Formulary

Math and Physics Calculators

Complex Numbers Calculator

A complex number has a real part and an imaginary part. The first is a real number, the second is an imaginary number. Imaginary numbers are represented as lying perpendicular to the number line of real numbers. The notation for a complex number is a + b i, where the imaginary unit i is equal to √ -1 .

Here are two calculators, the first for converting from Cartesian form to polar form and exponential form. The other for basic arithmetic in Cartesian form.

Number of decimal places on both calculators:

Conversion

Conversion of the forms of complex numbers, cartesian, to polar and exponentiation with →, the other was with ←. The angle φ is in rad, here you can convert angle units.

a = ρ * cos(φ) b = ρ * sin(φ)

Calculation

Calculate in cartesian form
( + i ) ( + i ) = + i

Basic arithmetic operations for complex numbers in Cartesian form, simply select an arithmetic symbol (+, -, *, /) and click Calculate. Result in polar form transfers the result to the upper calculator and returns the polar form.

Example of the conversion: -3-2i (enter -3 and -2) gives the results ρ=3.606 and φ=-2.554. The polar form is therefore 3.606 * (cos -2.554 + i sin -2.554) and the exponential form is 3.606 * e-2.554i. By pressing "Calculate value 1" or "Calculate value 2"" the Cartesian value of the first calculator becomes the first or second value of the second calculator.

Example of calculation in Cartesian form: (-5+8i) * (2-3i) = 14+31i. If you press "Convert result", this number is entered into the upper calculator and gives the values ρ=34.015 and φ=1.147.

The need for imaginary and, as a result, complex numbers arose from the desire to get results for all roots. Even roots of negative numbers have no solution in the real number space, so this had to be expanded accordingly. The first ideas for this came about in the middle of the 16th century, and they were finally introduced by Carl Friedrich Gauss in 1831 under their current name. In fact, imaginary numbers have a meaning beyond pure imagination; they provide meaningful results for many parts of physics.