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Complex Number Calculator

Perform arithmetic operations on complex numbers: addition, subtraction, multiplication, division, and find modulus.

Reviewed by Christopher FloiedPublished Updated

This free online complex number calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

Sum: real part

0

Sum: imaginary part

0

Product: real part

0

Product: imaginary part

0

|z₁| (modulus)

0

|z₂| (modulus)

0

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Complex Number Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Complex Number Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Complex Number Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Complex Number Calculator

The Complex Number Calculator performs arithmetic on complex numbers of the form a + bi, where i = √(-1). Complex numbers extend the real numbers to include solutions to equations like x² + 1 = 0, which have no real solutions. They are essential in electrical engineering (AC circuits, impedance), signal processing (Fourier transforms), quantum mechanics (wave functions), control theory (transfer functions), and fluid dynamics. Every complex number has a real part and an imaginary part, and can be visualized as a point in the complex plane. This calculator computes the sum, product, and modulus of two complex numbers, demonstrating the fundamental arithmetic operations that form the basis of complex analysis.

The Math Behind It

A complex number z = a + bi consists of a real part Re(z) = a and an imaginary part Im(z) = b, where i is the imaginary unit satisfying i² = -1. Complex numbers form a field under addition and multiplication. Addition is componentwise: (a+bi) + (c+di) = (a+c) + (b+d)i. Multiplication uses the distributive property and i² = -1: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac-bd) + (ad+bc)i. The modulus (absolute value) |z| = √(a² + b²) represents the distance from the origin in the complex plane. The complex conjugate of z = a + bi is z̄ = a - bi, and z·z̄ = a² + b² = |z|². Division uses conjugates: (a+bi)/(c+di) = (a+bi)(c-di)/((c+di)(c-di)) = ((ac+bd) + (bc-ad)i)/(c²+d²). Complex numbers can also be written in polar form: z = r(cos θ + i sin θ) = re^(iθ), where r = |z| and θ = arg(z). Polar form simplifies multiplication (multiply moduli, add arguments) and exponentiation (De Moivre's theorem). The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex root, making the complex numbers algebraically closed.

Formula Reference

Complex Addition

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

Variables: Add real and imaginary parts separately

Complex Multiplication

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

Variables: Use FOIL and i² = -1

Complex Modulus

|a+bi| = √(a² + b²)

Variables: Distance from origin in the complex plane

Worked Examples

Example 1: Addition and multiplication

Compute (3 + 2i) + (1 - 4i) and (3 + 2i)(1 - 4i)

Step 1:Sum: (3+1) + (2+(-4))i = 4 - 2i
Step 2:Product: real = 3(1) - 2(-4) = 3 + 8 = 11
Step 3:Product: imag = 3(-4) + 2(1) = -12 + 2 = -10
Step 4:Product: 11 - 10i

Sum = 4 - 2i, Product = 11 - 10i

Example 2: Modulus

Find |3 + 4i|

Step 1:Apply formula: |z| = √(a² + b²)
Step 2:|3 + 4i| = √(9 + 16) = √25 = 5

|3 + 4i| = 5

Common Mistakes & Tips

  • !Forgetting that i² = -1, which turns the bd term negative in multiplication
  • !Treating i as a variable rather than a specific constant (√-1)
  • !Confusing modulus with just the real or imaginary part
  • !Errors in division: must multiply by the conjugate of the denominator, not the numerator

Related Concepts

Complex Conjugate

Find the conjugate of a complex number and its properties

Complex Root

Find nth roots of complex numbers

Quadratic Formula

Complex numbers arise as roots when the discriminant is negative

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Complex Conjugate Calculator

Complex Root Calculator

Frequently Asked Questions

Why do we need complex numbers?

Complex numbers ensure every polynomial equation has a solution (Fundamental Theorem of Algebra). They are also essential in physics and engineering, particularly in electrical engineering, quantum mechanics, and signal processing.

What is the complex plane?

The complex plane (Argand diagram) represents complex numbers as points: the x-axis is the real part, the y-axis is the imaginary part. It allows geometric interpretation of complex arithmetic.

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