Complex Number Calculator

A complex number calculator performs arithmetic and geometric operations on numbers of the form a + bi, where i is the imaginary unit (i² = −1).

Our calculator supports all standard operations:

• addition
• subtraction
• multiplication
• division
• modulus
• argument
• conjugate
• polar form conversion
• De Moivre powers
• nth roots
• complex exponential

Complex numbers can be expressed in rectangular form (a + bi) or polar form (r∠θ).

The modulus r = √(a²+b²) is the distance from the origin, and the argument θ = atan2(b,a) is the angle.

Polar form makes multiplication, division, and powers much simpler: multiply moduli and add arguments.

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = −1. The real part is a and the imaginary part is b.

Complex numbers arose because equations like x² + 1 = 0 have no real solution, so mathematicians extended the number system to include √−1.

According to Wolfram MathWorld, the complex numbers form an algebraically closed field, meaning every non-constant polynomial has a root among them.

Geometrically, a + bi is a point (a, b) in the complex plane, with the horizontal axis for real values and the vertical axis for imaginary values.

• To add or subtract complex numbers, combine the real and imaginary parts separately: (a+bi) ± (c+di) = (a±c) + (b±d)i.
• Multiplication uses the FOIL method and the identity i² = −1: (a+bi)(c+di) = (ac−bd) + (ad+bc)i. For example, (3+4i)(1−2i) = 11 − 2i.
• Division multiplies numerator and denominator by the conjugate of the denominator, giving (a+bi)/(c+di) = [(ac+bd) + (bc−ad)i]/(c²+d²).

Khan Academy notes that multiplying by the conjugate clears the imaginary part from the denominator, since (c+di)(c−di) = c²+d² is always real. These four operations preserve the a + bi structure.

The modulus of a complex number z = a + bi is its distance from the origin in the complex plane, given by |z| = √(a²+b²). For instance, |3+4i| = √(9+16) = √25 = 5.

The argument is the angle θ from the positive real axis, computed as θ = atan2(b, a), typically reported between −π and π.

As NIST's Digital Library of Mathematical Functions describes, the modulus and argument together give the polar coordinates of z. The modulus is always non-negative, and multiplying a complex number by its conjugate yields the modulus squared: z·z* = |z|², which is always a real number.

To convert a complex number from rectangular form a + bi to polar form r∠θ, compute the modulus r = √(a²+b²) and the argument θ = atan2(b, a).

To go back from polar to rectangular, use a = r·cos θ and b = r·sin θ.

For example, 1 + i has r = √2 ≈ 1.414 and θ = 45°, so its polar form is √2 ∠45°.

Encyclopaedia Britannica explains that polar representation makes multiplication and division intuitive, since you multiply or divide the moduli and add or subtract the arguments. Rectangular form is best for addition and subtraction, while polar form simplifies products, quotients, powers, and roots.

De Moivre's theorem states that for z = r(cos θ + i sin θ), the nth power is z^n = rⁿ(cos nθ + i sin nθ). Raise the modulus to the power n and multiply the argument by n.

For example, (1+i) has r = √2 and θ = 45°, so (1+i)² = (√2)²(cos 90° + i sin 90°) = 2(0 + i) = 2i, which matches direct expansion.

Wolfram MathWorld presents De Moivre's formula as a direct consequence of Euler's formula e^(iθ) = cos θ + i sin θ. This method avoids the tedious repeated multiplication needed to expand high powers in rectangular form.

Every non-zero complex number has exactly n distinct nth roots. For z = r∠θ, the roots are z_k = ⁿ√r ∠(θ/n + 360°k/n) for k = 0, 1, …, n−1. They lie equally spaced on a circle of radius ⁿ√r in the complex plane.

For example, the three cube roots of 8 are 2∠0°, 2∠120°, and 2∠240°.

The special case of the nth roots of unity — the solutions of zⁿ = 1 — are catalogued in the OEIS and form the vertices of a regular n-gon on the unit circle.

As NIST DLMF notes, this equal spacing follows directly from adding 2π/n to the angle for each successive root.

The complex conjugate of z = a + bi is z* = a − bi, obtained by reversing the sign of the imaginary part. Geometrically it reflects the number across the real axis.

The conjugate has two key properties:

• z + z* = 2a is always real
• z·z* = a² + b² = |z|² is always a non-negative real number

Khan Academy highlights that conjugates are the standard tool for dividing complex numbers, because multiplying the denominator by its conjugate removes the imaginary part.

Conjugates also appear in physics and signal processing, where quantities like power and probability depend on multiplying a complex value by its conjugate to obtain a real magnitude.

Complex numbers are indispensable across engineering and science:

• In electrical engineering, alternating-current circuits are analyzed with phasors, where impedance is a complex quantity combining resistance and reactance.
• In signal processing, the Fourier transform represents signals as sums of complex exponentials e^(iωt), separating them into frequency components.
• Control systems use the complex plane to assess stability by locating poles and zeros.
• Quantum mechanics is fundamentally built on complex-valued wave functions.

Encyclopaedia Britannica notes that Euler's identity e^(iπ) + 1 = 0 elegantly links the constants e, i, π, 1, and 0. Fluid dynamics and fractal geometry, such as the Mandelbrot set, also rely on complex arithmetic.

• The most frequent error is forgetting that i² = −1, which changes the sign of the product term during multiplication; writing i² = 1 gives a wrong result.
• Another mistake is applying √(ab) = √a·√b to negative numbers — this rule fails, since √−1·√−1 = −1, not √1 = 1.
• When finding the argument, many students use plain arctan(b/a) and land in the wrong quadrant; use atan2(b, a) instead.
• People also drop roots when solving zⁿ = w, forgetting there are exactly n of them.
• Finally, mixing degrees and radians in polar calculations produces incorrect angles.

Always confirm which unit your formula expects before computing sines and cosines.