Algebra • Complex Analysis & Field Theory

Complex Number Calculator

The comprehensive complex number calculator for arithmetic operations (z₁ ± z₂, z₁ × z₂, z₁ / z₂), conjugate division, modulus |z| = √(a² + b²), argument phase, polar/exponential conversions, and De Moivre powers with live 2D Argand plane visualization.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Mathematical Solution & Proofs

Complex Number 1 (z₁)

a + bi

Real Part Re(z₁) : a

Imaginary Part Im(z₁) : b

Select Operation

Complex Number 2 (z₂)

c + di

Real Part Re(z₂) : c

Imaginary Part Im(z₂) : d

Worked Example Presets:

Complex Solution Standard Form

Rectangular Form (a + bi)

11 − 2i

Polar: 11.1803 ∠ -10.3°

Magnitude |z| 11.1803

Argument θ -0.1799 rad

Conjugate z̄ 11 + 2i

Reciprocal 1/z 0.088 + 0.016i

Exponential & Trigonometric Forms:

Exponential (re^iθ) 11.18 × e^-0.18i

Trigonometric (r cis θ) 11.18(cos -10.3° + i sin -10.3°)

Argand Plane (Complex Coordinate Plot)

Re × Im

z₁ z₂ Result z

Derivation Steps:

Direct Answer & Overview

Verified Educational Guide

How to Calculate and Simplify Complex Numbers

A complex number is an algebraic quantity of the form z = a + bi, where a is the real component, b is the imaginary component, and i is the imaginary unit satisfying i² = −1. Complex numbers can be computed in standard rectangular form, converted to polar form r(cos θ + i sin θ), or expressed in Euler exponential form r · e^(iθ).

Primary Mathematical Formula Rectangular, Polar & Exponential Form Interconversions

Standard Equation

ƒ(x)

Q.E.D.

z = a + bi = r(\cos \theta + i\sin \theta) = r e^{i\theta}, \quad r = \sqrt{a^2 + b^2}, \quad \theta = \operatorname{atan2}(b, a)

Multiplication scales magnitudes and adds angles: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂).

Exact Formula

Input Parameters

Required

Real Part a = Re(z): Cartesian coordinate along the horizontal real axis

Imaginary Part b = Im(z): Cartesian coordinate along the vertical imaginary axis

Expected Outputs

Calculated

Standard Form: Rectangular representation a + bi

Modulus & Argument: Polar coordinates r ∠ θ (magnitude and phase angle)

Complex Conjugate: Reflected coordinate z̄ = a − bi

Multiplicative Inverse: Reciprocal 1/z = (a − bi) / (a² + b²)

Worked Numerical Example

Instant Verification

Multiply (3 + 4i) × (1 − 2i)

→ FOIL expansion: 3(1) + 3(−2i) + 4i(1) + 4i(−2i) = 3 − 6i + 4i − 8(−1) = (3 + 8) + (−6 + 4)i = 11 − 2i

11 − 2i (Modulus: 11.1803, Argument: −10.30°)

Anatomy of Complex Numbers & The Argand Plane

The real number line cannot represent square roots of negative numbers, such as √(−1). To resolve this limitation and complete the algebraic closure of numbers (the Fundamental Theorem of Algebra), mathematics defines the imaginary unit i such that:

i² = −1 ⇔ i = √(−1)

A complex number z combines a real component and an imaginary component into a two-dimensional mathematical entity: z = a + bi.

Real Part: a

Re(z)

The horizontal displacement along the real x-axis.

Imaginary Part: b

Im(z)

The vertical displacement along the imaginary y-axis (scaled by i).

Modulus: |z|

r = √(a² + b²)

The Euclidean distance or length of the vector from the origin.

Conjugate: &zmacr;

a − bi

Reflection across the real axis; crucial for complex division.

Algebraic Operations: Add, Subtract, Multiply & Divide

Addition & Subtraction

Add or subtract real and imaginary components independently (component-wise vector addition):

(a + bi) ± (c + di) = (a ± c) + (b ± d)i

Multiplication

Expand algebraically with the FOIL method and substitute i² = −1:

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

Division (Conjugate Multiplication)

Multiply numerator and denominator by the conjugate of the divisor (c − di) to convert the denominator into a pure real number:

(a + bi) / (c + di) = [(a + bi)(c − di)] / (c² + d²) = [(ac + bd) + (bc − ad)i] / (c² + d²)

Complex Representation Systems Comparison

Depending on the mathematical or engineering task, complex numbers can be expressed in rectangular, polar, exponential, or phasor notation:

System	Mathematical Form	Key Parameters	Ideal Application	Computation Advantage
Cartesian (Rectangular)	z = a + bi	a = Re(z), b = Im(z)	Addition & Subtraction	Linear arithmetic, no trig functions
Polar (Trigonometric)	r(cos θ + i sin θ)	r = \|z\|, θ = Arg(z)	Geometric rotations & vectors	Explicit sinusoidal decomposition
Euler Exponential	z = r • e^iθ	r = magnitude, θ = radians	Multiplication, Powers, Calculus	Exponents add: z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}
Phasor (Engineering)	r ∠ θ	RMS amplitude, Phase angle	AC Electrical circuit impedance	Compact frequency-domain analysis

Modulus, Argument & Polar Conversion (Euler's Formula)

Through Euler's Formula (e^iθ = cos θ + i sin θ), any complex number can be translated between rectangular coordinates (a, b) and polar coordinates (r, θ):

Modulus (r)

r = √(a² + b²)

Always a positive real number representing Euclidean vector length.

