Complex Number Calculator | Rectangular ⇄ Polar (Phasor) Converter

Easily calculate real and imaginary numbers with our free online complex number calculator.

This calculator can handle complex numbers in any form, and converts between rectangular a+bi and polar r∠(θ) forms of complex numbers.

By selecting the Convert results to all forms checkbox, the rectangular/polar converter performs conversions and displays the result of a calculation in all forms:

  • a+bi (rectangular)
  • r∠(φ) (polar, also known as phasor in electrical engineering)
  • reiφ (exponential)
  • rcis(φ) (shorthand for cos(φ)+i*sin(φ), Euler's formula)

where r is the magnitude (or modulus or length) and φ is the argument (or phasor or angle) of the complex number.

As a powerful scientific complex number calculator, it can handle mathematical expressions ranging from simple forms like (1+2i) - (3-4i) to arbitrarily complicated forms such as sin(1+2i)/ln(3+4i) + atan(1+3i) - 4∠(1.8)

By selecting the Show work checkbox, the real and imaginary number calculator provides step-by-step calculations.

Powerful Features for Calculation on Complex Numbers

This real and imaginary number calculator provides the following functionalities.

Basic Operations

Perform basic operations of addition, subtraction, multiplication, and division on complex numbers.

Powers and Roots

Calculate powers and principal roots of complex numbers.

Advanced Mathematical Operations

Calculate expressions involving trigonometric, hyperbolic, exponential, and logarithmic functions that have real, imaginary, or complex numbers as arguments.

Special Function Evaluation

Evaluate functions like Γ(), ψ(), and ζ(), corresponding to the Gamma, psi, and zeta functions, respectively.

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Instructions for complex number calculator

Complex Numbers

What are complex numbers? Complex numbers are ordered pairs of real numbers (a,b), where a is called the real part and b is called the imaginary part. Instead of denoting a complex number with an ordered pair, it is customary to combine the pairs with a plus sign and denote the resulting complex number as a+bi, where i has the property that i2 = -1. In other words, i = √-1. Since there is no real number whose square is negative, i is called the imaginary unit. Imaginary numbers are complex numbers whose real parts are zero, such as bi.

Note: Both the real part and the imaginary part are real numbers. The imaginary part is so called because it is the coefficient of the imaginary unit i.

All the laws and rules, including the commutative, associative and distributive laws, which we use in conjunction with the real number system, are also applicable to the complex number system. This makes the basic binary operations of addition, subtraction, multiplication and division of complex numbers easy to do. This calculator for real and imaginary numbers uses the following rules to perform the four basic operations, as well as advanced calculations involving exponents, logarithms, trigonometric, hyperbolic, and other non-algebraic functions.

Basic Operations with Complex Numbers

Calculate Addition & Subtraction of Complex Numbers

To add or subtract complex numbers a+bi and c+di, simply add or subtract the corresponding real and imaginary parts.

(a+bi)+(c+di) = (a+c)+(b+d)i
(a+bi)-(c+di) = (a+c)-(b+d)i

Calculate Multiplication of Complex Numbers

To multiply complex numbers a+bi and c+di, use the usual laws of algebra keeping in mind that i2 = -1.

(a+bi)*(c+di) = (ac-bd)+(ad+bc)i

Calculate Division of Complex Numbers

To divide complex numbers a+bi and c+di, first multiply both the numerator and denominator by (c-di), and then use the usual laws of algebra keeping in mind that i*i = -1.

(a+bi) / (c+di) = (a+bi)(c-di) / (c2 + d2)
= [(ac+bd) + (-ad+bc)i] / (c2+d2)

Advanced Calculations with Complex Numbers

Calculate Exponentiation of Complex Numbers

The value of exp(a+bi) or equivalently ea+bi is calculated by using the celebrated Euler's formula, eθi = cosθ+i*sinθ. Replacing θ by b we get

ea+bi = ea*ebi
  = ea[cos(b)+i*sin(b)]

Calculate Logarithm of Complex Numbers

To find the natural logarithm, ln(a+bi), first express the complex number in the exponential form, ln(r*eiφ) (refer to conversion to polar forms), and use the laws of logarithm:

ln(a+bi) = ln(r*eiφ)
  = ln(r) + ln(eiφ)
  = ln(r) + φi

The above is called the principal natural logarithm of a+bi. Since the value of φ (argument or phasor) of a complex number is not unique (it's multivalued, differing by integer multiples of 2π), its logarithm is also not unique.