Complex Number Calculator | Rectangular ⇄ Polar (Phasor) Converter

Calculate and simplify expressions of real and imaginary numbers easily with our free online complex number calculator.

This real and imaginary number 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.

Parallel Sum

Calculate the parallel sum (also known as the reduced sum) using the parallel operator, ∥, as in: (1+2i) ∥ (3+4i) The parallel operator is frequently used in electrical engineering in conjunction with A/C circuits, as well as in other branches of science and engineering. Note that the binary parallel operator has higher precedence than addition and subtraction, but lower than multiplication, division, and exponentiation.

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 our 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 complex numbers easy to do. This calculator also performs calculations with algebraic and non-algebraic functions.

Basic Operations with Complex Numbers

This calculator for real and imaginary numbers uses the following formulas to perform the four basic operations of addition, subtraction, multiplication and division of complex numbers, as well as parallel sum.

Complex Number Addition & Subtraction Calculator

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

Complex Number Multiplication Calculator

To multiply complex numbers a+bi and c+di, the complex number multiplication calculator uses the usual laws of algebra and uses the fact that i2 = -1.

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

Complex Number Division Calculator

The complex number division calculator uses the result of the following algebraic simplification 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 noting that i*i = -1.

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

Complex Number Parallel Sum Calculator

The parallel sum of a+bi and c+di is calculated as:

(a+bi) ∥ (c+di) = (a+bi)(c+di)/[(a+bi)+(c+di)]

Advanced Calculations with Complex Numbers

This calculator for real and imaginary numbers uses the following formulas to perform advanced calculations involving exponents, logarithms, trigonometric, hyperbolic, and other non-algebraic functions of complex numbers.

Complex Number Exponentiation Calculator

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, the calculator uses the following formula to perform exponentiation.

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

Complex Number Logarithm Calculator

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. Employing the result of the following, the calculator computes the logarithm of a complex number:

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.

Complex Number Trigonometric Functions Calculator

The calculator evaluates trigonometric functions sin(a+bi) and cos(a+bi) by using the following formulas which contain hyperbolic functions sinh() and cosh():

sin(a+bi) = sin(a)*cosh(b) + cos(a)*sinh(b)i
cos(a+bi) = cos(a)*cosh(b) + sin(a)*sinh(b)i

Since the other trigonometric functions—tangent, cotangent, secant, and cosecant— are defined in terms of the sine and cosine functions, the calculator employs the following to evaluate these functions:

tan(a+bi) = sin(a+bi)/cos(a+bi)
cot(a+bi) = cos(a+bi)/sin(a+bi)
sec(a+bi) = 1/cos(a+bi)
csc(a+bi) = 1/sin(a+bi)

Complex Number Inverse Trigonometric Functions Calculator

The inverse trigonometric functions—arc sine, arc cosine, arc tangent, arc cotangent, arc secant, and arc cosecant— are denoted by sin⁻¹() or arcsin(), abbreviated as asin(), etc. The calculator finds arcsin(), arccos(), and arctan() as follows:

sin⁻¹(a+bi) = -i*ln[(-b+ai) + √(1−(a+bi)2)]
cos⁻¹(a+bi) = -i*ln[(a+bi) + √((a+bi)2-1)]
tan⁻¹(a+bi) = -i/2*ln[((i-(a+bi))/(i-(a+bi))]

For the other inverse trigonometric functions, the calculator uses the following formulas:

cot⁻¹(a+bi) = tan⁻¹(1/(a+bi))
sec⁻¹(a+bi) = cos⁻¹(1/(a+bi))
csc⁻¹(a+bi) = sin⁻¹(1/(a+bi))

Complex Number Hyperbolic Functions Calculator

To find the hyperbolic functions, the calculator uses the following formulas:

sinh(a+bi) = -i*sin(-b+ai)
cosh(a+bi) = cos(-b+ai)
tanh(a+bi) = sinh(a+bi)/cosh(a+bi)
coth(a+bi) = 1/tanh(a+bi)
sech(a+bi) = 1/cosh(a+bi)
csch(a+bi) = 1/sinh(a+bi)

Complex Number Inverse Hyperbolic Functions Calculator

To find the Inverse hyperbolic functions, the calculator uses the following formulas:

sinh⁻¹(a+bi) = ln[a+bi + √((a+bi)^2 + 1)]
cosh⁻¹(a+bi) = ln[a+bi + √((a+bi)^2 - 1)]
tanh⁻¹(a+bi) = 1/2 * ln[(1 + (a+bi)) / (1 - (a+bi))]
coth⁻¹(a+bi) = tanh(1/(a+bi))
sech⁻¹(a+bi) = cosh(1/(a+bi))
csch⁻¹(a+bi) = sinh(1/(a+bi))

Note that these inverse hyperbolic functions are also denoted by asinh(), acosh(), atanh(), acoth(), asech(), and acsch(), respectively.