# Cartesian Coordinate System **A two-dimensional coordinate system in which every point is uniquely identified by a pair of signed distances to a pair of perpendicular lines.** ![[CartesianCoordinateSystem.svg|550]] *A Cartesian coordinate plane* The *Cartesian coordinate system* uniquely identifies each point in two-dimensional Euclidean space using an ordered pair of *signed distances* to an ordered pair of *fixed perpendicular lines* known as the axes. The intersection of the two axes is known as the *origin* and is defined as $(0, 0)$. The Cartesian coordinate system can be generalised to any number of dimensions of Euclidean space. ## Higher dimensions The Cartesian coordinate system can be *generalised* to any $n$-dimensional Euclidean space. In an $n$-dimensional Euclidean space, points are specified with an ordered set of *$n$ signed distances* to an ordered set of fixed perpendicular *$n-1$ dimension hyperplanes* instead. ![[CartesianCoordinateSystem3D.svg|550]] *The Cartesian coordinate system in three dimensions with the $x$-$y$ Cartesian coordinate plane embedded* ## Notation In two-dimensional Cartesian coordinates, the horizontal and vertical axis are often denoted as the *$x$-axis* and the *$y$-axis*, respectively. Extending into three dimensions, the third axis is often denoted the *$z$-axis*. The coordinates themselves are referred to, in order, as the $x$, $y$, and $z$ coordinates. The $x$ and $y$ coordinates may also be referred to as the *abscissa* and the *ordinate*, respectively. Generic coordinates are thus notated as $(x,y)$ or $(x, y, z)$. The axes and coordinates may also be denoted using *subscripts*, as in $(x_{1}, x_{2}, \dots, x_{n})$, which is extendable to any $n$-dimensions. ### Quadrants, octants, and orthants In a two-dimensional Cartesian coordinate system, the two axes divide the plane into four infinite regions which are referred to as *quadrants*. They are commonly labelled using *Roman numerals*, starting with $\mathbf{I}$ for the quadrant where both coordinates are positive and continuing *anticlockwise*. ![[CartesianCoordinateSystemQuadrants.svg|550]] In a three-dimensional Cartesian coordinate system, the three axes divide the space into eight infinite regions instead, which are referred to as *octants*. The common notation method is to use a *list of coordinate signs*, such as $(+, -, +)$. The generalised analogue of quadrants and octants is an *orthant*. In $n$ dimensions, the axes divide the space into $2^{n}$ orthants which in higher dimensions are also notated as a list of coordinate signs.