Multiply complex numbers with FOIL method steps, polar multiplication, Argand diagram, and detailed breakdown.
Multiplying complex numbers is a fundamental operation that combines algebraic expansion with the key property i² = −1. Using the FOIL method (First, Outer, Inner, Last), the product (a+bi)(c+di) expands to ac + adi + bci + bdi², which simplifies to (ac−bd) + (ad+bc)i since i² = −1.
In polar form, complex multiplication has an elegant geometric interpretation: multiply the moduli and add the arguments. If z₁ = r₁∠θ₁ and z₂ = r₂∠θ₂, then z₁·z₂ = r₁r₂ ∠(θ₁+θ₂). This means multiplication scales the distance from the origin and rotates the angle — a powerful geometric operation that underlies rotation matrices, phasors in electrical engineering, and Fourier analysis.
This calculator breaks complex multiplication into complete FOIL steps, showing each partial product individually. You can see the First term (ac), Outer (adi), Inner (bci), and Last (bdi² = −bd) before they combine into the final result. This makes the process transparent and is ideal for learning.
Both rectangular and polar results are displayed, along with modulus verification (|z₁·z₂| = |z₁|·|z₂|). The Argand diagram plots z₁, z₂, and their product, visualizing the stretch-and-rotate nature of complex multiplication.
Presets include classic examples: conjugate pairs (z·z̄ = |z|²), pure imaginary products, squares, and mixed examples. The multiplication reference table summarizes all key formulas.
This tool is essential for students learning complex algebra, engineers working with phasors, and anyone needing to verify complex multiplications quickly.
Multiplying two complex numbers in rectangular form involves distributing four terms (FOIL), replacing i² with −1, and collecting real and imaginary parts. A single sign error in the i² substitution produces a completely wrong answer. This calculator multiplies any two complex numbers instantly, shows the FOIL expansion step, and displays the product in both rectangular and polar form. It also computes modulus and argument of the product, confirming that moduli multiply and arguments add — a key insight for students transitioning from rectangular to polar arithmetic.
(a+bi)(c+di) = (ac−bd) + (ad+bc)i; Polar: r₁·r₂ ∠ (θ₁+θ₂); |z₁·z₂| = |z₁|·|z₂|
Result: −7 + 22i
FOIL: First=8, Outer=10i, Inner=12i, Last=15i²=−15. Real: 8−15=−7. Imaginary: 10i+12i=22i. Product = −7+22i.
(a+bi)(c+di) expands to ac + adi + bci + bdi². Since i² = −1, the last term becomes −bd, giving the real part (ac−bd) and imaginary part (ad+bc)i. This is the FOIL (First, Outer, Inner, Last) method applied to complex numbers. When both numbers are purely real (b = d = 0), the formula reduces to ordinary multiplication. When both are purely imaginary (a = c = 0), the product bdi² = −bd is a negative real number, illustrating why the product of two imaginary numbers is real. This calculator shows the intermediate FOIL terms so students can see exactly where each piece of the result comes from.
In polar form, multiplication is (r₁∠θ₁)(r₂∠θ₂) = r₁r₂∠(θ₁+θ₂). This means multiplying by a complex number simultaneously scales (by the modulus) and rotates (by the argument). Multiplying by i (modulus 1, argument 90°) is a pure 90° rotation. Multiplying by 2i scales by 2 and rotates 90°. This rotation-and-scale interpretation is the foundation of conformal mappings in complex analysis, phase-shifting in signal processing, and 2D transformations in computer graphics. It also explains why the modulus of a product equals the product of moduli: |z₁z₂| = |z₁||z₂|.
Complex multiplication is the basic operation in mixing (modulation) of signals: multiplying a signal by e^{iωt} shifts its frequency by ω. This is how radio receivers tune to a specific station (heterodyne mixing) and how software-defined radios perform frequency translation. In quantum mechanics, multiplying a state vector by e^{iφ} (a phase factor) changes the global phase, which is unobservable, but relative phases between components produce measurable interference. The commutativity of complex multiplication (z₁z₂ = z₂z₁) simplifies analysis — a property lost when moving to quaternions, where multiplication is noncommutative.
Use FOIL: (a+bi)(c+di) = ac + adi + bci + bdi². Since i²=−1, this simplifies to (ac−bd) + (ad+bc)i.
FOIL stands for First, Outer, Inner, Last — the four products formed when multiplying two binomials: ac, ad, bc, bd. Use this as a practical reminder before finalizing the result.
Multiply moduli (r₁×r₂) and add arguments (θ₁+θ₂). Geometrically, this scales and rotates.
A complex number times its conjugate equals its modulus squared: (a+bi)(a−bi) = a²+b² = |z|². Keep this note short and outcome-focused for reuse.
Yes. z₁·z₂ = z₂·z₁. Both FOIL and polar methods confirm this.
It models rotation and scaling in 2D, is essential for AC circuits (phasors), signal processing, quantum mechanics, and computer graphics. Apply this check where your workflow is most sensitive.