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Complex Number worksheet

Grades 11–12 · CCSS HSN.CN.A.2, HSN.CN.A.3, HSN.CN.B.4

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Operation: Divide: z₁ ÷ z₂
- Enter z₁ as: Rectangular (a + bi)
- z₁ real part (a): 4
- z₁ imaginary part (b): 4
- z₂ real part (c): 1
- z₂ imaginary part (d): -2
Find: Result
Answer
2.
Problem 2.
- Operation: Subtract: z₁ − z₂
- Enter z₁ as: Polar (r and θ)
- z₁ modulus (r): 1
- z₁ angle (θ): 32 °
- z₂ real part (c): 2
- z₂ imaginary part (d): -3
Find: Result
Answer
3.
Problem 3.
- Operation: Add: z₁ + z₂
- Enter z₁ as: Polar (r and θ)
- z₁ modulus (r): 2
- z₁ angle (θ): 30 °
- z₂ real part (c): 1
- z₂ imaginary part (d): -3
Find: Result
Answer
4.
Problem 4.
- Operation: Subtract: z₁ − z₂
- Enter z₁ as: Polar (r and θ)
- z₁ modulus (r): 3
- z₁ angle (θ): 35 °
- z₂ real part (c): 2
- z₂ imaginary part (d): -2
Find: Result
Answer
5.
Problem 5.
- Operation: Roots: all k-th roots of z₁
- Enter z₁ as: Rectangular (a + bi)
- z₁ real part (a): 3
- z₁ imaginary part (b): 4
- Root index (k): 3
Find: Principal root
Answer
6.
Problem 6.
- Operation: Add: z₁ + z₂
- Enter z₁ as: Rectangular (a + bi)
- z₁ real part (a): 3
- z₁ imaginary part (b): 4
- z₂ real part (c): 1
- z₂ imaginary part (d): -2
Find: Result
Answer
7.
Problem 7.
- Operation: Add: z₁ + z₂
- Enter z₁ as: Polar (r and θ)
- z₁ modulus (r): 3
- z₁ angle (θ): 18 °
- z₂ real part (c): 2
- z₂ imaginary part (d): -3
Find: Result
Answer
8.
Problem 8.
- Operation: Add: z₁ + z₂
- Enter z₁ as: Rectangular (a + bi)
- z₁ real part (a): 3
- z₁ imaginary part (b): 5
- z₂ real part (c): 1
- z₂ imaginary part (d): -2
Find: Result
Answer
9.
Problem 9.
- Operation: Roots: all k-th roots of z₁
- Enter z₁ as: Rectangular (a + bi)
- z₁ real part (a): 2
- z₁ imaginary part (b): 3
- Root index (k): 3
Find: Principal root
Answer
10.
Problem 10.
- Operation: Divide: z₁ ÷ z₂
- Enter z₁ as: Rectangular (a + bi)
- z₁ real part (a): 3
- z₁ imaginary part (b): 3
- z₂ real part (c): 1
- z₂ imaginary part (d): -2
Find: Result
Answer

