Polar Form of Complex Numbers

If you use polar coordinates on z=x+iy, you get z=rcos(theta)+isin(theta), where
r=Sqrt[x^2+y^2] and theta=Arctan(y/x), theta in (-Pi, Pi].
Using Euler's equation this is

z=r*Exp[i Theta]

Note that r=Sqrt[x^2+y^2] is the modulus of z, |z|. Theta is sometimes called the argument of z, Arg(z)


So, the polar form of a complex number z is z=|z| Exp[iTheta] or
z=|z| Exp[iArg[z]]

In polar form, 1+iSqrt[3] is 2Exp[i Pi/3]. I am sure you could figure this out for yourself. Anyway, Mathematica can help you.

In[27]:=

  z=1+Sqrt[3]

Out[27]=

  1 + Sqrt[3]

In[28]:=

  Abs[z]

Out[28]=

  1 + Sqrt[3]

In[29]:=

  Arg[z]

Out[29]=

  0

Feel free to try this with some of your own values of z.

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