −6 − j6z₂ = −4 − j4
Re(z₁/z₂) =
3.Medium
ReIm11−1−122−2−233−3−3θz

The number z = 3·ej2.8 has its angle in radians. Find its imaginary part b.

r = 3θ = 2.8 rad
b =
4.Medium
ReIm11−1−122−2−233−3−3θz

Find the argument θ of z = −3 − j2 in degrees, as the principal value between −180° and 180°.

z = −3 − j2
θ = °
5.Hard
ReIm11−1−122−2−233−3−3z₁z₂

Solve z² − 6z + 11.25 = 0. The roots are a ± jb; find b > 0.

p = −6q = 11.25
b =
6.Hard
ReIm22−2−244−4−4z₀z₁z₂

The roots of zⁿ = w with n = 3 and w = 64∠180° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, 2. Find the argument of z₁ in degrees, between −180° and 180°.

w = 64∠180°n = 3k = 1
θ₁ = °

Answer key

  1. 1. Im(w) = 1
  2. 2. Re(z₁/z₂) = 1.5
  3. 3. b = 1.005
  4. 4. θ = −146.3 °
  5. 5. b = 1.5
  6. 6. θ₁ = 180 °

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