An ordered pair refers to a number written in a certain order. An ordered pair is used to show the position on a graph, where the "x" (horizontal) value is first, and the "y" (vertical) value is second. Also in the co-ordinate system, ordered pair is used to locate a point. Pairs in math are denoted by (,) and are generally considered to be ordered.
Ordered Pair Definition
An ordered pair consists of two numbers that are written in the fixed order. So, we can define an ordered pair as the pair of elements that occur in a particular order and are enclosed in brackets.
An ordered pair is a combination of the x coordinate and the y coordinate, having two values written in a fixed order within parentheses.
Ordered Pair Meaning
Ordered Pair = (x, y)
where, x = abscissa, the distance measure of a point from the x-axis.
and, y = ordinate, the distance measure of a point from the y-axis.
To graph a point, we have to draw a dot at the coordinates that correspond to the ordered pair. The x-coordinate tells us how many steps we have to take to on the x-axis. The y-coordinate tells us to have many steps to move on the y-axis.
Ordered Pair Example
The ordered pair (2, 5) means a pair of two integers, strictly in the order with 2 at the first place called the abscissa and 5 at the second place called the ordinate.
The ordered pair (2, 5) is not equal to the ordered pair (5, 2) because (2, 5) ≠ (5, 2). Therefore, in a pair, the order of elements is important.
Set of Ordered Pairs
The pair of elements that occur in a particular order and are enclosed in brackets is known as a set of ordered pairs. If ‘a’ and ‘b’ are two elements, then the two different pairs are (a, b), (b, a). In an ordered pair (a, b), a is called the first element and b is called the second element.
Consider, if A and B are two sets such that a ∈ A and b ∈ B, then by the ordered pair of elements we mean (a, b) where 'a' is called the Ist element and 'b' is called the IInd element of the ordered pair.
If we change the position of the elements in the ordered pair, then the ordered pair also get changed, i.e., it becomes (b, a) but (a, b) ≠ (b, a).
Properties of Ordered Pairs
Equality of Ordered Pairs:
Two ordered pairs are said to be equal if and only if the corresponding first elements are equal and the corresponding second elements are equal.
For example: Consider two ordered pairs (a, b) and (c, d). They are equal if a = c and b = d, i.e., (a, b) = (c, d).
Question: Find the values of x and y, if (2x - 3, y + 1) = (x + 5, 7)
Solution: We will solve by equality of ordered pairs
Given 2x - 3 = x + 5 and y + 1 = 7
⇒ 2x - x = 5 + 3
⇒ x = 8
and y = 7 - 1
⇒ y = 6
Hence x = 8 and y = 6.
Note: Both the elements of an ordered pair can be the same i.e., (2, 2), (5, 5).
Applications of Ordered Pair
The concept of an ordered pair is highly used in data comprehension as well for word problems and statistics.
Ordered pairs are used in the coordinate geometry to represent geometric figures and objects in an open space for visual comprehension. Various geometric shapes like circles, triangles, squares, rectangles, and polygons use the ordered pairs to represent the centre, vertices, and the length of the sides with coordinates.
1. Plot the point “P” with coordinates 6, 4.
Sol: As per the definition of ordered pair, the point P will be written as:
P = (6, 4)
The first number in the ordered pair shows the distance from the “x" axis which is 6.
The second number in the ordered pair shows the distance from the “y" axis which is 4.
To locate the point on the Cartesian plane, start from the origin. Take 6 steps towards the “x” axis (towards the right) starting from the origin. From here, move 4 steps towards the “y” axis (upwards).
As we know the order in which values are written in an ordered pair is very important. The ordered pair (6, 4) is different from the ordered pair (4, 6). Both represent two different points as shown below.
2. If (x, y) = (-1, 4) find the value of 3x + 2y - 4?
Sol: Given x = (-1) and y = 4. Substitute x, y value in given equation
3x + 2y - 4 = 3(-1) + 2(4) - 4 = 1.
The mathematician Rene Descartes and Pierre de Fermat invented analytic geometry in the 16th century and the Cartesian plane was designed.
Ordered pair is widely used in the field of computing and programming languages.
Ordered pairs are also known as 2-tuples, or sequences of length 2.