D.2 The geometric representation
We can represent complex numbers in ways other than the one in the definition. If you think about it, each number z = a + bi can be seen as an ordered pair (a,b). These can be visualized as vectors on the Cartesian plane.
The absolute value |z| = 
of a complex number z = a + bi represents the length of the vector (a,b) from the origin, while conjugation z = a−bi corresponds to reflecting the point across the real axis.
This geometric view gives us a new algebraic way to represent complex numbers.
To see why, recall the relation of the unit circle and the trigonometric functions on the plane.
This means that every complex number with unit absolute value can be written in the form cos(φ) + isin(φ). From the geometric representation, we can see that every complex number is uniquely determined by its...
