What is a complex number?

A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = -1. Complex numbers extend the real number line into a two-dimensional plane.

How do you multiply two complex numbers?

Multiply using the distributive property and simplify i² to -1: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. This calculator performs that expansion automatically.

How do you divide complex numbers?

Division multiplies both the numerator and denominator by the denominator's conjugate to eliminate i from the denominator, then simplifies. The calculator handles this automatically and shows the final simplified result.

What is the modulus of a complex number?

The modulus, written |z|, is the distance from the origin to the point (a, b) in the complex plane, calculated as √(a² + b²). It's the complex-number equivalent of absolute value.

What is the argument of a complex number?

The argument is the angle, measured in degrees here, between the positive real axis and the line from the origin to the point (a, b) in the complex plane. It's calculated using the arctangent of b/a, adjusted for the correct quadrant.

What is a complex conjugate?

The conjugate of a + bi is a - bi — the same real part with the sign of the imaginary part flipped. Conjugates are used to rationalize denominators during division and appear as pairs whenever a polynomial with real coefficients has complex roots.

What happens if I divide by 0 + 0i?

Division by zero is undefined for complex numbers just as it is for real numbers. The calculator will show an error if the second number is 0 + 0i.

Where are complex numbers used in real life?

Complex numbers are essential in electrical engineering (AC circuit analysis), signal processing (Fourier transforms), quantum mechanics, and control theory, and they appear as the roots of quadratic equations whenever the discriminant is negative.

Complex Number Calculator — Arithmetic, Modulus, Argument, and Conjugate

Our Complex Number Calculator handles the four basic operations — addition, subtraction, multiplication, and division — on any two complex numbers in the form a + bi. Every result also includes the modulus (distance from the origin in the complex plane), the argument (angle in degrees), and the conjugate, so you get a complete picture of the result, not just the raw arithmetic.

Quick Answer

A complex number has the form a + bi, where i² = -1. This calculator adds, subtracts, multiplies, or divides two complex numbers and also returns the result's modulus, argument, and conjugate — everything needed to fully describe the answer geometrically and algebraically.

First number

Second number

Enter two complex numbers, then click Calculate.

How to Use the Complex Number Calculator — Add, Subtract, Multiply, Divide

1
Enter the real and imaginary parts of your first complex number (a + bi).
2
Choose an operation: addition, subtraction, multiplication, or division.
3
Enter the real and imaginary parts of your second complex number (c + di).
4
Click 'Calculate' to see the result along with its modulus, argument, and conjugate.

Why Use Complex Number Calculator — Add, Subtract, Multiply, Divide?

Complex number arithmetic follows different rules than real numbers — multiplication and division in particular require distributing across i² = -1 and rationalizing the denominator, which is easy to get wrong by hand. Our calculator performs the exact arithmetic instantly and also surfaces the modulus and argument, the two values you need to convert a result into polar form or to reason about it geometrically in the complex plane — information that's tedious to compute manually but essential in electrical engineering, signal processing, and advanced algebra.

Frequently Asked Questions

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