Representation of complex numbers in polar form

Given a complex number $z$ we have two ways of represent it. It is possible to have:

• Binomic form: given a complex number its binomic form is the most usual one, $z=a+bi$, and it is also possible to identify it with a pair of numbers given in Cartesian coordinates. This pair is $(a,b)$ and they allow us to draw $z$ in the complex plane. The procedure is as follows:
  • Draw the first component of the pair on the axis OX (eix real) (real axis). This is the real part of $z$.
  • Draw the second component of the pair on the axis OY (imaginary axis). This is the imaginary part of $z$.
  • Mark the point where the straight lines parallel to the axes OX and OY, and going through $a$ and $b$, intersect.

  • Join the origin of the complex plane with this point. This is the complex number $z$.

• Polar form: The complex number written in polar form is $z=|z{|}_{\alpha }$ and it is identified by the pair $(|z|,\alpha )$ which are its polar coordinates. They will also allow us to draw the number in the complex plane. The procedure is:

  • Draw an angle $\alpha$ that starts from the origin of the complex plane.
  • Take the module of $z$, $|z|$, draw it. This length is the one that determines the number $z$ in the complex plane.