Add, multiply, divide complex numbers and combine parallel impedances. Rectangular to polar conversion, phasors, De Moivre powers and roots with R + jX steps.
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What is a Complex Number?
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = −1. The real part is a, the imaginary part is b. Despite the name 'imaginary', complex numbers are no more fictitious than negative numbers — they are the natural completion of arithmetic that lets every polynomial equation have a solution.
Geometrically, every complex number is a point in the 2D plane: a is the horizontal coordinate, b the vertical. Addition is vector addition; multiplication is a combined rotation and scaling. This geometric picture is why complex numbers are indispensable in electrical engineering, signal processing, quantum mechanics, fluid dynamics, and computer graphics.
Forms of Complex Numbers
Rectangular Form (a + bi)
The rectangular (or Cartesian) form writes a complex number as a sum of its real and imaginary parts: z = a + bi. This is the natural form for addition and subtraction, where you simply combine real parts with real parts and imaginary parts with imaginary parts.
Example: 3 + 4i has real part 3 and imaginary part 4
Polar Form (r∠θ)
The polar form expresses a complex number by its distance from the origin (modulus r) and angle from the positive real axis (argument θ): z = r∠θ, which equals r·(cos θ + i·sin θ) and, more compactly, r·e^(iθ). This is the natural form for multiplication, division and powers.
The modulus r = √(a² + b²) is the length of the arrow from the origin to the point; the argument θ = atan2(b, a) is the angle it makes with the positive real axis, measured counter-clockwise. Polar form makes the geometry of multiplication obvious: multiplying by r·e^(iθ) means 'scale by r, rotate by θ'.
Example: 5∠53.13° is equivalent to 3 + 4i
Converting Between Forms
Rectangular to Polar: r = √(a² + b²), θ = atan2(b, a)
Polar to Rectangular: a = r·cos(θ), b = r·sin(θ)
Complex Number Operations
Addition and Subtraction
Add or subtract real parts and imaginary parts separately. Geometrically this is vector addition — the same parallelogram rule that adds force vectors in physics.
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication
In rectangular form: (a + bi)(c + di) = (ac − bd) + (ad + bc)i — expand the product like binomials and remember i² = −1.
In polar form: (r₁∠θ₁)·(r₂∠θ₂) = (r₁·r₂)∠(θ₁ + θ₂). Multiplication adds angles and multiplies lengths — a remarkable identity that makes complex numbers the natural tool for rotations.
Division
To divide in rectangular form, multiply numerator and denominator by the conjugate of the denominator. This turns the denominator into a real number (a² + b²) and lets you separate the real and imaginary parts.
In polar form: (r₁∠θ₁) ÷ (r₂∠θ₂) = (r₁/r₂)∠(θ₁ − θ₂). Division subtracts angles and divides lengths.
Complex Conjugate
The conjugate of a + bi is a − bi. Geometrically, it is the reflection across the real axis. Conjugates are how you 'realize' a denominator and how you compute |z|² without a square root.
Important: z · z̄ = a² + b² = |z|² (always a non-negative real number)
Powers and Roots
De Moivre's Theorem: (r·e^(iθ))ⁿ = rⁿ·e^(inθ). Raise the modulus to the n-th power; multiply the argument by n.
Every non-zero complex number has exactly n distinct n-th roots, spaced evenly around a circle of radius ⁿ√r at angle increments of 360°/n. This is why polynomial equations of degree n have exactly n complex roots (the Fundamental Theorem of Algebra).
Complex Functions
All the familiar real functions extend to complex numbers, often with surprising new behaviour:
Exponential: e^(a+bi) = eᵃ·(cos b + i·sin b) — Euler's formula in disguise
Natural logarithm: ln(r·e^(iθ)) = ln(r) + iθ — multi-valued, since θ is defined modulo 2π
Trigonometric: sin(z) = (e^(iz) − e^(−iz))/(2i), cos(z) = (e^(iz) + e^(−iz))/2 — these can take values outside [−1, 1] for complex inputs
Complex Number Calculator
Applications of Complex Numbers
Complex numbers are not academic curiosities — they run modern technology:
Electrical Engineering: AC voltages and currents are encoded as complex phasors; impedance combines resistance and reactance as Z = R + jX (engineers use j to avoid confusion with current i)
Signal Processing: the Fourier transform is fundamentally a sum of complex exponentials; the FFT — the algorithm behind MP3, JPEG, Wi-Fi and MRI — multiplies and adds complex numbers billions of times per second
Quantum Mechanics: every wave function is complex-valued, and the probability of finding a particle is |ψ|² = ψ·ψ̄
Control Theory: poles of the transfer function in the complex plane determine whether a system is stable; engineers literally draw root-locus plots in the complex plane
Fluid Dynamics: 2D potential flow uses complex analysis; the Joukowski transform maps a circle to an aerofoil shape used in early aircraft wings
Computer Graphics: the Mandelbrot and Julia sets are iterations zₙ₊₁ = zₙ² + c; quaternions (a 4D extension) drive every 3D rotation in modern games
Number Theory: the Riemann zeta function ζ(s) is defined on complex s; its non-trivial zeros control the distribution of prime numbers
Tips for Using the Calculator
Use rectangular form for adding or subtracting; the formulas are simplest there
