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Complex Number Calculator

Math Calculators

Complex arithmetic, modulus, argument and conjugate

Quick Answer

Complex Number Calculator: Complex arithmetic, modulus, argument and conjugate. Add/subtract: (a±c) + (b±d)i.

z₁ = a + bi

+i

z₂ = c + di

+i

z₁ × z₂ =

11 − 2i

|z| = 11.180 · arg(z) = -10.3°

Modulus |z|11.180
Argument-10.3°
Conjugate11 + 2i
Polar form11.18∠-10.3°

Steps

  1. 1z₁ = 3 + 4i，z₂ = 1 − 2i
  2. 2(3×1 − 4×-2) + (3×-2 + 4×1)i = 11 − 2i（i² = −1）
  3. 3|z| = √(re² + im²) = 11.180；arg(z) = atan2(im, re) = -10.30°

Complex plane

ReImz₁z₂z

z₁ (blue), z₂ (green), result z (orange). |z| is the distance from origin; arg(z) the angle.

Operation rules

OperationRuleGeometric meaning
Add(a+c) + (b+d)ivector addition
Subtract(a−c) + (b−d)ivector subtraction
Multiply(ac−bd) + (ad+bc)imoduli multiply, angles add
Divide(z₁·z̄₂)/|z₂|²moduli divide, angles subtract
Multiplying by i rotates a point 90° counterclockwise — that is why i×i = −1 (two 90° turns = 180°). Conjugate product z·z̄ = |z|² is always a non-negative real.

💡 i² = −1. Multiply: treat i like a variable then replace i². Divide: multiply top & bottom by the conjugate (c − di) — the denominator becomes the real |z₂|². Polar form re^(iθ) makes powers trivial: zⁿ = rⁿ∠nθ (De Moivre).

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How to Use

  1. Enter values or expressions: type directly into the Complex Number Calculator — keyboard friendly.
  2. Instant results: everything computes live as you type, no button needed.
  3. Follow the steps: the result panel shows full working and formula substitution.
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Key Points

  • Complex Number Calculator is a free online math tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: Complex arithmetic, modulus, argument and conjugate.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Complex arithmetic (z₁ = a+bi, z₂ = c+di)

Add/subtract: (a±c) + (b±d)i
Multiply: (ac − bd) + (ad + bc)i
Divide: z₁/z₂ = z₁·conj(z₂) ÷ |z₂|²
Modulus |z| = √(a²+b²); argument arg(z) = atan2(b, a)

Example: (3+4i)×(1−2i) = 11−2i; |3+4i| = 5.

FAQ

How do I use the Complex Number Calculator?
Open the page and fill in the input fields — results update automatically. Complex arithmetic, modulus, argument and conjugate.
Are the results accurate?
Yes. All calculations use standard mathematical formulas and are verified for accuracy.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required — just open and use.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. All calculations run locally in your browser. Nothing is uploaded or stored.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated:

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Results provided by this tool are for reference only and do not constitute professional advice.