Definition:Cartesian Plane/Ordered Pair
Identification of Point in Plane with Ordered Pair
Let $Q$ be a point on the Cartesian plane.
Construct two straight lines through $Q$:
- one perpendicular to the $x$-axis, intersecting the $x$-axis at the point $x$
- one perpendicular to the $y$-axis, intersecting the $y$-axis at the point $y$.
The point $Q$ is then uniquely identified by the ordered pair $\tuple {x, y}$.
$x$ Coordinate
Let $x$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $x$-axis.
Then $x$ is known as the $x$ coordinate.
If $Q$ is in the positive direction along the real number line that is the $x$-axis, then $x$ is positive.
If $Q$ is in the negative direction along the real number line that is the $x$-axis, then $x$ is negative.
$y$ Coordinate
Let $y$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $y$-axis.
Then $y$ is known as the $y$ coordinate.
If $Q$ is in the positive direction along the real number line that is the $y$-axis, then $y$ is positive.
If $Q$ is in the negative direction along the real number line that is the $y$-axis, then $y$ is negative.
Also known as
The ordered pair $\tuple {x, y}$ which determine the location of $P$ in the cartesian plane can be referred to as the rectangular coordinates or (commonly) just coordinates of $P$.
Linguistic Note
It's an awkward word coordinate.
It really needs a hyphen in it to emphasise its pronunciation (loosely and commonly: coe-wordinate), and indeed, some authors spell it co-ordinate.
However, this makes it look unwieldy.
An older spelling puts a diaeresis indication symbol on the second "o": coördinate.
But this is considered archaic nowadays and few sources still use it.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (next): Chapter $\text I$. Coordinates: $1$. Coordinate Network
- 1934: D.M.Y. Sommerville: Analytical Geometry of Three Dimensions ... (previous) ... (next): Chapter $\text I$: Cartesian Coordinate-system: $1.1$. Cartesian coordinates
- 1936: Richard Courant: Differential and Integral Calculus: Volume $\text { II }$ ... (previous) ... (next): Chapter $\text I$: Preliminary Remarks on Analytical Geometry and Vector Analysis: $1$. Rectangular Co-ordinates and Vectors: $1$. Coordinate Axes
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{III}$: Gentleman, Soldier and Mathematician
- 1958: P.J. Hilton: Differential Calculus ... (previous) ... (next): Chapter $1$: Introduction to Coordinate Geometry
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 40$. Construction with Ruler and Compasses
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.9$: Cartesian Product
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Rectangular co-ordinates
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Graphical Representation of Complex Numbers