Convert the general complex number z=x+iy to modulus-argument form.

Modulus-argument form implies that we should express z in terms of its straight line distance from the origin and the angle this straight line would make with the x axis. This is expressed as z = r e^i*theta where r is the modulus and theta is the argument. We thus wish to express r and theta in terms of x and y.

This problem is best solved visually by considering an Argand diagram for the general complex number, z. In this way we can see that z is represented as a point x units along the x-axis and y units along the y-axis, forming a right angled triangle with the vertical and horizontal.

As with any right angled triangle, using Pythagoras, we can see that the length of the hypotenuse (I.e. the distance from the origin, the modulus) is given by r = sqrt(x²+y²). Similarly, using basic trigonometry we can also see that the angle between this line and the x axis (theta, the argument) is given by theta = arctan(y/x).

This means that in modulus-argument form:

z = sqrt(x²+y²) exp(i*arctan(y/x))