Complex Number Calculator

About the Complex Number Calculator

Complex numbers extend the real number system by introducing the imaginary unit i, where i² = −1. Every complex number can be written in rectangular form z = a + bi or in polar form z = r·e^(iθ), where r is the modulus and θ is the argument. This calculator handles all fundamental complex number operations with both representations.

Addition and subtraction of complex numbers work component-wise: (a+bi) ± (c+di) = (a±c) + (b±d)i. Multiplication uses the FOIL method and i² = −1: (a+bi)(c+di) = (ac−bd) + (ad+bc)i. Division multiplies numerator and denominator by the conjugate of the divisor, and the conjugate simply negates the imaginary part.

In polar form, multiplication becomes elegantly simple: multiply the moduli and add the arguments. Division divides moduli and subtracts arguments. Powers use De Moivre's theorem: z^n = r^n · (cos(nθ) + i·sin(nθ)).

This calculator shows all results in both rectangular and polar form, with modulus, argument in degrees and radians, and an interactive Argand diagram plotting z₁, z₂, and the result. The operations table shows every operation simultaneously, so you can compare add, subtract, multiply, divide, and conjugate results at a glance.

Presets cover common examples from textbooks and engineering applications. Whether you are studying electrical circuit phasors, signal processing, quantum mechanics, or pure algebra, complex number arithmetic is essential.

When This Page Helps

Complex number arithmetic — especially multiplication, division, and exponentiation — requires distributing real and imaginary parts, rationalizing denominators, or converting between rectangular and polar forms. A single sign error produces a completely wrong answer. This calculator performs addition, subtraction, multiplication, division, and integer powers of two complex numbers, showing each result in both rectangular (a + bi) and polar (r∠θ) form with full intermediate steps. It also computes modulus, conjugate, and argument, making it a one-stop reference for any complex-number task in algebra, electrical engineering, or signal processing.

How to Use the Inputs

1. Choose the operation: add, subtract, multiply, divide, conjugate, or power.
2. Enter real and imaginary parts for z₁ (and z₂ if applicable).
3. For power operations, enter the integer exponent.
4. Read the result in both rectangular and polar form.
5. View the Argand diagram to see z₁, z₂, and the result plotted.
6. Check the all-operations table for a comprehensive summary.

Example Calculation

Tips & Best Practices

• In polar form, multiplication = multiply moduli + add arguments.
• The conjugate of z = a+bi is z* = a−bi; z·z* = |z|² is always real.
• Division by zero (z₂ = 0) is undefined — the calculator handles this.
• Powers use De Moivre's theorem: z^n = r^n · cis(nθ).
• The Argand diagram Origin is at the center of the plot.

Rectangular vs. Polar Form

Every complex number z = a + bi can also be written in polar form z = r(cosθ + i sinθ) = r e^{iθ}, where r = √(a²+b²) is the modulus and θ = atan2(b,a) is the argument. Rectangular form is convenient for addition and subtraction (just add real and imaginary parts), while polar form simplifies multiplication (multiply moduli, add arguments) and division (divide moduli, subtract arguments). Euler's formula e^{iθ} = cosθ + i sinθ bridges the two representations and is arguably the most beautiful equation in mathematics when evaluated at θ = π.

Complex Numbers in Electrical Engineering

Electrical engineers use complex numbers (called phasors) to analyze AC circuits. A sinusoidal voltage V(t) = V₀ cos(ωt + φ) is represented as the complex phasor Ṽ = V₀ e^{iφ}. Impedance Z = R + jX (using j instead of i) combines resistance R and reactance X into a single complex number. Ohm's law in phasor form, Ṽ = Ĩ Z, turns differential equations into algebraic ones. Multiplying phasors corresponds to amplitude scaling and phase shifting, and dividing phasors gives the transfer function of a filter or amplifier. Without complex arithmetic, AC circuit analysis would require solving coupled trigonometric equations — far more cumbersome.

Fundamental Theorem of Algebra

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has at least one complex root. Equivalently, every degree-n polynomial factors completely into n linear factors over the complex numbers. This is why complex numbers were invented: to provide solutions to equations like x² + 1 = 0 that have no real solutions. The theorem guarantees that the complex numbers are algebraically closed — you never need to invent "hyper-imaginary" numbers to solve polynomial equations. Roots of real polynomials come in conjugate pairs (a+bi and a−bi), which is why the real and imaginary parts of physical systems' transfer function poles always exhibit mirror symmetry about the real axis.

Frequently Asked Questions

• What is a complex number?

  A complex number z = a + bi has a real part a and an imaginary part b, where i = √(−1).

• What are rectangular and polar forms?

  Rectangular: z = a + bi. Polar: z = r·(cos θ + i·sin θ) = r·e^(iθ), where r = |z| and θ = arg(z).

• How do you multiply complex numbers?

  Use FOIL: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i, since i²=−1.

• What is an Argand diagram?

  An Argand diagram plots complex numbers on a 2D plane with the real part on the x-axis and the imaginary part on the y-axis.

• What is the modulus of a complex number?

  The modulus |z| = √(a² + b²) is the distance from the origin to the point z in the complex plane.

• What is De Moivre's theorem?

  (cos θ + i·sin θ)^n = cos(nθ) + i·sin(nθ). It simplifies computing powers and roots of complex numbers.