What are the different forms of complex numbers and how do you convert between them?

Complex numbers have three primary forms: the general form, z=a+ib; the polar form, z=r(cosθ+isinθ); and the exponential form, z=rexp(iθ). To convert from the general form to either form you need to find r and θ: r is known as the modulus of z, by referring to an Argand diagram the modulus of z is the length of the line z=a+ib, so to find the modulus you use Pythagoras. θ is called the argument of z and is found by looking at the trigonometry of the line; the two components of z are the opposite and adjacent so you can use tanθ=b/a and rearrange for θ. To work in reverse it is best to use the polar form of the complex number as you simply set a=rcosθ and b=rsinθ.