Complex Number Calculator

Background.

A complex number is an ordered pair of real numbers (a, b) written in the form a + bi, where i is the imaginary unit satisfying i² = −1. The complex number calculator accepts the real part a and the imaginary part b as inputs and returns the modulus, argument, and complex conjugate. These three quantities are the foundation of all complex arithmetic, appearing in electrical engineering, quantum mechanics, control theory, signal processing, and fluid dynamics. The modulus measures the distance from the origin in the complex plane, the argument measures the angle with the positive real axis, and the conjugate reflects the number across the real axis. These quantities are mutually dependent and together specify a unique complex number.

Engineering students and professionals are the primary audience for this calculator. In electrical engineering, AC circuit analysis represents impedance as a complex number Z = R + jX, where the modulus |Z| is the total impedance magnitude and the argument is the phase angle between voltage and current. In control theory, the poles of a transfer function are complex numbers whose modulus determines oscillation frequency and whose argument determines damping ratio. Signal processing represents sinusoids as complex exponentials via Euler's formula, and the Fourier transform maps time-domain signals into complex frequency-domain coefficients. In every case, the modulus and argument are the physical quantities of interest.

The geometric interpretation of complex numbers was developed by Caspar Wessel in 1799 and independently by Jean-Robert Argand in 1806, though it is now called the Argand diagram or complex plane. The horizontal axis represents the real part and the vertical axis represents the imaginary part. This representation transforms algebraic operations into geometric ones: addition becomes vector addition, multiplication becomes scaling and rotation, and conjugation becomes reflection. The modulus is the Euclidean norm in this plane, and the argument is the polar angle. These geometric insights explain why complex numbers are indispensable in any field that combines magnitude and phase information.

Complex numbers also unify many results in real analysis. The Fundamental Theorem of Algebra, first proved by Carl Friedrich Gauss in 1799, states that every non-constant polynomial with complex coefficients has at least one complex root. This means the complex numbers are algebraically closed, a property the real numbers lack. Without complex numbers, the characteristic equations of linear differential equations would sometimes have no solutions, and the Fourier transform would be restricted to even functions. The calculator provides the basic geometric parameters that make this algebraic closure computationally accessible and helps students bridge the gap between abstract algebra and geometric intuition.

In applied mathematics, the polar form z = r(cos θ + i sin θ) simplifies multiplication and division. To multiply two complex numbers, multiply their moduli and add their arguments. To divide, divide their moduli and subtract their arguments. This is the foundation of phasor arithmetic in electrical engineering and of the convolution theorem in signal processing. The calculator converts rectangular coordinates (a, b) into polar coordinates (r, θ), enabling users to perform these operations without manual trigonometric lookup or coordinate transformation tables.

What is complex number calculator?

A complex number is a number of the form z = 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. The set of all complex numbers is denoted by ℂ and forms a two-dimensional vector space over the real numbers. Every complex number can be represented as a point (a, b) in the complex plane, also called the Argand diagram. The horizontal coordinate is the real part, and the vertical coordinate is the imaginary part. The modulus or absolute value of z = a + bi is |z| = √(a² + b²), which is the Euclidean distance from the origin to the point (a, b). The argument or phase of z is the angle θ that the vector from the origin to (a, b) makes with the positive real axis, measured in radians and restricted to the principal range (−π, π]. The complex conjugate of z is z̄ = a − bi, obtained by reflecting z across the real axis. These three operations—modulus, argument, and conjugation—are the most frequently computed quantities in elementary complex analysis and form the basis for all advanced complex arithmetic.

How to use this calculator.

1. 01Enter the real part a of the complex number.
2. 02Enter the imaginary part b.
3. 03The calculator computes the modulus r = √(a² + b²).
4. 04The calculator computes the argument θ = atan2(b, a) in radians and degrees.
5. 05The calculator computes the complex conjugate a − bi.
6. 06Review the modulus to determine the distance from the origin.
7. 07Review the argument to determine the polar angle for phasor or rotation applications.

The formula.

The modulus of a complex number z = a + bi is derived from the Pythagorean theorem. In the complex plane, the point (a, b) forms a right triangle with legs of length |a| and |b| and hypotenuse of length |z|. Therefore |z|² = a² + b², and taking the non-negative square root yields |z| = √(a² + b²). This definition generalizes the absolute value on the real numbers: when b = 0, the formula reduces to |a|, which is the standard real absolute value. The modulus satisfies the same fundamental properties as the real absolute value: non-negativity, multiplicativity (|zw| = |z||w|), and the triangle inequality (|z + w| ≤ |z| + |w|). The argument is defined as the unique angle θ in the interval (−π, π] such that a = r cos θ and b = r sin θ, where r = |z|. This definition requires the two-argument arctangent function, atan2(b, a), which correctly determines the quadrant of the angle. When a > 0, θ = arctan(b/a). When a < 0 and b ≥ 0, θ = arctan(b/a) + π. When a < 0 and b < 0, θ = arctan(b/a) − π. When a = 0 and b > 0, θ = π/2. When a = 0 and b < 0, θ = −π/2. When a = b = 0, the argument is undefined, but the calculator returns 0 by convention to avoid exceptional cases. The complex conjugate z̄ = a − bi is the reflection of z across the real axis. It satisfies z·z̄ = a² + b² = |z|², which is a real number. This identity is the basis for rationalizing denominators: to divide by z, multiply numerator and denominator by z̄ to obtain a real denominator |z|². The conjugate also appears in the polar form: if z = re^(iθ), then z̄ = re^(−iθ). The real and imaginary parts can be recovered from z and z̄ via a = (z + z̄)/2 and b = (z − z̄)/(2i). Dimensional analysis confirms that the modulus carries the same units as the real and imaginary parts. If a and b are measured in ohms, then |z| is in ohms, and the argument is dimensionless, measured in radians. This consistency is essential in electrical engineering, where impedance, admittance, and transfer functions are all complex quantities with physical units. The calculator preserves this dimensional structure by computing only dimensionless ratios and magnitudes from the inputs.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST Digital Library of Mathematical Functions (2024). "Elementary Algebra." §1.9. https://dlmf.nist.gov/1.9
2. [2]Ahlfors, L.V. (1979). Complex Analysis, 3rd ed. McGraw-Hill. ISBN 978-0070006577.
3. [3]Needham, T. (1997). Visual Complex Analysis. Oxford University Press. ISBN 978-0198534464.
4. [4]Oppenheim, A.V. and Schafer, R.W. (2009). Discrete-Time Signal Processing, 3rd ed. Pearson. ISBN 978-0131988422.
5. [5]Hayt, W.H., Buck, J.A., and Durbin, S.M. (2011). Engineering Electromagnetics, 8th ed. McGraw-Hill. ISBN 978-0073380664.