An ordered pair, as its name suggests, is a pair of elements that have specific importance for the order of their placements. Ordered pairs are usually used in coordinate geometry to represent a point on a coordinate plane. Also, they are used to represent elements of a relation.
Let us learn more about ordered pairs along with its definition, meaning, properties, and more.
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1. What is an Ordered Pair? 2. Ordered Pair in Coordinate Geometry 3. Graphing Ordered Pairs 4. Ordered Pairs in Different Quadrants 5. Ordered Pair in Sets 6. Equality Property of Ordered Pairs 7. FAQs on Ordered Pair
An ordered pair is a pair formed by two elements that are separated by a comma and written inside the parantheses. For example, (x, y) represents an ordered pair, where ‘x’ is called the first element and ‘y’ is called the second element of the ordered pair. These elements have specific names according to what context they are being used and they can be either variables or constants. The order of the elements has a certain importance in an ordered pair. It means (x, y) may not be equal to (y, x) all the time.
(2, 5), (a, b), (0, -5), etc are some examples of ordered pairs.
An ordered pair in coordinate geometry is used to represent the position of a point on the coordinate plane with respect to the origin. A coordinate plane is formed by two perpendicular intersecting lines among which one is horizontal (x-axis) and the other line is vertical (y-axis). The intersecting point of both axes is the origin. Every point on the coordinate plane is represented by an ordered pair (x, y) where the first element x is called the x-coordinate and the second element y is called the y-coordinate. We can see more differences between the elements of the ordered pair used in geometry here.
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First Element of Ordered Pair Second Element of Ordered Pair It is called x-coordinate. It is called y-coordinate. Another name for this is “abscissa”. Another name for this is “ordinate”. It represents the horizontal distance of the point from the origin. It represents the vertical distance of the point from the origin. This number is one of the numbers on the x-axis. This number is one of the numbers on the y-axis. It represents the distance of the point from the y-axis. It represents the distance of the point from the x-axis. Example: If (2, 4) is a point on the coordinate plane, then 2 is the distance of the point from the y-axis. 
Example: If (2, 4) is a point on the coordinate plane, then 2 is the distance of the point from the y-axis. 
Now, we understood the difference between the x-coordinate and y-coordinate of an ordered pair in coordinate geometry. Let us now see the steps to graph ordered pairs.
- Step 1: Always start from the origin and move horizontally by |x| units to the right if x is positive and to the left if x is negative. Stay there.
- Step 2: Start from where you have stopped in Step 1 and move vertically by |y| units to up if y is positive and to down if y is negative. Stay there.
- Step 3: Place a dot exactly at the point where you have stopped in Step 2 and that dot represents the ordered pair (x, y)
In these steps, |x| and |y| represent the absolute values of x and y respectively.
Example: Graph the ordered pair (4, -3).
Let’s start from the origin, move to the right by 4 units (as 4 is positive) and then move down by 3 units (as 3 is negative).
The order of the elements in an ordered pair is important and hence the name “ordered” pair. For example, (4, -3) and (-3, 4) are located at different positions on the plane as shown below.
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We can see in the above figure that the coordinate plane is divided into 4 parts by the x and y axes. Each of these 4 parts is known as a quadrant. The signs of x and y in an ordered pair (x, y) of a point differs depending upon the quadrant and they are shown in the table below.
Quadrant Ordered Pair Signs I x > 0, y > 0 II x < 0, y > 0 III x < 0, y < 0 IV x > 0, y < 0
For example, (2, -4) refers to 2 on the x-axis (positive) and -4 on the y-axis (negative). So (2, -4) is a point in quadrant IV.
Till now we have seen that ordered pairs are used in coordinate geometry to locate a point. But they are used in set theory as well in a different context. The set of all possible ordered pairs from a set A to a set B is called the cartesian product. For example, if A = {1, 2, 3} and B = {a, b, c}, then the cartesian product is A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c)} and it is a set formed by all ordered pairs (x, y) where x is in A and y is in B. Any subset of the cartesian product is called a relation. For example, {(1, a), (1, b), (3, c)} is a relation.
Examples:
- If (2, 4) belongs to a relation “divides”, it means that 2 divides 4.
- If (4, 2) belongs to a relation “greater than”, it means that 4 is greater than 2.
- If (x, y) belongs to a relation “is a sister of”, it means that x is a sister of y.
For any two ordered pairs (x, y) and (a, b) (either in coordinate geometry or in relations), if (x, y) = (a, b) then x = a and y = b. i.e., if two ordered pairs are equal, then their corresponding elements are equal. This is called “equality property of ordered pairs”. For example:
- If (x, y) = (2, -3) then x = 2 and y = -3.
- If (x + 1, y – 2) = (-3, 5) then x + 1 = -3 and y – 2 = 5.
Important Notes on Ordered Pairs:
- An ordered pair (x, y) is used to represent the location of a point in coordinate geometry where x is the horizontal distance and y is the vertical distance of the point.
- An ordered pair (x, y) represents an element of a relation R which is denoted by xRy (x “is related to” y).
- If (x, y) = (a, b) then x = a and y = b.
Related Topics:
- Ordered Pair Calculator
- Introduction to Graphing
- Graphing Functions
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