Complex Numbers Calculator
Perform all complex number operations with step-by-step solutions. Add, subtract, multiply, divide, find modulus, argument, convert to polar form, and apply De Moivre's theorem. Perfect for A-Level Further Maths.
Select Operation
Add two complex numbers
📐Key Formulas
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Add real parts together and imaginary parts together
Subtraction
(a + bi) - (c + di) = (a - c) + (b - d)i
Subtract real parts and imaginary parts separately
Multiplication
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Use FOIL and remember i² = -1
Division
(a + bi)/(c + di) = [(a + bi)(c - di)]/[c² + d²]
Multiply by the conjugate of the denominator
Modulus
|a + bi| = √(a² + b²)
Distance from the origin on the Argand diagram
Argument
arg(a + bi) = tan⁻¹(b/a)
Angle from positive real axis (adjust for quadrant)
⚠️Common Mistakes
❌ Forgetting i² = -1 in multiplication
✓ Always replace i² with -1 when expanding
Example: (2 + i)(3 + i) = 6 + 2i + 3i + i² = 6 + 5i - 1 = 5 + 5i
❌ Wrong quadrant for argument
✓ Check the signs of a and b to determine the correct quadrant
Example: arg(-1 + i) = 3π/4, not π/4 (second quadrant)
❌ Not multiplying by the conjugate when dividing
✓ Always multiply both numerator and denominator by the conjugate of the denominator
Example: (1 + i)/(1 - i) = (1 + i)²/[(1 - i)(1 + i)] = (2i)/2 = i
❌ Confusing conjugate with negative
✓ Conjugate changes only the sign of the imaginary part, not the real part
Example: Conjugate of (3 + 4i) is (3 - 4i), not (-3 - 4i)
Understanding Complex Numbers
What are Complex Numbers?
A complex number is a number of the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = -1.
Complex numbers extend the real number system to include solutions to equations like x² + 1 = 0, which has no real solutions but has complex solutions x = ±i.
The Argand Diagram
Complex numbers can be visualized on an Argand diagram, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Each complex number corresponds to a unique point in this plane.
The modulus |z| is the distance from the origin, and the argument arg(z) is the angle from the positive real axis.
Worked Examples
Example 1: Multiplying Complex Numbers
Calculate (3 + 2i)(1 + 4i)
Step 1: Use FOIL method
Step 2: = (3)(1) + (3)(4i) + (2i)(1) + (2i)(4i)
Step 3: = 3 + 12i + 2i + 8i²
Step 4: = 3 + 14i + 8(-1) [since i² = -1]
Step 5: = 3 + 14i - 8
Answer: -5 + 14i
Example 2: Dividing Complex Numbers
Calculate (5 + 3i) ÷ (1 + 2i)
Step 1: Multiply by conjugate: (5 + 3i)(1 - 2i) / (1 + 2i)(1 - 2i)
Step 2: Numerator: 5 - 10i + 3i - 6i² = 5 - 7i + 6 = 11 - 7i
Step 3: Denominator: 1 + 4 = 5
Step 4: = (11 - 7i) / 5
Answer: 2.2 - 1.4i
Example 3: Finding Modulus and Argument
Find |z| and arg(z) for z = 3 + 4i
Modulus: |z| = √(3² + 4²)
= √(9 + 16) = √25 = 5
Argument: arg(z) = tan⁻¹(4/3)
= 0.927 rad = 53.13°
Answer: |z| = 5, arg(z) = 53.13°
Example 4: De Moivre's Theorem
Calculate (1 + i)⁴
Step 1: Convert to polar: r = √2, θ = π/4
Step 2: z = √2(cos 45° + i sin 45°)
Step 3: z⁴ = (√2)⁴(cos 180° + i sin 180°)
Step 4: = 4(−1 + 0i)
Answer: -4
Exam Tips for Complex Numbers
💡 Always remember i² = -1
This is the most fundamental rule. After expanding, replace every i² with -1.
💡 Check the quadrant for argument
tan⁻¹(b/a) gives the reference angle. Adjust based on the signs of a and b.
💡 Multiply by the conjugate to divide
This makes the denominator real: (c + di)(c - di) = c² + d².
💡 Use polar form for powers
De Moivre's theorem makes calculating z^n much easier than repeated multiplication.
💡 Complex numbers have 2 square roots
The two roots are negatives of each other. Don't forget the second root!
💡 Sketch on an Argand diagram
Visualizing the numbers helps check your argument is in the correct quadrant.
Frequently Asked Questions
What is a complex number?
A complex number is a number of the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = -1. Complex numbers extend the real numbers and are essential in advanced mathematics.
How do you multiply complex numbers?
Use the FOIL method: (a + bi)(c + di) = ac + adi + bci + bdi². Then substitute i² = -1 and collect like terms to get (ac - bd) + (ad + bc)i.
What is the modulus of a complex number?
The modulus |z| is the distance from the origin on the Argand diagram, calculated as |z| = √(a² + b²). For example, |3 + 4i| = √(9 + 16) = 5.
What is De Moivre's theorem used for?
De Moivre's theorem [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) is used to calculate powers of complex numbers and find roots of complex numbers efficiently.
How do you find the square roots of a complex number?
Convert to polar form z = r(cos θ + i sin θ), then √z = √r(cos(θ/2) + i sin(θ/2)). The second root is found by adding π to the angle, or simply negate the first root.
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