# 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