# Definition:Cartesian Plane

## Definition

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.

### Cartesian Coordinate 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 $\tuple {x, y}$.

Hence:

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

### $x$ 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.

## Quadrants

For ease of reference, the cartesian plane is often divided into four quadrants by the axes:

### First Quadrant

Quadrant $\text{I}: \quad$ The area above the $x$-axis and to the right of the $y$-axis is called **the first quadrant**.

That is, **the first quadrant** is where both the $x$ coordinate and the $y$ coordinate of a point are positive.

### Second Quadrant

Quadrant $\text{II}: \quad$ The area above the $x$-axis and to the left of the $y$-axis is called **the second quadrant**.

That is, **the second quadrant** is where the $x$ coordinate of a point is negative and the $y$ coordinate of a point is positive.

### Third Quadrant

Quadrant $\text{III}: \quad$ The area below the $x$-axis and to the left of the $y$-axis is called **the third quadrant**.

That is, **the third quadrant** is where both the $x$ coordinate and the $y$ coordinate of a point are negative.

### Fourth Quadrant

Quadrant $\text{IV}: \quad$ The area below the $x$-axis and to the right of the $y$-axis is called **the fourth quadrant**.

That is, **the fourth quadrant** is where the $x$ coordinate of a point is positive and the $y$ coordinate of a point is negative.

Note that the axes themselves are generally not considered to belong to any quadrant.

## Also known as

The **cartesian coordinate plane** is often seen referred to as the **$x y$-plane**, or (without the hyphen) the **$x y$ plane**.

Some sources refer to it as the **Euclidean plane**, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ that term is reserved for the abstract geometry consisting of $\R^2$ together with the set of straight lines.

## Historical Note

The Cartesian plane was supposedly invented by René Descartes in around the year $1637$.

However, a study of the literature will reveal that this idea originated considerably earlier, perhaps going back as far as Nicole Oresme.

## Sources

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