Complex Numbers Calculator
Perform all complex number operations with step-by-step solutions. Add, subtract, multiply, divide, find modulus, argument, convert to polar form, apply De Moivre's theorem, find nth roots, roots of unity, and evaluate Euler's formula. Interactive Argand diagram included.
Select Operation
Add two complex numbers
How helpful was this?
Help other students find great tools
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.
Powers of i — The Four-Number Cycle
The powers of i repeat in a cycle of 4. To find i^n, divide n by 4 and use the remainder.
| Power | i&sup0; | i¹ | i² | i³ | i⁴ | i⁵ | i⁶ | i⁷ |
|---|---|---|---|---|---|---|---|---|
| Value | 1 | i | -1 | -i | 1 | i | -1 | -i |
| n mod 4 | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 |
Quick rule: i^100 → 100 ÷ 4 = 25 remainder 0 → i^100 = i^0 = 1
Forms of Complex Numbers
| Form | Notation | When to Use | Convert To |
|---|---|---|---|
| Cartesian | a + bi | Addition, subtraction, identifying parts | r = √(a²+b²), θ = tan¹(b/a) |
| Polar (trig) | r(cosθ + isinθ) | Multiplication, division, powers, roots | a = rcosθ, b = rsinθ |
| Exponential | re^(iθ) | Advanced theory, Euler's formula, compact notation | Same as polar (Euler's formula) |
| Modulus-Argument | [r, θ] | Shorthand for polar, common in exam notation | Equivalent to polar form |
Common Mistakes to Avoid
❌ Forgetting i² = -1 in multiplication
✅ Always substitute i² = -1 when expanding
(2+i)(3+i) = 6+2i+3i+i² = 6+5i-1 = 5+5i
❌ Wrong quadrant for argument
✅ Check signs of a, b for the correct quadrant
arg(-1+i) = 3π/4, not π/4 (second quadrant)
❌ Not multiplying by conjugate to divide
✅ Multiply top & bottom by the conjugate of the denominator
(1+i)/(1-i) × (1+i)/(1+i) = (2i)/2 = i
❌ Confusing conjugate with negative
✅ Only change sign of the imaginary part
Conjugate of (3+4i) is (3-4i), not (-3-4i)
❌ Forgetting the second square root
✅ Complex numbers have exactly n nth roots
√(4i) = ±(√2 + √2·i)
❌ Wrong formula for modulus (a²+b² instead of √)
✅ Modulus = √(a²+b²)
|3+4i| = √(9+16) = 5, not 25
Worked Examples
Multiplying Complex Numbers
Calculate (3 + 2i)(1 + 4i) using FOIL
Step 1: First: (3)(1) = 3
Step 2: Outer: (3)(4i) = 12i
Step 3: Inner: (2i)(1) = 2i
Step 4: Last: (2i)(4i) = 8i² = -8
Step 5: Combine: 3 + 12i + 2i - 8 = -5 + 14i
Answer: -5 + 14i
Division Using Conjugate
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 = 11-7i
Step 3: Denominator: 1+4 = 5
Answer: 2.2 - 1.4i
De Moivre's Theorem: (1+i)&sup4;
Apply De Moivre's theorem
Step 1: Convert: r = √2, θ = π/4
Step 2: z = √2(cos 45° + i sin 45°)
Step 3: z&sup4; = (√2)&sup4;(cos 180° + i sin 180°)
Step 4: = 4(-1 + 0i)
Answer: -4
Finding Cube Roots of 8
Find all three cube roots of 8
Step 1: z = 8 = 8(cos 0 + i sin 0), so r=8, θ=0
Step 2: r^(1/3) = 2, angles: 0°, 120°, 240°
z&sub0;: 2(cos 0°) = 2
z&sub1;: 2(cos 120°+i sin 120°) = -1+√3i
z&sub2;: 2(cos 240°+i sin 240°) = -1-√3i
Roots: 2, -1+√3i, -1-√3i
Euler's Formula: e^(iθ) = cosθ + isinθ
Euler's formula is one of the most important results in mathematics. It connects the exponential function with trigonometric functions, providing a bridge between algebra and geometry.
