Complex Number Calculator
Perform calculations with complex numbers in the form a + bi.
Quick Examples
Complex Numbers
A complex number is written as z = a + bi, where a is the real part and b is the imaginary part.
Polar Form
Where r = |z| = √(a² + b²) is the modulus, and θ = arctan(b/a) is the argument.
Operations
- Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
- Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
- Modulus: |z| = √(a² + b²)
What Is a Complex Number?
A complex number extends the familiar number line into a second dimension. Every complex number has two parts: a real part you already know (like 3) and an imaginary part built from the imaginary unit i, defined by the property i² = −1. Written together as z = a + bi, the pair (a, b) can be plotted as a point on a plane, where the horizontal axis is real and the vertical axis is imaginary.
Complex numbers were first taken seriously in the 1500s, when mathematicians solving cubic equations kept running into square roots of negative numbers. Rather than discarding these "impossible" quantities, they found the algebra still worked — and eventually discovered that complex numbers describe real phenomena beautifully. Today they are essential in electrical engineering (alternating-current analysis), signal processing, quantum mechanics, and control theory. Wherever waves, oscillations, or rotations appear, complex numbers are usually hiding nearby.
Key Formulas
The conjugate flips the sign of the imaginary part and is the workhorse of complex division: multiplying the top and bottom of a fraction by the conjugate of the denominator turns the denominator into the real number a² + b². The argument θ = arctan(b/a) needs quadrant care — most calculators use the two-argument atan2(b, a) to place the angle correctly. De Moivre's theorem, zⁿ = rⁿ(cos nθ + i·sin nθ), makes powers and roots of complex numbers straightforward in polar form.
Step-by-Step: Multiplying (2 + 3i)(1 − i)
- Step 1 — Expand like a binomial product: (2 + 3i)(1 − i) = 2·1 + 2·(−i) + 3i·1 + 3i·(−i).
- Step 2 — Simplify each term: this gives 2 − 2i + 3i − 3i².
- Step 3 — Combine the imaginary terms: −2i + 3i = i, leaving 2 + i − 3i².
- Step 4 — Replace i² with −1: −3i² = −3(−1) = +3, so the result is 2 + i + 3 = 5 + i.
Worked Examples
Example 1: Addition
(3 + 4i) + (1 − 2i): add the real parts, 3 + 1 = 4; add the imaginary parts, 4 + (−2) = 2. Result: 4 + 2i.
Example 2: Division
(6 + 8i) ÷ (1 + 2i): multiply top and bottom by the conjugate (1 − 2i). Numerator: (6 + 8i)(1 − 2i) = 6 − 12i + 8i − 16i² = 6 − 4i + 16 = 22 − 4i. Denominator: 1² + 2² = 5. Result: (22 − 4i)/5 = 4.4 − 0.8i.
Example 3: Modulus and argument
For z = 3 + 4i: |z| = √(3² + 4²) = √25 = 5. For z = 1 + i: θ = arctan(1/1) = 45°, so 1 + i = √2·(cos 45° + i·sin 45°).
Common Mistakes
- Treating i² as 1: i² = −1. Forgetting the minus sign is the single most common error in complex arithmetic.
- Using plain arctan for the argument: arctan(b/a) alone cannot distinguish quadrants II and III; use atan2(b, a) or adjust by hand.
- Expanding squares incorrectly: (a + bi)² = a² + 2abi − b², not a² − b². The cross term 2abi is easy to drop.
- Confusing |z| with a + b: the modulus is √(a² + b²), never the plain sum of the parts.
Frequently Asked Questions
Why not just use real numbers? Some equations, like x² + 1 = 0, have no real solution. Complex numbers complete the number system so that every polynomial equation has a solution — this is the Fundamental Theorem of Algebra.
What is the conjugate good for? Division, computing the modulus (|z|² = z·z̄), and clearing imaginary parts from denominators.
Can complex numbers be ordered, like 2 + 3i > 1 + i? No — there is no ordering of the complex numbers that plays nicely with arithmetic, unlike the reals.
Are complex numbers used outside mathematics? Constantly: AC circuit analysis, audio and image processing (Fourier transforms), quantum physics, and fluid dynamics all rely on them.
The Powers of i: A Four-Step Cycle
The powers of the imaginary unit cycle every four steps, which makes simplifying high powers of i a division problem:
| Power | Value | Example |
|---|---|---|
| i¹ | i | — |
| i² | −1 | — |
| i³ | −i | i² · i = −i |
| i⁴ | 1 | i² · i² = (−1)(−1) = 1 |
| i⁵ … | i | the cycle repeats |
To find iⁿ, divide n by 4 and keep the remainder: i²³ has remainder 3 (23 = 4·5 + 3), so i²³ = i³ = −i.
De Moivre's Theorem in Action: Powers and Roots
De Moivre's theorem turns powers into angle-multiplication: (r(cos θ + i·sin θ))ⁿ = rⁿ(cos nθ + i·sin nθ). Run it backward and it gives every nth root: the n roots of z sit evenly spaced on a circle of radius r^(1/n) at angles (θ + 360°k)/n for k = 0, 1, …, n − 1.
The cube roots of 8
For z = 8: r = 8, θ = 0°, and r^(1/3) = 2. The three roots are 2(cos 0° + i·sin 0°) = 2, 2(cos 120° + i·sin 120°) = −1 + i√3, and 2(cos 240° + i·sin 240°) = −1 − i√3. Cubing any of them returns 8 — check (−1 + i√3)³ and the algebra closes back to 8.
Euler's Formula
The exponential and trigonometric forms meet in Euler's formula: e^(iθ) = cos θ + i·sin θ, so every complex number can be written z = re^(iθ). At θ = π it gives Euler's identity, e^(iπ) + 1 = 0, tying together e, i, π, 1, and 0. The exponential form is why multiplying complex numbers adds their angles: re^(iθ)·se^(iφ) = rse^(i(θ+φ)).
Where Complex Numbers Actually Get Used
- AC circuits: impedance Z = R + iX combines resistance (R) and reactance (X); the modulus is the magnitude of opposition to current, the argument the phase shift. Ohm's law then works unchanged: V = IZ.
- Signal processing: the Fourier transform decomposes sounds and images into complex exponentials — MP3, JPEG, and Wi-Fi all rest on this.
- Control engineering: system stability is judged by where complex eigenvalues sit in the plane — poles in the left half-plane mean a stable system.
- Quantum mechanics: wavefunctions are complex-valued; the observable probability is |ψ|², the modulus squared.
What is the principal value of the argument? The convention that reports arg z in (−180°, 180°] (or (−π, π]), so arg(−1) = 180°, never −180°. Angles differing by a full turn describe the same direction, so this convention picks one canonical answer.
How many square roots does a complex number have? Two — a non-zero complex number always has exactly two square roots, opposite each other across the origin (for example, 2i has roots 1 + i and −1 − i).
What is the argument of 0? Strictly, it is undefined — 0 + 0i has modulus 0 and no direction, so no angle. Calculators usually display 0 as a placeholder, but mathematically arg(0) does not exist.