Since Cartesian form and polar form are equivalent ways to write the same complex number $z$, there are rules for how to convert between these two forms of representing complex numbers.

If you draw a complex number $z$ in the complex plane, you can use the Pythagorean Theorem and trigonometry to find the norm and the argument of $z$.

In the complex plane, a complex number $z=a+bi$ will form a right triangle with legs of lengths $a$ and $b$. In this geometric picture, the norm of $z$—which is defined as the distance from $z$ to the origin—is the length of the hypotenuse in the triangle. That means you can use the Pythagorean theorem to find the norm of complex numbers.

You can find the argument $𝜃$ by using either sine, cosine or tangent:

Every expression above yields two values for the argument $𝜃$. You find the correct argument by using two of the formulas and choosing the repeated value. It is also possible to look at the arguments of $a$ and $b$, or plot the number and choose the angle which lies in the correct quadrant.

The right triangle in the complex plane formed by a complex number $z$ can also be used to change from polar form $z=\left(r,𝜃\right)$ to Cartesian form $z=a+bi$.

This last form for complex numbers connects complex numbers with trigonometric functions. This is really important when you work with the exponential form of complex numbers.