Editing Cartesian coordinate system (section)

===One dimension===
{{Main|Number line}}

An [[affine line]] with a chosen Cartesian coordinate system is called a ''number line''. Every point on the line has a real-number coordinate, and every real number represents some point on the line.

There are two [[degree of freedom|degrees of freedom]] in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinct [[real number]]s (most commonly zero and one). Other points can then be uniquely assigned to numbers by [[linear interpolation]]. Equivalently, one point can be assigned to a specific real number, for instance an ''origin'' point corresponding to zero, and an [[Curve orientation|oriented]] length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers.{{efn|Consider the two [[ray (geometry)|rays]] or half-lines resulting from splitting the line at the origin. One of the half-lines can be assigned to positive numbers, and the other half-line to negative numbers.}} Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a {{math|+}} or {{math|−}} sign chosen based on direction).

A [[geometric transformation]] of the line can be represented by a [[function of a real variable]], for example [[translation (geometry)|translation]] of the line corresponds to addition, and [[scaling (geometry)|scaling]] the line corresponds to multiplication. Any two Cartesian coordinate systems on the line can be related to each-other by a [[linear function]] (function of the form {{nobr|<math>x \mapsto ax + b</math>)}} taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an [[Affine transformation|affine map]] from one line to the other taking each point on one line to the point on the other line with the same coordinate.

{{Anchor|Cartesian coordinates in two dimensions}}