Answer key: Complex Number worksheet

1. Result: −0.8 + 2.4i
1. Multiply top and bottom by the conjugate of z₂. That turns the bottom into a plain number, c² + d²
  $c² + d²$ $= 1^{2} + {(−2)}^{2}$ $= 5$
2. The new top is z₁ times the conjugate of z₂
  $(4 + 4i) (1 + 2i)$ $= −4 + 12i$
3. Divide the real part by that plain number, c² + d²
  $Re$ $= \frac{−4}{5}$ $= −0.8$
4. Divide the imaginary part by it too
  $Im$ $= \frac{12}{5}$ $= 2.4$
5. Put the two parts together for the answer
  $(4 + 4i) ÷ (1 − 2i)$ $= −0.8 + 2.4i$
2. Result: −1.151952 + 3.529919i
1. Find the across (real) part: the size r times the cosine of the angle
  $a$ $= 1 \times cos 32 °$ $= 0.848048$
2. Find the up (imaginary) part: the size r times the sine of the angle
  $b$ $= 1 \times sin 32 °$ $= 0.529919$
3. Put the two parts together to write z₁ as a + bi
  $z₁$ $= 0.848048 + 0.529919i$
4. Subtract the real parts, since they line up with each other
  $Re$ $= 0.848048 - 2$ $= −1.151952$
5. Subtract the imaginary parts the same way
  $Im$ $= 0.529919 - (−3)$ $= 3.529919$
6. Put the two parts together for the answer
  $(0.848048 + 0.529919i) − (2 − 3i)$ $= −1.151952 + 3.529919i$
3. Result: 2.732051 − 2i
1. Find the across (real) part: the size r times the cosine of the angle
  $a$ $= 2 \times cos 30 °$ $= 1.732051$
2. Find the up (imaginary) part: the size r times the sine of the angle
  $b$ $= 2 \times sin 30 °$ $= 1$
3. Put the two parts together to write z₁ as a + bi
  $z₁$ $= 1.732051 + i$
4. Add the real parts, since they line up with each other
  $Re$ $= 1.732051 + 1$ $= 2.732051$
5. Add the imaginary parts the same way
  $Im$ $= 1 + (−3)$ $= −2$
6. Put the two parts together for the answer
  $(1.732051 + i) + (1 − 3i)$ $= 2.732051 − 2i$
4. Result: 0.457456 + 3.720729i
1. Find the across (real) part: the size r times the cosine of the angle
  $a$ $= 3 \times cos 35 °$ $= 2.457456$
2. Find the up (imaginary) part: the size r times the sine of the angle
  $b$ $= 3 \times sin 35 °$ $= 1.720729$
3. Put the two parts together to write z₁ as a + bi
  $z₁$ $= 2.457456 + 1.720729i$
4. Subtract the real parts, since they line up with each other
  $Re$ $= 2.457456 - 2$ $= 0.457456$
5. Subtract the imaginary parts the same way
  $Im$ $= 1.720729 - (−2)$ $= 3.720729$
6. Put the two parts together for the answer
  $(2.457456 + 1.720729i) − (2 − 2i)$ $= 0.457456 + 3.720729i$
5. Principal root: 1.628937 + 0.520175i
1. Find the size (modulus) of z₁: its distance from 0, using Pythagoras on the two parts
  $r$ $= \sqrt{3^{2} + 4^{2}}$ $= 5$
2. Find the angle (argument). atan2 uses the signs of both parts, so it points the right way
  $θ$ $= atan2 (4, 3)$ $= 53.130102$
3. Every root has the same size: the 3-th root of the original size r
  $ρ$ $= 5^{\frac{1}{3}}$ $= 1.709976$
4. Divide the angle by 3, adding a full turn each time for the other roots (j = 0 to 2). They sit 120° apart
  $φ₀$ $= \frac{53.130102}{3}$ $= 17.710034$
5. Root w₀ (the principal root) at angle 17.710034°
  $w₀$ $= 1.628937 + 0.520175i$
6. Root w₁ at angle 137.710034°
  $w₁$ $= −1.264953 + 1.150614i$
7. Root w₂ at angle 257.710034°
  $w₂$ $= −0.363984 − 1.670788i$
6. Result: 4 + 2i
1. Add the real parts, since they line up with each other
  $Re$ $= 3 + 1$ $= 4$
2. Add the imaginary parts the same way
  $Im$ $= 4 + (−2)$ $= 2$
3. Put the two parts together for the answer
  $(3 + 4i) + (1 − 2i)$ $= 4 + 2i$
7. Result: 4.85317 − 2.072949i
1. Find the across (real) part: the size r times the cosine of the angle
  $a$ $= 3 \times cos 18 °$ $= 2.85317$
2. Find the up (imaginary) part: the size r times the sine of the angle
  $b$ $= 3 \times sin 18 °$ $= 0.927051$
3. Put the two parts together to write z₁ as a + bi
  $z₁$ $= 2.85317 + 0.927051i$
4. Add the real parts, since they line up with each other
  $Re$ $= 2.85317 + 2$ $= 4.85317$
5. Add the imaginary parts the same way
  $Im$ $= 0.927051 + (−3)$ $= −2.072949$
6. Put the two parts together for the answer
  $(2.85317 + 0.927051i) + (2 − 3i)$ $= 4.85317 − 2.072949i$
8. Result: 4 + 3i
1. Add the real parts, since they line up with each other
  $Re$ $= 3 + 1$ $= 4$
2. Add the imaginary parts the same way
  $Im$ $= 5 + (−2)$ $= 3$
3. Put the two parts together for the answer
  $(3 + 5i) + (1 − 2i)$ $= 4 + 3i$
9. Principal root: 1.451857 + 0.493404i
1. Find the size (modulus) of z₁: its distance from 0, using Pythagoras on the two parts
  $r$ $= \sqrt{2^{2} + 3^{2}}$ $= 3.605551$
2. Find the angle (argument). atan2 uses the signs of both parts, so it points the right way
  $θ$ $= atan2 (3, 2)$ $= 56.309932$
3. Every root has the same size: the 3-th root of the original size r
  $ρ$ $= {3.605551}^{\frac{1}{3}}$ $= 1.533406$
4. Divide the angle by 3, adding a full turn each time for the other roots (j = 0 to 2). They sit 120° apart
  $φ₀$ $= \frac{56.309932}{3}$ $= 18.769977$
5. Root w₀ (the principal root) at angle 18.769977°
  $w₀$ $= 1.451857 + 0.493404i$
6. Root w₁ at angle 138.769977°
  $w₁$ $= −1.153228 + 1.010643i$
7. Root w₂ at angle 258.769977°
  $w₂$ $= −0.298628 − 1.504046i$
10. Result: −0.6 + 1.8i
1. Multiply top and bottom by the conjugate of z₂. That turns the bottom into a plain number, c² + d²
  $c² + d²$ $= 1^{2} + {(−2)}^{2}$ $= 5$
2. The new top is z₁ times the conjugate of z₂
  $(3 + 3i) (1 + 2i)$ $= −3 + 9i$
3. Divide the real part by that plain number, c² + d²
  $Re$ $= \frac{−3}{5}$ $= −0.6$
4. Divide the imaginary part by it too
  $Im$ $= \frac{9}{5}$ $= 1.8$
5. Put the two parts together for the answer
  $(3 + 3i) ÷ (1 − 2i)$ $= −0.6 + 1.8i$