Complex Numbers

Grade 12 · algebra · 101 practice problems · read aloud

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🌟 Complex Numbers: Beyond the Real

Complex numbers extend our number system to include solutions to equations like x² + 1 = 0. They're written as a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1). Complex numbers are essential in electrical engineering, quantum physics, and signal processing.

🔢 Step-by-Step Operations

Addition/Subtraction

Combine real parts with real parts, imaginary with imaginary.

Multiplication

Use distributive property and remember i² = -1.

Division

Multiply numerator and denominator by the conjugate of the denominator.

📚 Worked Examples

Example 1: Multiplication

Problem: (3 + 2i)(1 - 4i)

Step 1: FOIL method: 3(1) + 3(-4i) + 2i(1) + 2i(-4i)

Step 2: Simplify: 3 - 12i + 2i - 8i²

Step 3: Replace i² with -1: 3 - 10i - 8(-1)

Step 4: Final: 11 - 10i

Example 2: Division

Problem: (2 + 3i) ÷ (1 - 2i)

Step 1: Multiply by conjugate: [(2 + 3i)(1 + 2i)] ÷ [(1 - 2i)(1 + 2i)]

Step 2: Numerator: 2 + 4i + 3i + 6i² = 2 + 7i - 6 = -4 + 7i

Step 3: Denominator: 1 - (2i)² = 1 - (-4) = 5

Step 4: Final: (-4 + 7i)/5 = -4/5 + (7/5)i

🚨 Common Mistakes

  • Forgetting i² = -1: Always replace i² with -1 immediately
  • Misapplying conjugates: Only use conjugates for division, not other operations
  • Sign errors: Be careful with negative signs when multiplying complex numbers
  • Incorrect simplification: Don't combine real and imaginary parts

💡 Tips & Tricks

  • FOIL carefully: First, Outer, Inner, Last for multiplication
  • Conjugate shortcut: (a + bi)(a - bi) = a² + b²
  • Visualize: Think of complex numbers as points in the complex plane
  • Check work: Multiply your answer by the divisor to verify division

🎯 Practice Suggestions

  • Start with basic addition/subtraction before moving to multiplication/division
  • Practice with both integer and fractional coefficients
  • Create your own problems and check with a calculator
  • Work on identifying when to use each operation method
  • Try applying complex numbers to solve quadratic equations with negative discriminants

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 Convert the complex number 3 + 4i to polar form r(cosθ + isinθ) = ?

Hint: To convert a complex number from rectangular to polar form, calculate the magnitude using the Pythagorean theorem and determine the angle using inverse trigonometric functions.

Show the answer

Answer: 5(cos(0.9273) + isin(0.9273))

  1. Calculate the modulus r The modulus r is the distance from the origin to the point (3,4) in the complex plane. Formula: r = sqrt(a^2 + b^2) where a = 3 and b = 4 r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
  2. Calculate the argument θ The argument θ is the angle between the positive real axis and the line connecting the origin to the point (3,4). Formula: θ = arctan(b/a) where a = 3 and b = 4 θ = arctan(4/3) = arctan(1.3333...)
  3. Determine the correct quadrant Since both the real part (3) and imaginary part (4) are positive, our complex number is in the first quadrant. This means we can use the arctan value directly without adjustment.
  4. Calculate the numerical value of θ θ = arctan(4/3) = arctan(1.3333...) Using a calculator: θ ≈ 0.9273 radians
  5. Write the polar form Substitute r = 5 and θ = 0.9273 into the polar form: r(cosθ + isinθ) = 5(cos(0.9273) + isin(0.9273)) Therefore, the polar form of 3 + 4i is 5(cos(0.9273) + isin(0.9273))

To convert the complex number 3 + 4i to polar form r(cosθ + isinθ), follow these steps:

2 Convert the complex number 4 - 4i to polar form r(cosθ + isinθ) = ?

Hint: To convert a complex number to polar form, first calculate the modulus r using the Pythagorean theorem, then determine the angle θ based on the quadrant of the complex number.

