The coordinate system most frequently used to quantitatively describe a n-dimensional space.
In 2 dimensions, i.e. a plane, it describes any point as a function of 2 perpendicular unit vectors (1,0) and (0,1) and in 3 dimensions as a function of 3 perpendicular unit vectors (1,0,0), (0,1,0) and (0,0,1). Functions in 2 dimensions are often conveniently described using the so-called theory of functions. When using this type of mathematical description, the imaginary number
i = √(-1) is introduced to label the y-axis.
a + ib is then actually a 2 dimensional vector with a x-axis component of 'a' and a y-axis component of 'b'.
The 'a' is called the real part and the 'b' the imaginary part of the function, an expression that is frequently encountered in MRI, where the real image is a pixel-wise representation of 'a' and the imaginary image a pixel-wise representation of 'b', with 'a' and 'b' the components of the xy-magnetization along the x- and y-axis, respectively.
(Renatus Cartesius/Rene Descartes, 1596-1650, French philosopher and mathematician)

A coordinate system, which uses the distance from the coordinate system center and positional angles to identify points in space rather than orthogonal independent unit vectors as in the Cartesian coordinate system. Polar and partly polar (cylindrical) coordinate systems are widely used to describe spin motion in NMR experiments.
It is important to know how to compute the coordinates of a point in the polar coordinate system when they are given in a Cartesian system and vice versa. The length of the vector r pointing from the coordinate origin to a point in 2D space is given as
r = √(x² + y²).
while the polar or phase angle f is obtained by performing the operation
f = arctan (y/x),
where the arctan function is the inverse of the tangent function.