Switch to polar form for multiplication, division, powers and roots — angles and moduli decouple cleanly
Remember atan2(b, a), not atan(b/a), when computing the argument — atan can't tell quadrants apart
Always specify whether your angles are in degrees or radians; mixing them up is the #1 source of wrong answers
The conjugate z̄ is the secret weapon for division and for finding magnitudes without a square root
Frequently Asked Questions
The imaginary unit i is defined by i² = −1, the rule that lets you take square roots of negative numbers. It wasn't invented to be mystical; it was forced on mathematicians by 16th-century algebra. Cardano (1545) discovered a cubic formula that found the real roots of x³ + ax + b = 0 — but only by taking the square root of a negative number along the way, even when all three roots were ordinary real numbers. The 'imaginary' quantities had to be valid intermediate steps because the answer at the end was unquestionably real. Bombelli formalized arithmetic with these numbers in 1572; Euler introduced the letter i in 1777; Gauss (1831) gave them the geometric interpretation as points in a 2D plane, after which their respectability was secured. So i isn't a trick — it's the natural completion of the real numbers, just as negative numbers are the natural completion of the positive integers. The name 'imaginary' is a historical accident; physicists and engineers calculate with i every day without anyone calling Maxwell's equations imaginary.
They guarantee that every polynomial equation has solutions. Over the real numbers, x² + 1 = 0 has none — you can't square a real number to get −1. Over the complex numbers, x² + 1 = 0 has two solutions: x = i and x = −i. The Fundamental Theorem of Algebra, proved by Gauss in 1799, says that every polynomial of degree n with complex coefficients has exactly n complex roots (counted with multiplicity). No more 'sometimes solvable, sometimes not'. This algebraic closure has dramatic consequences. It makes the eigenvalue problem always solvable (eigenvalues are roots of the characteristic polynomial), which is why linear algebra works smoothly over ℂ. It lets you factor any polynomial into linear factors, which underpins partial-fraction decomposition in calculus. And it gives differential equations a unified theory — every linear ODE has solutions of the form e^(λt) where λ is complex; if λ is real, you get exponential growth or decay, if λ is purely imaginary you get oscillation, and if it has both parts you get damped oscillation. The complex numbers unify what looked like three separate phenomena.
Multiplication scales and rotates. If z₁ has modulus r₁ and argument θ₁, and z₂ has modulus r₂ and argument θ₂, then z₁·z₂ has modulus r₁·r₂ and argument θ₁ + θ₂. So multiplying by z is the same as 'scale every point by |z|, then rotate the whole plane by arg(z) about the origin'. This is why multiplying by i — which has modulus 1 and argument 90° — rotates the plane by a quarter turn: it sends 1 to i, i to −1, −1 to −i, and back to 1. Multiplying by e^(iθ) is a pure rotation by angle θ. This single fact is the foundation of why complex numbers are everywhere in physics and engineering: anything that involves rotation, oscillation or waves becomes algebra once you switch to the complex plane. A rotating engine, an AC voltage, a quantum wave function — they all become multiplications by e^(iωt).
Euler's formula says e^(iθ) = cos θ + i·sin θ. Set θ = π and you get e^(iπ) = cos π + i·sin π = −1 + 0i = −1, so e^(iπ) + 1 = 0 — five of the most important constants of mathematics (0, 1, e, i, π) linked in one equation, and one of the most-voted 'most beautiful equation' in maths history. The geometric interpretation is direct: e^(iθ) traces the unit circle as θ varies, so e^(iπ) is the point at angle π (180°) on the unit circle, which is exactly −1. The deeper meaning is that the exponential function and trigonometric functions are the same function on the complex plane, just viewed differently. This is why d/dt of e^(iωt) = iω·e^(iωt) describes circular motion at angular frequency ω. It is also why engineers replace cos(ωt) with the 'real part of e^(iωt)': differentiation becomes multiplication by iω, integration becomes division — algebra replaces calculus. Schrödinger's equation, Maxwell's equations, AC circuit analysis, and the Fourier transform all rest on this single identity.
Because each form makes one operation simple and the other awkward. To add (3 + 4i) + (1 + 2i) in rectangular form, you just add components: (3+1) + (4+2)i = 4 + 6i. In polar form you would have to convert to rectangular, add, then convert back — three operations instead of one. Conversely, to multiply (3 + 4i)·(1 + 2i) in rectangular form, you have to FOIL the binomial and remember i² = −1: (3·1 − 4·2) + (3·2 + 4·1)i = −5 + 10i. In polar form (5∠53.13°)·(√5∠63.43°) = 5√5∠116.56°, just multiply moduli and add angles. Powers and roots compound this: computing (1 + i)¹⁰⁰ in rectangular form is a nightmare, but in polar form it's (√2∠45°)¹⁰⁰ = 2⁵⁰∠4500° = 2⁵⁰∠180° = −2⁵⁰. The rule of thumb: if the operation is linear (sum, difference), use rectangular. If it involves rotations (product, quotient, power, root), use polar. Calculators that show both forms exist precisely so you don't have to convert by hand.
