Complex Number Calculator

What Is a Complex Number?

A complex number is a number built from two parts: a real part and an imaginary part. You’ll usually see it written as z = a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. At first, “imaginary” can sound like “not real,” but complex numbers are extremely practical. They let mathematicians and engineers represent rotation, oscillation, waves, and phase in a way that makes calculations cleaner and more consistent.

The key idea is that a complex number behaves like a point (or a vector) on a 2D plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. So the number 3 + 4i corresponds to the point (3, 4). Once you think of complex numbers this way, operations like magnitude and angle become as natural as distance and direction.

Why Complex Numbers Matter in Real Problems

Complex numbers show up anytime you’re dealing with circular motion, periodic signals, or systems with both size and direction. Electrical engineering uses complex numbers to model AC circuits because voltage and current can be out of phase. Signal processing uses them to represent frequency components. Control systems use them to describe stability and resonant behavior. Even in pure math, complex numbers unlock elegant solutions to polynomial equations and provide a powerful toolkit for calculus.

The reason they’re so useful is that complex multiplication naturally encodes rotation and scaling. That means you can represent “grow by a factor and rotate by an angle” with one multiplication, rather than separate geometry steps. Polar form makes that connection even clearer.

Rectangular Form: z = a + bi

Rectangular form (also called standard form or Cartesian form) is the most direct way to write a complex number. You can read the real part and imaginary part immediately: Re(z) = a and Im(z) = b. This form is perfect for addition and subtraction because you simply add or subtract the matching parts: real with real and imaginary with imaginary.

Addition and Subtraction

If z₁ = a + bi and z₂ = c + di, then: z₁ + z₂ = (a + c) + (b + d)i and z₁ − z₂ = (a − c) + (b − d)i. Visually, these operations are like adding or subtracting vectors on the complex plane.

Multiplication: Where the Magic Starts

Multiplication is where complex numbers begin to feel different from regular numbers. The product of (a + bi)(c + di) expands like any binomial: (a + bi)(c + di) = ac + adi + bci + bdi². The last term uses i² = −1, so bdi² becomes −bd. Group the real and imaginary parts and you get: (ac − bd) + (ad + bc)i.

This looks like algebra, but geometrically it represents scaling and rotation together. That’s why complex multiplication is foundational in engineering, graphics, and wave math.

Division and the Role of the Conjugate

Dividing by a complex number is easiest when you use the complex conjugate. The conjugate of a + bi is a − bi. It mirrors the point across the real axis. The special property is that multiplying a number by its conjugate removes the imaginary part: (a + bi)(a − bi) = a² + b², a real number.

To divide (a + bi) by (c + di), multiply top and bottom by (c − di): the denominator becomes c² + d², and the numerator expands into a clean real + imaginary result. This calculator does that automatically and will flag division-by-zero cases when c and d are both 0.

Magnitude and Argument: Turning z into Distance and Direction

Once a complex number is seen as a point (a, b), its magnitude is just distance from the origin: |z| = √(a² + b²). This is also called the modulus. Magnitude answers, “How big is this complex number?”

The argument (often written arg(z)) is the angle from the positive real axis to the vector (a, b). It is computed using atan2(b, a), which automatically chooses the correct quadrant. The calculator lets you view this angle in degrees or radians depending on your work.

Polar Form: r∠θ and r(cosθ + i sinθ)

Polar form rewrites the same complex number using magnitude and angle: z = r(cosθ + i sinθ), often abbreviated as r∠θ. Here r is the magnitude and θ is the argument. Converting between rectangular and polar is a normal part of complex arithmetic.

From rectangular to polar: r = √(a² + b²) and θ = atan2(b, a). From polar to rectangular: a = r cosθ and b = r sinθ. This calculator supports both directions, and it keeps your angle unit consistent.

Why Polar Form Makes Powers and Roots Easier

Polar form shines when you need powers or roots. De Moivre’s theorem says: (r∠θ)^n = r^n ∠ (nθ) for integer n. Instead of expanding (a + bi)^n by repeated multiplication, you can jump straight to the answer by raising r and multiplying the angle.

Integer Powers

For an integer n, the calculator computes z₁^n using polar math for stability and clarity, then converts back to rectangular form. Negative powers are handled by taking the reciprocal first, so z^(-n) = 1/(z^n) when z ≠ 0.

n-th Roots and Why There Are Many

Complex roots are different from real roots because angles wrap around. The n-th roots of z = r∠θ are: r^(1/n) ∠ ((θ + 2πk)/n) for k = 0, 1, 2, …, n−1. That means you usually get n distinct roots evenly spaced around a circle.

This calculator lists all roots in a table, showing each root in rectangular form and polar form. This is useful in algebra and engineering, especially when solving equations like z^n = w.

Choosing Degrees vs Radians

Degrees are common in everyday geometry and many applied contexts. Radians are the natural unit for calculus and many physics formulas. The calculator’s angle unit toggle controls both polar inputs and polar outputs, so you can stay consistent.

A good mental check: typical triangle angles look like 30°, 45°, 60°, 90°. Typical radian angles look like π/6, π/4, π/3, π/2. If you enter 60 while you’re in radians, you’re entering a huge angle, and conversions will not match your expectations.

Common Use Cases You Can Do Quickly

• AC circuits: represent impedance and phase shifts as complex values.
• Signal processing: represent frequency components and phase in Fourier methods.
• 2D rotations: rotate a point by multiplying by a unit complex number cis(θ).
• Quadratic and polynomial problems: handle negative discriminants and complex roots.
• Control systems: analyze poles and stability using complex plane locations.

Practical Checks for Clean Results

Complex arithmetic is accurate, but your interpretation depends on good inputs. A few fast checks prevent confusion:

• If you divide by z₂, make sure z₂ is not 0 + 0i.
• If you compute roots, keep n as a positive integer (n ≥ 2) to get a meaningful root list.
• If a result looks “rotated,” confirm whether you’re reading the angle in degrees or radians.
• If you need engineering-style output, increase precision. If you need readability, keep precision modest.

Limitations and Safe-Use Notes

This calculator is designed for education, planning, and general math work. Results use standard floating-point arithmetic, so extremely large magnitudes, very high powers, or very large n values may overflow or lose precision. If you’re using results for critical engineering decisions, validate the model, units, and assumptions and cross-check with trusted tools or domain-specific software.