Definition:Cartesian Product/Cartesian Space/Two Dimensions
Definition
Let $S$ be a set.
The cartesian $2$nd power of $S$ is:
- $S^2 = S \times S = \set {\tuple {x_1, x_2}: x_1, x_2 \in S}$
The set $S^2$ called a cartesian space of $2$ dimensions.
Cartesian Plane
When $S$ is the set of real numbers $\R$, the cartesian product takes on a special significance.
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:
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:
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.
Also see
Source of Name
This entry was named for René Descartes.
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Definition $1.4$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.7$. Pairs. Product of sets: Example $22$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets