Algebra • Complex Numbers & Arithmetic

Complex Number Multiplication Calculator

Multiply complex numbers z₁ × z₂ with step-by-step binomial FOIL expansion, exact algebraic proofs, polar angle addition, and real-time Argand plane visualization.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Mathematical Solution & Proofs

FOIL & Rotational Scaling Solver Algebraic Expansion & Vector Argand Plane

Preset:

Complex Inputs (z₁ × z₂)

First Number z₁ = a + bi |z₁| = 3.6056

Real Part (a)

Imaginary Part (b)

Second Number z₂ = c + di |z₂| = 4.1231

Real Part (c)

Imaginary Part (d)

Product Result: z₁ × z₂

Standard Rectangular Form

−5 + 14i

Polar Form (r ∠ θ) 14.866 ∠ 109.65° 1.9138 rad

Euler Form (r · e^(iθ)) 14.866 · e^(1.9138i) z* = −5 − 14i

Modulus Property |z₁·z₂| = |z₁|·|z₂| 3.606 × 4.123 = 14.866

Argument Property θ = θ₁ + θ₂ 33.69° + 75.96° = 109.65°

Complex Argand Plane Rotational Scaling

z₁ Vector 3 + 2i

z₂ Vector 1 + 4i

z₁ × z₂ −5 + 14i

Geometric Meaning: Multiplication on the complex plane scales the length (modulus) by 4.123× and rotates counter-clockwise by +75.96°.

Step-by-Step FOIL Algebraic Proof

Verified Binomial Expansion

Step 1: Set Up the Binomial Product

Write out the two complex numbers in standard binomial notation:

(3 + 2i) × (1 + 4i)

Step 2: Apply the FOIL Method (First, Outer, Inner, Last)

Multiply each term systematically:

= (3 × 1) + (3 × 4i) + (2i × 1) + (2i × 4i)

= 3 + 12i + 2i + 8i²

Step 3: Substitute the Fundamental Identity i² = −1

The imaginary unit squared transforms the last term into a purely real value:

= 3 + 12i + 2i + 8(−1) = 3 + 12i + 2i − 8

Step 4: Group Real & Imaginary Parts and Conclude

Collect real terms (ac − bd) and imaginary terms (ad + bc)i:

= (3 − 8) + (12 + 2)i = −5 + 14i

Direct Answer & Overview

Verified Educational Guide

How to Multiply Complex Numbers

To multiply two complex numbers z₁ = a + bi and z₂ = c + di, expand them as binomials using the FOIL method, replace i² with −1, and group the real and imaginary parts. The resulting product is (ac − bd) + (ad + bc)i. Geometrically on the Argand plane, the magnitudes multiply (|z₁ · z₂| = |z₁| · |z₂|) and the angles add (Arg(z₁ · z₂) = Arg(z₁) + Arg(z₂)).

Primary Mathematical Formula Algebraic Standard Form Formula

Standard Equation

ƒ(x)

Q.E.D.

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

Valid for all real numbers a, b, c, d with defining imaginary identity i² = −1.

Exact Formula

1. What Is Complex Number Multiplication?

In mathematics, the set of complex numbers ℂ extends the standard one-dimensional real line into a two-dimensional geometric plane. Every complex number is expressed in standard rectangular form as z = a + bi, where a is the real part (Re(z)), b is the imaginary part (Im(z)), and i is the imaginary unit governed by the defining algebraic relation:

i² = −1 ⇔ i = √(−1)

Unlike vector multiplication in ℝ² (which is conventionally split into the scalar dot product and the orthogonal cross product), complex multiplication is an internal algebraic operation. Multiplying two complex numbers produces another unique complex number. It behaves simultaneously like binomial polynomial expansion in algebra and an elegant combination of dilation (stretching/shrinking) and rotation in geometry.

2. Algebraic Formula Derivation & The FOIL Method

To derive the universal formula for multiplying two complex numbers z₁ = a + bi and z₂ = c + di, we treat both quantities as algebraic binomials and apply the distributive law of multiplication over addition:

1. Initial Product: (a + bi) × (c + di)

2. FOIL Distributive Expansion: a(c + di) + bi(c + di) = ac + adi + bci + bdi²

3. Substitute i² = −1: ac + adi + bci + bd(−1) = ac + adi + bci − bd

4. Group Real & Imaginary: (ac − bd) + (ad + bc)i

Why does the minus sign appear in (ac − bd)?

Students frequently ask why a negative sign arises when multiplying positive numbers. The minus sign in −bd originates entirely from the definition i × i = i² = −1. When multiplying the two imaginary terms (bi)(di) = bd · i², the product of the imaginary units collapses back into the real axis with a negated sign.