Show the answer

Answer: 4√2(cos(7π/4) + isin(7π/4))

  1. Calculate the modulus r = √(a² + b²) = √(4² + (-4)²) = √(16 + 16) = √32 = 4√2
  2. Determine the angle θ using tanθ = b/a = -4/4 = -1
  3. Since the complex number is in the 4th quadrant (positive real, negative imaginary), θ = 2π - π/4 = 7π/4
  4. Write in polar form: r(cosθ + isinθ) = 4√2(cos(7π/4) + isin(7π/4))

The answer is 4√2(cos(7π/4) + isin(7π/4)).

3 Convert the complex number 5 + 5i to polar form r(cosθ + isinθ) = ?

Hint: To convert a complex number to polar form, calculate the modulus using the Pythagorean theorem, then determine the angle based on the quadrant location.

Show the answer

Answer: 5√2(cos(π/4) + isin(π/4))

  1. Identify real and imaginary parts: a = 5, b = 5
  2. Calculate modulus r = √(a² + b²) = √(5² + 5²) = √(25 + 25) = √50 = 5√2
  3. Calculate reference angle: tanθ = b/a = 5/5 = 1, so reference angle = π/4
  4. Determine actual angle: Since both coordinates are positive (Quadrant I), θ = π/4
  5. Write in polar form: r(cosθ + isinθ) = 5√2(cos(π/4) + isin(π/4))

The answer is 5√2(cos(π/4) + isin(π/4)).

4 Convert the complex number 6 + 8i to polar form r(cosθ + isinθ) = ?

Hint: To convert a complex number to polar form, first calculate the modulus using the Pythagorean theorem, then determine the angle using inverse trigonometric functions while considering the quadrant.

Show the answer

Answer: 10(cos(53.13°) + isin(53.13°))

  1. Identify the real and imaginary parts: a = 6, b = 8
  2. Calculate the modulus r = √(a² + b²) = √(6² + 8²) = √(36 + 64) = √100 = 10
  3. Calculate the angle θ = tan⁻¹(b/a) = tan⁻¹(8/6) = tan⁻¹(4/3) ≈ 53.13°
  4. Since both real and imaginary parts are positive, the complex number is in Quadrant I, so θ remains 53.13°
  5. Write in polar form: r(cosθ + isinθ) = 10(cos(53.13°) + isin(53.13°))

The answer is 10(cos(53.13°) + isin(53.13°)).

5 Convert the complex number 5 - 7i to polar form r(cosθ + isinθ) = ?

Hint: To convert a complex number to polar form, first calculate the modulus using the Pythagorean theorem, then determine the angle based on the quadrant location of the point.

Show the answer

Answer: √74(cos(5.352) + isin(5.352))

  1. Identify the real and imaginary parts: a = 5, b = -7
  2. Calculate the modulus r = √(a² + b²) = √(5² + (-7)²) = √(25 + 49) = √74
  3. Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(7/5) ≈ 0.9505 radians
  4. Determine the actual angle θ: Since the point (5, -7) is in Quadrant IV, θ = 2π - 0.9505 ≈ 6.2832 - 0.9505 = 5.3327 radians
  5. Write in polar form: r(cosθ + isinθ) = √74(cos(5.3327) + isin(5.3327))

The answer is √74(cos(5.3327) + isin(5.3327)).

6 Convert the complex number 2 - 7i to polar form r(cosθ + isinθ) = ?

Hint: To convert a complex number to polar form, calculate the modulus using the Pythagorean theorem and determine the angle based on the quadrant location.

Show the answer

Answer: √53(cos(5.300) + isin(5.300))

  1. Identify real and imaginary parts: a = 2, b = -7
  2. Calculate modulus r = √(a² + b²) = √(2² + (-7)²) = √(4 + 49) = √53
  3. Calculate reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(7/2) = tan⁻¹(3.5) ≈ 1.2925 radians
  4. Determine actual angle θ: Since the point (2, -7) is in Quadrant IV, θ = 2π - 1.2925 ≈ 6.2832 - 1.2925 = 4.9907 radians
  5. Write in polar form: r(cosθ + isinθ) = √53(cos(4.9907) + isin(4.9907))

The answer is √53(cos(4.9907) + isin(4.9907)) or approximately √53(cos(5.300) + isin(5.300))

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