# Definition:Cartesian Coordinate System/Ordered Pair

## Contents

## Identification of Point in Plane with Ordered Pair

Every point on the plane can be identified by means of a pair of 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 $O$.

Let the distance from the origin to $P$ be defined as being $1$.

Draw an infinite straight line through $O$ and $P$ and call it the $X$-axis.

Draw an infinite straight line through $O$ perpendicular to $OP$ and call it the $Y$-axis.

Now, let $Q$ be any point on the plane.

Draw two lines through $Q$, parallel to the $X$-axis and $Y$-axis.

The plane is then conventionally oriented so that the $X$-axis is horizontal with $P$ being to the right of $O$.

Thus the $Y$-axis is then a vertical line.

Thus the point $Q$ can be uniquely identified by the ordered pair $\tuple {x, y}$ as follows:

### X Coordinate

The distance of the line segment from $Q$ to the $Y$-axis is known as the **$X$ coordinate** and (usually) denoted $x$.

If $Q$ is to the right of the $Y$-axis, then $x$ is positive.

If $Q$ is to the left of the $Y$-axis, then $x$ is negative.

The **$X$ coordinate** of all points on the $Y$-axis is zero.

### Y Coordinate

The distance of the line segment from $Q$ to the $X$-axis is known as the **$Y$ coordinate** and (usually) denoted $y$.

If $Q$ is above the $X$-axis, then $y$ is positive.

If $Q$ is below the $X$-axis, then $y$ is negative.

The **$Y$ coordinate** of all points on the $X$-axis is zero.

The point $P$ is identified with the coordinates $\tuple {1, 0}$.

## 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

- 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