Ordered Pairs and Coordinates

The Cartesian plane works because of one simple agreement: every point gets a name made of two numbers, in a fixed order. That name is the ordered pair $(x, y)$. This lesson nails down the language , abscissa, ordinate, equality of pairs , that you will use for the rest of the chapter and beyond.

Definitions

An ordered pair $(x, y)$ is a pair of real numbers in which the first number $x$ and the second number $y$ play different roles. The first slot is for the abscissa; the second is for the ordinate.

For a point $P$ in the plane:

• The abscissa of $P$ is its signed horizontal distance from the $y$-axis. If $P$ is to the right of the $y$-axis, the abscissa is positive; to the left, negative; on the $y$-axis, zero. We usually call this $x$.
• The ordinate of $P$ is its signed vertical distance from the $x$-axis. If $P$ is above the $x$-axis, the ordinate is positive; below, negative; on the $x$-axis, zero. We usually call this $y$.

The pair $(x, y)$ is the Cartesian coordinates of $P$ (or simply the coordinates of $P$).

Concept and rules

Existence and uniqueness. Each point in the plane corresponds to exactly one ordered pair, and each ordered pair corresponds to exactly one point. This one-to-one correspondence is the entire reason coordinate geometry works.

Equality of ordered pairs. Two ordered pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equal if and only if $x_1 = x_2$ and $y_1 = y_2$. Both coordinates must match. This is much stricter than equality of sets: the set $\{2, 3\}$ equals the set $\{3, 2\}$, but the ordered pair $(2, 3)$ does not equal $(3, 2)$.

Why order matters. The two slots of $(x, y)$ play geometrically different roles. The first slot says how far horizontally; the second says how far vertically. Switching the slots means switching horizontal and vertical, which lands at a different point.

Reading coordinates from a grid. Given a point on a grid:

1. Drop a perpendicular from the point to the $x$-axis. Read the value where it lands: that is the abscissa.
2. Drop a perpendicular from the point to the $y$-axis. Read the value where it lands: that is the ordinate.
3. Write the pair: (abscissa, ordinate).

Writing coordinates from a description. "$3$ units to the right of and $2$ units below the origin": the abscissa is $3$, the ordinate is $-2$. Coordinates: $(3, -2)$.

Coordinates of special points.

• Origin: $(0, 0)$.
• Any point on the $x$-axis: $(a, 0)$ for some real $a$.
• Any point on the $y$-axis: $(0, b)$ for some real $b$.

Coordinates of a point on the line $y = x$. If $y = x$, then both coordinates are equal. So $(2, 2), (-3, -3), (\sqrt{2}, \sqrt{2})$ are all on this line. We will explore this line carefully in Chapter 4.

Notation note. Some textbooks write $P = (x, y)$, others $P(x, y)$. Both mean the same thing. The brackets are parentheses, not square brackets , square brackets often mean a closed interval, not a coordinate.

Worked examples

Example 1. What is the abscissa and ordinate of $P = (-7, 4)$?

Abscissa $= -7$, ordinate $= 4$. The point is $7$ units to the left of the $y$-axis and $4$ units above the $x$-axis.

Example 2. Are $(3, -2)$ and $(-2, 3)$ the same point?

No. The first has abscissa $3$, ordinate $-2$; the second has abscissa $-2$, ordinate $3$. These are different points.

Example 3. A point lies on the $x$-axis at $5$ units to the left of the origin. Find its coordinates.

On the $x$-axis: ordinate $= 0$. Five units left of the origin: abscissa $= -5$. Coordinates: $(-5, 0)$.

Example 4. Find $a$ and $b$ if $(a + 1, b - 2) = (3, 5)$.

By equality of ordered pairs: $a + 1 = 3$ and $b - 2 = 5$. So $a = 2$ and $b = 7$.

Example 5. A point has equal abscissa and ordinate, both negative. Give one example, and state the line it lies on.

Example: $(-4, -4)$. The line is $y = x$. All such points satisfy "abscissa equals ordinate" and they form a single straight line through the origin.

Try it yourself

1. State the abscissa and ordinate of each: $(2, 5), (-3, 7), (0, -4), (6, 0), (-1, -1)$.
2. Are $(4, 3)$ and $(3, 4)$ the same point?
3. Find $x$ and $y$ if $(x - 2, y + 1) = (5, 4)$.
4. Find the coordinates of a point on the $x$-axis with abscissa $-6$.
5. Find the coordinates of a point on the $y$-axis with ordinate $\sqrt{2}$.
6. State whether the points $(-3, 5)$ and $(3, -5)$ have the same abscissa.
7. Write the coordinates of any three different points on the line "abscissa equals ordinate".
8. Find the coordinates of the point that is $4$ units above the origin.
9. If $(a + 3, 7) = (8, 7)$, find $a$.
10. Give an example of two ordered pairs that have the same abscissa but different ordinates.

Pitfalls / Insight

• An ordered pair is not a set. $(3, 5) \ne (5, 3)$, even though $\{3, 5\} = \{5, 3\}$.
• Use the right separator. A coordinate is written with a comma between the two entries, not a colon or a semicolon.
• Coordinates can be irrational or negative. The point $(\sqrt{2}, -\pi)$ is just as honest as $(1, 2)$.

Insight. Every word in this lesson , abscissa, ordinate, ordered pair, coordinates , is just a way to express the simple fact: each point in the plane has a unique two-number ID, and that ID respects order. Internalise the order, and the rest of coordinate geometry is bookkeeping.