Definition:Cartesian Product/Cartesian Space/Real Cartesian Space

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Definition

Let $n \in \N_{>0}$.

Then $\R^n$ is the cartesian product defined as follows:

$\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$


Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:

$\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$


Cartesian Plane

A general point $Q = \tuple {x, y}$ in the Cartesian plane

The Cartesian plane is a Cartesian coordinate system of $2$ dimensions.

Every point on the plane can be identified uniquely by means of an ordered pair of real coordinates $\tuple {x, y}$, as follows:


Identify one distinct point on the plane as the origin $O$.

Select a point $P$ on the plane different from $O$.

Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis.


Identify the $x$-axis with the real number line such that:

$0$ is identified with the origin $O$
$1$ is identified with the point $P$

The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$.

It is conventional to locate $P$ to the right of $O$, so as to arrange that:

to the right of the origin, the numbers on the $x$-axis are positive
to the left of the origin, the numbers on the $x$-axis are negative.


Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis.

Identify the point $P'$ on the $y$-axis such that $OP' = OP$.

Identify the $y$-axis with the real number line such that:

$0$ is identified with the origin $O$
$1$ is identified with the point $P'$


The orientation of the $y$-axis is determined by the position of $P'$ relative to $O$.

It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.

Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right:

going vertically "up" the page or screen from the origin, the numbers on the $y$-axis are positive
going vertically "down" the page or screen from the origin, the numbers on the $y$-axis are negative.


Cartesian 3-Space

The Cartesian $3$-space is a Cartesian coordinate system of $3$ dimensions.


Definition by Axes

A general point in Cartesian $3$-Space

Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:

Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis.

Recall the identification of the point $P$ with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.


Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis.

Identify the point $P''$ on the $z$-axis such that $OP'' = OP$.

Identify the $z$-axis with the real number line such that:

$0$ is identified with the origin $O$
$1$ is identified with the point $P$


The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$.

It is conventional to locate $P''$ as follows.

Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.

Then $P''$ is then one unit above the $x$-$y$ plane.


Let the $x$-$y$ plane be identified with the plane of the page or screen.

The orientation of the $z$-axis is then:

coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive
going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.


Definition by Planes

X-y-z-planes.png

Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:

Identify one distinct point in space as the origin $O$.

Let $3$ distinct planes be constructed through $O$ such that all are perpendicular.

Each pair of these $3$ planes intersect in a straight line that passes through $O$.

Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection.


Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively.


Select a point $P$ on the $x$-axis different from $O$.

Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Identify the point $P'$ on the $y$-axis such that $OP' = OP$.

Identify the point $P''$ on the $z$-axis such that $OP'' = OP$.


The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$.

It is conventional to locate $P''$ as follows.

Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.

Then $P''$ is then one unit above the $x$-$y$ plane.


Let the $x$-$y$ plane be identified with the plane of the page or screen.

The orientation of the $z$-axis is then:

coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive
going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.


Countable-Dimensional Real Cartesian Space

The countable cartesian product defined as:

$\displaystyle \R^\omega := \R \times \R \times \cdots = \prod_\N \R$

is called the countable-dimensional real cartesian space.

Thus, $\R^\omega$ can be defined as the set of all real sequences:

$\R^\omega = \left\{{\left({x_1, x_2, \ldots}\right): x_1, x_2, \ldots \in \R}\right\}$


Also known as

The real cartesian space of order $n$ is sometimes seen as the (real) cartesian $n$-space.

Some sources call this euclidean $n$-space -- however, on $\mathsf{Pr} \infty \mathsf{fWiki}$ this term is reserved for the associated metric space.


Also see


It can be shown that:

$\R^2$ is isomorphic to any infinite flat plane in space
$\R^3$ is isomorphic to the whole of space itself.


Source of Name

This entry was named for René Descartes.


Sources