Argument (θ)

θ = atan2(b, a)

Principal argument θ ∈ (−π, π] radians.

Exponential Form

z = r e^iθ

The most compact notation for multiplication, division, and powers.

Powers & Roots with De Moivre's Theorem

Multiplying a complex number by itself n times in rectangular form requires tedious binomial expansions. In polar form, De Moivre's Theorem makes calculating powers instantaneous:

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) = rⁿ e^inθ

Similarly, the n distinct nth roots of z are uniformly distributed around a circle of radius r^1/n at angles (θ + 2kπ) / n for k = 0, 1, ..., n − 1.

Step-by-Step Worked Examples

Complex Multiplication Level: Intermediate

Calculate (3 + 4i) × (1 − 2i).

1. FOIL Expansion: 3(1) + 3(−2i) + 4i(1) + 4i(−2i) = 3 − 6i + 4i − 8i².

2. Substitute i² = −1: 3 − 2i − 8(−1) = 3 − 2i + 8.

3. Combine like terms: (3 + 8) − 2i = 11 − 2i.

Result: 11 − 2i (Modulus = √(121 + 4) = √125 ≈ 11.1803).

Conjugate Division Level: Intermediate

Calculate (5 − 2i) / (2 + 3i).

1. Identify denominator conjugate: Denominator is (2 + 3i), so its conjugate is (2 − 3i).

2. Multiply numerator and denominator by (2 − 3i):

Top: (5 − 2i)(2 − 3i) = 10 − 15i − 4i + 6i² = 10 − 19i − 6 = 4 − 19i.

Bottom: (2 + 3i)(2 − 3i) = 2² + 3² = 4 + 9 = 13.

Result: (4/13) − (19/13)i ≈ 0.3077 − 1.4615i.

Common Calculation Pitfalls & Conjugate Errors

Pitfall 1: Double-Negative Sign Error on i²

Remember that i² = −1. When multiplying (+bi)(−di) = −bdi² = −bd(−1) = +bd. A double-negative sign error here invalidates the entire real part.

Pitfall 2: Forgetting Quadrant Adjustments on atan2

When calculating the angle θ for numbers in Quadrants II or III (a < 0), simple arctan(b/a) gives the wrong angle. Always use the 4-quadrant atan2(b, a) function.

Engineering, Quantum & Real-World Applications

AC Circuit Impedance

Electrical engineering uses phasor notation Z = R + jX where resistance is real and reactance (capacitive/inductive) is imaginary.

Quantum State Vectors

Quantum mechanics represents state wavefunctions as complex probability amplitudes on the Bloch sphere with interference phases.

Fourier Transforms & DSP

Digital Signal Processing decomposes audio, radar, and medical imaging into frequency spectra using complex exponentials.

Fact-Checked & Verified • Computational Accuracy Standards

Updated September 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

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Frequently Asked Questions

What is a complex number and what does i represent?

A complex number is an algebraic quantity expressed in the standard rectangular form z = a + bi, where a is the real part (Re(z)), b is the imaginary part (Im(z)), and i is the imaginary unit defined by the fundamental identity i² = -1 (or i = √(-1)). Complex numbers extend the real number system to allow roots of negative polynomials and model 2D rotations.

How do you multiply and divide two complex numbers?

To multiply two complex numbers (a + bi)(c + di), expand using FOIL and substitute i² = -1 to obtain (ac − bd) + (ad + bc)i. To divide (a + bi) / (c + di), multiply both numerator and denominator by the complex conjugate of the denominator (c − di). This transforms the denominator into a pure positive real number (c² + d²), yielding [(ac + bd) + (bc − ad)i] / (c² + d²).

How do you convert a complex number from rectangular to polar and exponential form?

Given rectangular coordinates z = a + bi: 1) Compute the modulus (magnitude) r = |z| = √(a² + b²); 2) Compute the principal argument (phase angle) θ = atan2(b, a); 3) In polar (trigonometric) form, write z = r(cos θ + i sin θ) or r cis θ; 4) In Euler's exponential form, write z = r e^(iθ).

What is De Moivre's Theorem for powers of complex numbers?

De Moivre's Theorem states that for any real angle θ and integer exponent n, [r(cos θ + i sin θ)]^n = r^n(cos(nθ) + i sin(nθ)), or in exponential form, (r e^(iθ))^n = r^n e^(inθ). This allows powers and roots of complex numbers to be computed in a single step without binomial expansion.

What is an Argand plane?

An Argand plane (or complex plane C) is a two-dimensional geometric coordinate system where the horizontal axis represents the real component Re(z) and the vertical axis represents the imaginary component Im(z). Every complex number z = a + bi corresponds to a unique point (a, b) or position vector emanating from the origin.

Where are complex numbers used in the real world?

Complex numbers are vital in electrical engineering (alternating current circuit impedance and phasor analysis), signal processing (Fourier and Laplace transforms), quantum physics (wavefunctions and Schrödinger dynamics), fluid dynamics (aerodynamic airfoil mapping), and computer graphics (quaternions for 3D rotation).