The special case e^(iπ) + 1 = 0 is known as Euler's identity, linking five fundamental constants: e, i, π, 1, and 0.
For any angle θ, the point e^(iθ) lies on the unit circle at angle θ from the positive real axis. This means the exponential form re^(iθ) is equivalent to the polar form r(cosθ + isinθ).
Why it matters
- •Simplifies multiplication in polar form: just multiply moduli and add arguments
- •Makes De Moivre's theorem intuitive: (re^(iθ))^n = r^n e^(inθ)
- •Essential in signal processing, quantum mechanics, and electrical engineering
- •Provides the most compact notation for complex numbers: z = re^(iθ)
Roots of Unity
The nth roots of unity are the n solutions to the equation z^n = 1. They are equally spaced on the unit circle, forming a regular n-gon.
A key property: the sum of all nth roots of unity equals zero. This elegant result follows from the fact that the roots are symmetrically distributed around the origin.
The primitive root ω = e^(2πi/n) generates all other roots by repeated multiplication: ω, ω², ..., ω^(n-1), ω^n = 1.
Reference Values
| n | Roots | Shape |
|---|---|---|
| 2 | 1, -1 | Line segment |
| 3 | 1, -½+½√3i, -½-½√3i | Triangle |
| 4 | 1, i, -1, -i | Square |
| 6 | 1, ½+½√3i, -½+½√3i, -1, -½-½√3i, ½-½√3i | Hexagon |
When to Use Each Operation
“I need to Add or subtract”
Combine real and imaginary parts separately
add / subtract“I need to Multiply”
FOIL method, or use polar form for large powers
multiply“I need to Divide”
Multiply by conjugate of denominator
divide“I need to Distance from origin”
Modulus |z| = √(a²+b²)
modulus“I need to Angle from real axis”
Argument arg(z) = tan⁻¹(b/a)
argument“I need to Raise to a power”
Convert to polar, use De Moivre's theorem
demoivre“I need to Find roots”
Convert to polar, use nth root formula
nthroot“I need to Roots of zⁿ = 1”
Roots of unity: equally spaced on unit circle
unity“I need to Exponential form”
Euler's formula: e^(iθ) = cosθ + isinθ
eulerExam 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.
Count your roots
A degree-n equation has exactly n roots. Don't stop early!
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.
What is Euler's formula?
Euler's formula states that e^(iθ) = cos θ + i sin θ, connecting exponential and trigonometric functions. The special case e^(iπ) + 1 = 0 links five fundamental mathematical constants in one equation.
What are the nth roots of a complex number?
The nth roots of z are the n values w such that wⁿ = z. They are found using De Moivre's formula: w_k = r^(1/n)[cos((θ+2πk)/n) + i sin((θ+2πk)/n)] for k = 0, 1, ..., n-1. The roots are equally spaced on a circle.
What are roots of unity?
The nth roots of unity are the n solutions to zⁿ = 1. They lie equally spaced on the unit circle and always sum to zero. For example, the cube roots of unity are 1, -½+√3/2·i, and -½-√3/2·i, forming an equilateral triangle.
How do you plot complex numbers on an Argand diagram?
On an Argand diagram, the horizontal axis is the real part and the vertical axis is the imaginary part. Plot z = a + bi at coordinates (a, b). The distance from the origin is |z| and the angle from the positive real axis is arg(z).
What is the difference between polar and Cartesian form?
Cartesian form (a + bi) uses real and imaginary parts. Polar form r(cos θ + i sin θ) uses distance from origin (r) and angle (θ). Polar is better for multiplication, division, powers, and roots; Cartesian is better for addition and subtraction.
Explore More Free Tools
All our tools are 100% free with step-by-step learning