3. Geometric Interpretation on the Argand Plane (Rotational Scaling)

While the algebraic FOIL method is mechanically efficient, the true elegance of complex multiplication emerges on the Argand diagram. When represented geometrically as position vectors from the origin:

1. Length Dilation (Modulus Scaling)

The length of the resulting product vector equals the algebraic product of the lengths of the two factor vectors:

|z₁ · z₂| = |z₁| · |z₂|

If |z₂| > 1, the vector stretches; if |z₂| < 1, the vector contracts; if |z₂| = 1, length is invariant.

2. Angular Rotation (Argument Addition)

The counter-clockwise orientation angle of the product equals the arithmetic sum of the individual angles:

Arg(z₁ · z₂) = Arg(z₁) + Arg(z₂)

Multiplication by z₂ acts as a rotation operator by angle θ₂ followed by a radial scaling factor r₂.

Special Case: Multiplying by the Imaginary Unit i (+90° Rotation)

The imaginary unit i has a modulus of |i| = 1 and an argument of θ = 90° (π/2 rad). Therefore, multiplying any arbitrary complex number z = a + bi by i leaves its length unchanged while rotating the vector exactly 90 degrees counter-clockwise:

z · i = (a + bi) · i = ai + bi² = −b + ai

Notice that the coordinates map from (a, b) → (−b, a), which is precisely the standard 90° counter-clockwise rotation transformation matrix in 2D linear algebra.

4. Polar & Euler Exponential Multiplication

While rectangular coordinates (a + bi) make addition simple, polar and exponential forms make multiplication trivial. According to Euler's formula:

e^(iθ) = cos θ + i sin θ &implies; z = r e^(iθ)

When multiplying two complex numbers in Euler form, z₁ = r₁ e^(iθ₁) and z₂ = r₂ e^(iθ₂), we apply standard exponent laws:

z₁ × z₂ = (r₁ e^(iθ₁))(r₂ e^(iθ₂)) = (r₁ · r₂) · e^(i(θ₁ + θ₂))

Using trigonometric angle-addition identities, this directly matches the polar trigonometric representation:

z₁ × z₂ = r₁ r₂ [ cos(θ₁ + θ₂) + i sin(θ₁ + θ₂) ]

This property is the foundation of De Moivre's Theorem, which states that for any integer n: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).

5. Step-by-Step Worked Manual Examples

Below are three comprehensive, step-by-step worked solutions demonstrating standard binomial multiplication, complex conjugate multiplication, and pure imaginary scaling.

Example 1: Standard Binomial Product (2 + 5i) × (4 − 3i)

Step 1: Set up FOIL expansion:

= (2 × 4) + (2 × (−3i)) + (5i × 4) + (5i × (−3i))

Step 2: Simplify individual products:

= 8 − 6i + 20i − 15i²

Step 3: Replace i² with −1:

= 8 + 14i − 15(−1) = 8 + 14i + 15

Step 4: Group real and imaginary components:

= (8 + 15) + 14i = 23 + 14i

Example 2: Complex Conjugate Product (z · z̄) (4 + 7i) × (4 − 7i)

Step 1: Apply the difference of squares identity (x + y)(x − y) = x² − y²:

= (4)² − (7i)²

Step 2: Evaluate square of the imaginary term:

= 16 − 49i² = 16 − 49(−1) = 16 + 49

Step 3: Conclude:

= 65 (= |z|² = 4² + 7² = 16 + 49)

Example 3: Pure Imaginary Scaling (0 + 3i) × (0 + 5i)

Step 1: Direct multiplication:

= (3i) × (5i) = 15 · i²

Step 2: Substitute i² = −1:

= 15(−1) = −15 + 0i (Pure real negative number)

6. Algebraic Properties & 2×2 Matrix Representation

The complex numbers ℂ satisfy all formal mathematical field axioms under multiplication:

Property	Mathematical Statement	Physical / Geometric Consequence
Commutativity	z₁ · z₂ = z₂ · z₁	Order of rotation and dilation does not affect the final coordinate.
Associativity	(z₁ · z₂) · z₃ = z₁ · (z₂ · z₃)	Compound rotations can be grouped in any order.
Distributivity	z₁ · (z₂ + z₃) = z₁z₂ + z₁z₃	Linear mapping holds across vector vector addition.
Multiplicative Identity	z · (1 + 0i) = z	Rotation by 0° and scaling by 1 preserves coordinates.
Multiplicative Inverse	z · (1/z) = 1 (for z ≠ 0)	Inverse is given by z̄ / \|z\|².

