The complex number system is an extension of  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part system with imaginary unit i .

A complex number is of the form x + iy ( or x + yi ), where x and y are real numbers. x is called the  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part and y is called the  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part of the complex number.

The numbers of the form α+iβ, β≠0 are called  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part.

If we consider x axis as real axis and y axis as imaginary axis to represent a complex number, then the xy -plane is called  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part plane.

The  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part of the complex number x + iy is defined as the complex number x − iy.

A  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is always a constant.

The fixed point is the  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part and the constant distant is the  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part of the circle.

The  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part of a complex number P is z = r (cosθ + isinθ ).

The angleθ is called the  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part of the complex number z denoted by θ=arg(z) .

The  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part property For any two complex numbers z1 and z2 is z1+z2=z1+z2.

The  argumentassociativecentrecirclecommutativecomplexconjugateimaginary numbersimaginary partpolar formradiusreal numberreal part property For any three complex numbers z1,z2,z3 is (z1 z2) z3 = z1 (z2 z3).