Because polar form is multi-valued. A complex number z = r·e^(iθ) can equally be written as r·e^(i(θ + 2πk)) for any integer k — adding any whole number of full turns to the angle gives the same point. When you take the n-th root, you divide the angle by n: ⁿ√z = ⁿ√r · e^(i(θ + 2πk)/n). For k = 0, 1, 2, …, n−1 you get n different angles (and hence n different points); for k = n you come back to where you started. So every non-zero complex number has exactly n distinct n-th roots, equally spaced on a circle of radius ⁿ√r at angle increments of 360°/n. The cube roots of 1, for instance, are 1, e^(i·120°) ≈ −0.5 + 0.866i, and e^(i·240°) ≈ −0.5 − 0.866i — they form an equilateral triangle. This is why polynomial equations like z⁵ = 32 have five solutions, not one. The Fundamental Theorem of Algebra is built on top of this fact: a polynomial xⁿ − c factors completely over the complex numbers because c has exactly n n-th roots.
Every AC voltage or current — a sinusoid V(t) = V₀·cos(ωt + φ) — is replaced by a complex phasor V = V₀·e^(iφ), and the time dependence e^(iωt) is suppressed. This trick converts differential equations into algebra. Resistors have impedance R (purely real); inductors have impedance jωL (purely imaginary, positive); capacitors have impedance 1/(jωC) (purely imaginary, negative). Engineers use j instead of i because i is already reserved for current. Ohm's law works in phasors too: V = I·Z, where Z is the complex impedance. The angle of Z tells you the phase shift between voltage and current — a purely resistive circuit has Z's angle = 0 (in phase), a purely inductive one has +90° (current lags), and a purely capacitive one has −90° (current leads). The magnitude of Z is the apparent resistance to AC; the real part is the part that dissipates power, the imaginary part is the part that just shuttles energy back and forth. None of this would be tractable without complex numbers — and the entire global power grid runs on calculations done in the complex plane.
Enter the first impedance as Complex Number 1 and the second as Complex Number 2, using rectangular form Z = R + jX (resistance as the real part, reactance as the imaginary part — positive for inductive, negative for capacitive). Choose the 'Parallel (Z₁ ∥ Z₂)' operation and press Calculate. The tool computes Z₁·Z₂/(Z₁+Z₂) — the canonical formula for two impedances in parallel — in one step, so you no longer have to chain a multiply, an add and a divide by hand and copy intermediate phasors between runs. The result panel decomposes the answer for you: it shows the rectangular form (the new R + jX), the polar form (magnitude and phase), and a dedicated R / X line giving the equivalent resistance R = Re and reactance X = Im directly. For example, a 3 + 4j Ω branch in parallel with a 1 + 2j Ω branch gives roughly 1.36 + 1.48j Ω. To combine three or more impedances, run the parallel operation pairwise: combine the first two, paste that result back into Complex Number 1, and parallel it with the third. The same workflow handles parallel admittances or any A·B/(A+B) combination.
To convert rectangular to polar, enter your number as a + bi in rectangular form, pick any operation that returns it unchanged (for instance Conjugate twice, or simply Add with 0 + 0i in the second box) and read the 'Polar form:' line in the Calculation panel — it shows the modulus r and the argument θ, in degrees or radians depending on the 'Use degrees for θ' checkbox. Even simpler, every single-value result already prints both the rectangular and polar forms plus the magnitude |Z| and phase, so any calculation doubles as a converter. To compute a power like (1 + i)ⁿ, enter 1 + 1i, choose the Power (zⁿ) operation, type the exponent n, and the tool applies De Moivre's theorem internally — raising the modulus to n and multiplying the angle by n — so even (1 + i)¹⁰⁰ comes back instantly without manual binomial expansion.
Three big uses. (1) Realizing denominators: dividing complex numbers in rectangular form requires multiplying top and bottom by the conjugate of the denominator, which turns (c + di)·(c − di) = c² + d², a real number, freeing the real and imaginary parts in the result. Without conjugates, division would be terrible. (2) Computing magnitudes without a square root: |z|² = z·z̄ = a² + b². In quantum mechanics, the probability density |ψ|² is computed exactly this way, multiplying the wave function by its conjugate. (3) Symmetry of polynomial roots: if a polynomial with real coefficients has a complex root α + βi, it must also have the conjugate root α − βi (because conjugating the whole equation gives the same equation back). This is why quadratics with negative discriminants always have conjugate pair roots, and why the roots of any real polynomial come in conjugate pairs unless they happen to be real. The conjugate is also how 'real signal processing' works — discrete signals' FFTs satisfy X[N−k] = X[k]̄, halving the storage needed.
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