2×2 Real Matrix Representation of Complex Multiplication

Every complex number z = a + bi can be isomorphically mapped to a real 2 × 2 matrix representing simultaneous scaling by r = √(a² + b²) and rotation by angle θ:

z = a + bi &longleftrightarrow M(z) = [ a −b ; b a ]

Multiplying two such matrices M(z₁) × M(z₂) produces exactly the matrix corresponding to (ac − bd) + (ad + bc)i, demonstrating why complex arithmetic is widely used to program 2D camera rotations in computer graphics engines without trigonometric function calls.

7. Common Pitfalls & Sign Traps

⚠ Pitfall 1: Forgetting that i² = −1 (Sign Flip Trap)

The most ubiquitous error is writing (2i)(3i) = 6i or +6. Multiplying imaginary terms yields 6i² = 6(−1) = −6. Always write out i² in your intermediate step before substituting −1.

⚠ Pitfall 2: Double Negative Confusion when Multiplying Subtracted Imaginary Parts

When multiplying (a − bi)(c − di), the last term is (−bi)(−di) = +bd · i² = −bd. Students frequently see two minus signs, conclude that the product is positive, and forget that the i² flips the sign a third time.

⚠ Pitfall 3: Confusing Complex Multiplication with Vector Dot Product

The vector dot product (a, b) · (c, d) = ac + bd is a scalar real quantity. Complex multiplication produces a new 2D vector (ac − bd, ad + bc). Notice the subtraction in the real term and the non-zero imaginary term.

8. Real-World Applications: AC Circuits, DSP, & Computer Graphics

Electrical AC Circuits (Ohm's Law)

In alternating current analysis, AC voltage V and current I are sinusoidal phasors. Complex impedance Z = R + jX multiplies with current to determine voltage: V = I × Z. The magnitude multiplication gives peak voltage, and the argument addition indicates phase lead/lag.

Digital Signal Processing (DSP)

The Discrete Fourier Transform (DFT and FFT) relies on multiplying discrete audio and radar sample arrays by complex exponential twiddle factors W_N^k = e^(−j2πk/N) to deconstruct mixed signals into distinct frequency harmonics.

Computer Graphics & Quaternions

Unit complex numbers e^(iθ) perform 2D game character and camera rotations without costly sine and cosine evaluations. In 3D space, this generalizes to 4-dimensional hypercomplex numbers known as quaternions to prevent gimbal lock.

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

How do you multiply two complex numbers together?

To multiply two complex numbers z₁ = a + bi and z₂ = c + di, expand them as binomials using the FOIL method (First, Outer, Inner, Last): (a + bi)(c + di) = ac + adi + bci + bdi². Next, substitute the defining identity i² = −1, which transforms bdi² into −bd. Finally, combine the real parts and imaginary parts into standard form: (ac − bd) + (ad + bc)i.

Why does multiplying complex numbers rotate vectors on the Argand plane?

In the polar representation z = r(cos θ + i sin θ) = r e^(iθ), multiplication yields (r₁ e^(iθ₁))(r₂ e^(iθ₂)) = (r₁ r₂) e^(i(θ₁ + θ₂)). The moduli multiply (dilating or contracting the distance from the origin), while the arguments add together (θ₁ + θ₂). This angle addition creates a pure counter-clockwise rotation around the origin.

What happens when you multiply a complex number by its conjugate?

Multiplying a complex number z = a + bi by its conjugate z̄ = a − bi produces the square of its modulus: z · z̄ = (a + bi)(a − bi) = a² − (bi)² = a² + b² = |z|². The imaginary parts (+abi − abi) cancel completely, always yielding a non-negative real number.

What is the effect of multiplying a complex number by i?

Multiplying any complex number z by the imaginary unit i rotates it exactly 90° (π/2 radians) counter-clockwise on the complex plane without changing its magnitude. For example, (a + bi) · i = ai + bi² = −b + ai. The modulus remains |z · i| = |z| · 1 = |z|.

Is complex number multiplication commutative and associative?

Yes. The set of complex numbers ℂ forms a field under addition and multiplication. Complex multiplication satisfies the commutative property (z₁ · z₂ = z₂ · z₁), the associative property (z₁ · (z₂ · z₃) = (z₁ · z₂) · z₃), and distributes over addition (z₁ · (z₂ + z₃) = z₁ · z₂ + z₁ · z₃).

How do you multiply complex numbers in polar or Euler exponential form?

In polar form, multiply the magnitudes and add the angles: (r₁ ∠ θ₁) · (r₂ ∠ θ₂) = (r₁ · r₂) ∠ (θ₁ + θ₂). In Euler exponential notation, applying standard index laws gives: (r₁ e^(iθ₁)) · (r₂ e^(iθ₂)) = (r₁ · r₂) e^(i(θ₁ + θ₂)). This formulation is especially advantageous in electrical engineering and signal processing when computing alternating currents and waveforms.