Complex number calculator


Rectangular form:
z = 3

Angle notation (phasor):
z = 3 ∠ 0°

Polar form:
z = 3 × (cos 0° + i sin 0°)

Exponential form:
z = 3 × ei 0

Polar coordinates:
r = |z| = 3 ... magnitude (modulus, absolute value)
θ = arg z = 0 rad = 0° = 0π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 3
Real part: x = Re z = 3
Imaginary part: y = Im z = 0

This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i2 = −1 or j2 = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

Basic operations with complex numbers

We hope that work with the complex number is quite easy because you can work with imaginary unit i as a variable. And use definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.



12 + 6-5i = 18-5i





To multiply two complex numbers, use distributive law, avoid binomials, and apply i2 = -1.



The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the denominator's complex conjugate. This avoids imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it real), multiply with conjugate c-di:


6i / (4+3i) = 0.72+0.96i

Absolute value or modulus

The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142136
|6i| = 6
abs(2+5i) = 5.3851648

Square root


sqrt(9i) = 2.1213203+2.1213203i
sqrt(10-6i) = 3.2910412-0.9115656i
pow(-32,1/5)/5 = -0.4

Square, power, complex exponentiation

Our calculator can power any complex number to any integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:
i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.03994-9022199.58262i
i^i = 0.2078795764


Square Root of a value or expression.
the sine of a value or expression. Autodetect radians/degrees.
the cosine of a value or expression. Autodetect radians/degrees.
tangent of a value or expression. Autodetect radians/degrees.
Power one complex number to another integer/real/complex number
The natural logarithm of a value or expression
The base-10 logarithm of a value or expression
abs or |1+i|
The absolute value of a value or expression
Phase (angle) of a complex number


cube root: cuberoot(1-27i)
roots of Complex Numbers: pow(1+i,1/7)
phase, complex number angle: phase(1+i)
cis form complex numbers: 5*cis(45°)
The polar form of complex numbers: 10L60
complex conjugate calculator: conj(4+5i)
equation with complex numbers: (z+i/2 )/(1-i) = 4z+5i
system of equations with imaginary numbers: x-y = 4+6i; 3ix+7y=x+iy
De Moivre's theorem - equation: z^4=1
multiplication of three complex numbers: (1+3i)(3+4i)(−5+3i)
Find the product of 3-4i and its conjugate.: (3-4i)*conj(3-4i)
operations with complex numbers: (3-i)^3

Complex numbers in word problems:

  • Mappings of complex numbers
    complex_numbers Find the images of the following points under mappings: z=3-2j w=2zj+j-1
  • The modulus
    abs_value Find the modulus of the complex number 2 + 5i
  • Im>0?
    ReIm Is -10i a positive number?
  • Linear imaginary equation
    cplx_function Given that 2(z+i)=i(z+i) "this is z star" Find the value of the complex number z.
  • Modulus and argument
    complex_numbers Find the mod z and argument z if z=i
  • Stadium
    sphere_segment A domed stadium is in the shape of spherical segment with a base radius of 150 m. The dome must contain a volume of 3500000 m³. Determine the height of the dome at its centre to the nearest tenth of a meter.
  • Log
    exp_log Calculate value of expression log |3 +7i +5i2| .
  • Goniometric form
    complex_fnc Determine the goniometric form of a complex number z = √ 110 +4 i.
  • Complex number coordinates
    complex_numbers Which coordinates show the location of -2+3i
  • Subtracting complex in polar
    imac Given w =√2(cosine (p/4) + i sine (pi/4) ) and z = 2 (cosine (pi/2) + i sine (pi/2) ), what is w - z expressed in polar form?
  • Imaginary numbers
    Complex_conjugate_picture.svg Find two imaginary numbers whose sum is a real number. How are the two imaginary numbers related? What is its sum?
  • Let z1=x1+y1i
    cplx_function Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)
  • Bearing
    compass A plane flew 50 km on a bearing 63°20' and the flew on a bearing 153°20' for 140km. Find the distance between the starting point and the ending point.

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