What Is a Complex Number Calculator?
A complex number calculator performs arithmetic with values that contain a real part and an imaginary part. A complex number is commonly written as a + bi, where a is real, b is the coefficient of the imaginary part, and i satisfies i² = −1. This calculator returns each answer in rectangular and polar form, explains the arithmetic, and plots the inputs and result on an Argand diagram.
Complex numbers appear in electrical engineering, control systems, signal processing, quantum mechanics, fluid dynamics, and algebra. Rectangular coordinates are convenient when adding components. Polar notation is especially useful when multiplication, division, magnitude, or phase angle matters.
Rectangular and Polar Forms of Complex Numbers
Rectangular form identifies a point with its real coordinate a and imaginary coordinate b. Polar form describes that same point with magnitude r and argument θ. The magnitude is the distance from the origin, so r = √(a² + b²). The principal argument is found with θ = atan2(b, a), which preserves the correct quadrant.
How Rectangular and Polar Forms Connect
The same point can be described by its coordinates or by its distance and angle from the origin.
To convert back, use a = r cos θ and b = r sin θ. A polar magnitude cannot be negative because it represents distance. Angles outside one complete turn are valid inputs, but the calculator reports a normalized principal argument between −180° and 180°, using 180° on the negative real axis.
How to Add and Subtract Complex Numbers
Addition and subtraction operate independently on matching components. For z₁ = a + bi and z₂ = c + di, addition gives (a + c) + (b + d)i. Subtraction gives (a − c) + (b − d)i. The order matters in subtraction, which is why the result steps also show the reverse difference.
For example, (3 + 4i) + (1 − 2i) equals 4 + 2i. Subtracting the second number from the first gives 2 + 6i. If an input begins in polar form, convert it to rectangular coordinates before combining its real and imaginary components.
How to Multiply and Divide Complex Numbers
Multiplication expands two binomials and replaces i² with −1. The rectangular formula is (ac − bd) + (ad + bc)i. In polar form, the same operation has a useful geometric meaning: multiply the magnitudes and add the arguments. Multiplying 5∠30° by 2∠−45° therefore produces magnitude 10 and angle −15°.
Division can be understood by multiplying the numerator and denominator by the denominator's conjugate. Its familiar algebraic formula divides by c² + d². The implementation uses a scaled equivalent algorithm so very large or very small finite components are less likely to overflow or underflow. Division is undefined only when both components of the denominator are zero.
How to Convert Rectangular Form to Polar Form
Enter the real and imaginary components, choose Convert Form, and submit. The calculator uses a stable hypotenuse calculation for magnitude and a quadrant-aware arctangent for the angle. For 3 + 4i, the magnitude is 5 and the argument is approximately 53.130102°. The polar result is 5∠53.130102°.
For polar-to-rectangular conversion, enter r and θ and choose whether θ is measured in degrees or radians. The calculator evaluates cosine and sine, then removes tiny floating-point artifacts near expected zeros. The full-precision numeric values remain available internally; rounding affects display only.
Complex Number Formulas and Examples
| Operation | Rectangular formula | Polar insight |
|---|---|---|
| Addition | (a+c) + (b+d)i | Convert to rectangular first |
| Subtraction | (a−c) + (b−d)i | Convert to rectangular first |
| Multiplication | (ac−bd) + (ad+bc)i | Multiply magnitudes; add angles |
| Division | ((ac+bd)+(bc−ad)i)/(c²+d²) | Divide magnitudes; subtract angles |
| Conjugate | a−bi | Same magnitude; opposite argument |
The conjugate reflects a point across the real axis. A number multiplied by its conjugate produces the nonnegative real value a² + b², which equals the squared magnitude. This identity is central to simplifying complex division.
Complex Number Calculator FAQ
How do I convert a complex number to polar form?
Calculate r = √(a² + b²), then find θ = atan2(b, a). The two-argument arctangent is important because it identifies the correct quadrant instead of losing sign information.
How do I convert polar form to rectangular form?
Use a = r cos θ and b = r sin θ, then write the result as a + bi. Confirm whether the supplied angle is in degrees or radians before evaluating the trigonometric functions.
How do I multiply complex numbers?
Expand the two rectangular expressions and use i² = −1, producing (ac − bd) + (ad + bc)i. Alternatively, multiply polar magnitudes and add polar arguments.
How do I divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator, then collect real and imaginary terms. The denominator cannot be 0 + 0i because division by zero is undefined.
What is the argument of a complex number?
The argument is the directed angle from the positive real axis to the number's vector in the complex plane. A principal argument chooses one standard representative, normally within a single turn.
Why is the argument of zero undefined?
Zero lies at the origin and has no direction from the origin to itself. Its magnitude is zero, but every angle would describe the same location, so no unique argument exists.
Related Calculators
Quadratic Formula Calculator
Solve real and complex quadratic roots.
Polynomial Roots Calculator
Find roots of higher-degree polynomials.
Vector Calculator
Calculate vector operations and angles.
Scientific Notation Converter
Convert very large and small values.
Angle Calculator
Work with common angle relationships.
Matrix Multiplier
Multiply matrices with clear steps.