Math Lesson 16.1.1 - Ordered Pair

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Welcome to our Math lesson on Ordered Pair, this is the first lesson of our suite of math lessons covering the topic of Relation and Function, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Ordered Pair

We learnt in previous tutorials that a graph shows an infinite number of points that are very close to each other, so we cannot see the spaces between these points but only a solid line. All the above-mentioned points represent pairs of coordinates written in brackets, where the first coordinate of any number pair is always the horizontal one while the second coordinate is the vertical one. For example, if one says A(2, -3) is a point of a graph, we understand that the x-coordinate of the point A is x_A = 2 and the y-coordinate of the same point is y_A = -3.

In mathematics, an ordered pair is a set of two numbers usually written in the form (a, b). These two numbers are taken from different sets, usually determined by values in the two perpendicular number axes - the horizontal and vertical ones. The order of the two numbers is important in the sense that (a, b) is different from (b, a) unless a = b. As stated earlier, ordered pairs are commonly used to specify a location on a coordinate plane.

Thus, if we consider the following set of ordered pairs given

we understand that four points from a given graph in the coordinates system are given, where the x-coordinates of these four points are -1, 0, 1 and 2 respectively while the corresponding y-coordinates are 3, -1, 4 and -3.

Ordered pairs are not always connected through lines. In many cases, they are disconnected points in the coordinate system. This occurs when the data set shown on the graph represents the relationship between the variables through scattered points, as in the figure below.

Example 1

1. Show the following ordered pairs on the coordinates system
   (-2, 2), (-1, 0), (2, 5), (1, 3), (-4, -2), (6, 0)
2. Calculate the distance between the remotest points determined by the ordered pairs above.

Solution 1

1. Given that the first number of an ordered pair represents the x-coordinate and the second number the y-coordinate of the corresponding point, we obtain the following figure for all points determined by each number pair.
2. From a first glance at the figure, there is a doubt whether the remotest points are (-4, -2) and (6, 0) or (-4, -2) and (2, 5). We use the distance formula
   d = √(x₂ - x₁ )² + (y₂ - y₁ )²
   to calculate the distance between each of the pairs of numbers. Thus, for the first option (-4, -2) and (6, 0) we have
   d₁ = √(6 - (-4))² + (0 - (-2))²
   = √(6 + 4)² + (0 + 2)²
   = √10² + 2²
   = √100 + 4
   = √104
   and for the second option (-4, -2) and (2, 5) we have
   d₂ = √(2 - (-4))² + (5 - (-2))²
   = √(2 + 4)² + (5 + 2)²
   = √6² + 7²
   = √36 + 49
   = √85
   Since √104 > √85, then the remotest points are those with coordinates (-4, -2) and (6, 0).

You have reached the end of Math lesson 16.1.1 Ordered Pair. There are 9 lessons in this physics tutorial covering Relation and Function, you can access all the lessons from this tutorial below.

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