Complex Numbers Calculator
Add, subtract, multiply, or divide complex numbers, compute powers of i, convert between rectangular and polar form, or raise a complex number to a power — with a visual Argand diagram, step-by-step working, and clear explanations of every operation.
Background
A complex number has the form 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 line into a two-dimensional plane (the complex or Argand plane), where every complex number can be plotted as a point or vector. This lets us represent quantities — like solutions to quadratic equations with negative discriminants — that have no place on the real number line alone.
How to use this calculator
- Choose an operation: arithmetic on two complex numbers, powers of i, converting to/from polar form, or raising a complex number to a power.
- Enter the real and imaginary parts of your complex number(s) — or the exponent if computing a power of i.
- Click Calculate to see the result, an Argand diagram plotting the numbers on the complex plane, and a full step-by-step explanation.
- Use the quick example chips to instantly load common textbook problems.
How complex number operations work
Addition and subtraction. Combine real parts together and imaginary parts together: (a+bi) ± (c+di) = (a±c) + (b±d)i.
Multiplication. Use FOIL just like multiplying binomials, then simplify using i² = −1: (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
Division. Multiply both the numerator and denominator by the conjugate of the denominator to eliminate i from the bottom: (a+bi)/(c+di) = (a+bi)(c−di) / (c²+d²).
Powers of i. Powers of i cycle every 4: i¹=i, i²=−1, i³=−i, i⁴=1, then it repeats. To find iⁿ for large n, divide n by 4 and use the remainder.
Polar form. Every complex number a+bi can be written as r(cos θ + i sin θ), where r = √(a²+b²) is the modulus (distance from origin) and θ = atan2(b,a) is the argument (angle from positive real axis).
De Moivre's Theorem. Raising a complex number to a power is much easier in polar form: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) — raise the modulus to the power and multiply the angle by the power.
Formula & Equations Used
Imaginary unit: i² = −1
Addition/Subtraction: (a+bi) ± (c+di) = (a±c) + (b±d)i
Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Division (via conjugate): (a+bi)/(c+di) = [(ac+bd) + (bc−ad)i] / (c²+d²)
Modulus: r = √(a²+b²)
Argument: θ = atan2(b, a)
De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
Example Problems & Step-by-Step Solutions
Example 1 — Multiplying complex numbers
Multiply (2+3i)(1−i).
Step 1: FOIL: 2(1) + 2(−i) + 3i(1) + 3i(−i) = 2 − 2i + 3i − 3i²
Step 2: Since i² = −1: −3i² = −3(−1) = +3
Result: 2 + 3 + (−2+3)i = 5 + i
Example 2 — Dividing complex numbers
Divide (4+2i) ÷ (1+i).
Step 1: Multiply top and bottom by the conjugate (1−i): (4+2i)(1−i) / (1+i)(1−i)
Step 2: Denominator: 1² + 1² = 2. Numerator: 4−4i+2i−2i² = 4−2i+2 = 6−2i
Result: (6−2i)/2 = 3 − i
Example 3 — Finding a high power of i
Find i²⁹.
Step 1: Powers of i cycle every 4: i¹=i, i²=−1, i³=−i, i⁴=1.
Step 2: Divide 29 by 4: 29 = 4(7) + 1, so the remainder is 1.
Result: i²⁹ = i¹ = i
Example 4 — De Moivre's Theorem
Find (1+i)⁴.
Step 1: Convert to polar: r = √2, θ = 45° (π/4 radians).
Step 2: By De Moivre's: r⁴ = (√2)⁴ = 4, and 4θ = 180°.
Result: 4(cos 180° + i sin 180°) = 4(−1 + 0i) = −4
Frequently Asked Questions
Why does i² equal −1?
This is the defining property of the imaginary unit — it's not derived, it's the starting definition. Mathematicians introduced i specifically to give a name to √(−1), since no real number squares to a negative value. Everything else about complex numbers follows from this one rule.
What is a complex conjugate and why does division need it?
The conjugate of a+bi is a−bi (same real part, flipped sign on the imaginary part). Multiplying a complex number by its conjugate always produces a real number: (a+bi)(a−bi) = a²+b². This is exactly why we multiply by the conjugate when dividing — it clears the imaginary part from the denominator.
What do the modulus and argument represent geometrically?
If you plot a+bi as a point on the complex plane (real part on the x-axis, imaginary part on the y-axis), the modulus r is the straight-line distance from the origin to that point, and the argument θ is the angle that line makes with the positive real axis — just like converting Cartesian coordinates to polar coordinates.
Why is De Moivre's Theorem useful?
Multiplying out (a+bi) raised to a high power by repeated FOIL is extremely tedious and error-prone. In polar form, raising to a power becomes simple: just raise r to that power and multiply the angle. This turns a multi-step algebra problem into two quick calculations.
How do I find any power of i quickly?
Since powers of i repeat in a cycle of 4 (i, −1, −i, 1), you only need to find the remainder when the exponent is divided by 4. A remainder of 1 means the answer is i, 2 means −1, 3 means −i, and 0 means 1.
Where do complex numbers actually get used?
Beyond solving quadratics with negative discriminants, complex numbers are essential in electrical engineering (AC circuit analysis), signal processing, quantum mechanics, and control systems — anywhere oscillating or rotating quantities need to be tracked using both magnitude and phase.
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