complex calculator online

What is a complex calculator online?

A complex calculator online is a fast way to perform arithmetic with numbers that contain both a real part and an imaginary part. A complex number is usually written as a + bi, where a is real, b is real, and i is the imaginary unit with the property i² = -1. These numbers are essential in electrical engineering, control systems, signal processing, quantum mechanics, and advanced algebra.

Instead of doing long manual steps every time, this tool gives you instant results for common operations such as complex addition, complex multiplication, complex division, conjugates, modulus, powers, and polar conversion.

How to use this complex number calculator

• Enter the real and imaginary parts for z₁.
• Enter the real and imaginary parts for z₂.
• Select an operation from the dropdown list.
• If you choose power, set the exponent n.
• Click Calculate to get your result instantly.

Results are shown in rectangular form (x + yi) where appropriate, and polar results include both radians and degrees.

Core formulas used

1) Addition and subtraction

If z₁ = a + bi and z₂ = c + di:

• z₁ + z₂ = (a + c) + (b + d)i
• z₁ - z₂ = (a - c) + (b - d)i

2) Multiplication

Multiply like binomials and use i² = -1:

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

3) Division

Multiply numerator and denominator by the conjugate of the denominator:

• (a + bi)/(c + di) = ((ac + bd) + (bc - ad)i) / (c² + d²)

Division is undefined when c² + d² = 0 (that is, when z₂ = 0 + 0i).

4) Conjugate and modulus

• Conjugate: conj(a + bi) = a - bi
• Modulus: |a + bi| = √(a² + b²)

5) Polar form and argument

Any nonzero complex number can be written as:

• z = r(cos θ + i sin θ), where r = |z| and θ = arg(z)

This imaginary number calculator uses atan2(b, a) for a quadrant-correct angle.

Quick examples

Example A: Complex multiplication

Let z₁ = 3 + 4i and z₂ = 1 - 2i. Then: (3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i² = 3 - 2i + 8 = 11 - 2i.

Example B: Modulus

For z = 3 + 4i: |z| = √(3² + 4²) = √25 = 5.

Example C: Conjugate

If z = -7 + 2i, then conj(z) = -7 - 2i.

Why this tool is useful

• Students: verify homework and exam practice quickly.
• Engineers: work with impedance, phasors, and AC circuit analysis.
• Developers: test algorithm outputs for FFT, DSP, and control models.
• Self-learners: understand the geometry of complex numbers with immediate feedback.

Common input tips

• Use decimals freely (e.g., 2.75, -0.5).
• Leave a blank field? It is treated as 0 internally.
• For power operations, fractional exponents are allowed.
• Use polar conversion when you need magnitude/phase form.

FAQ

Is this a scientific complex number calculator?

It supports the most commonly needed operations for algebra, engineering, and STEM coursework: addition, subtraction, multiplication, division, modulus, conjugate, polar form, and powers.

Can I use this as a polar form calculator?

Yes. Choose “Polar form of z₁” or “Polar form of z₂” to get magnitude and angle in both radians and degrees.

Does it support very small values?

Yes. Tiny floating-point artifacts are cleaned for readability, while preserving numerical accuracy for normal use.