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# Ordinal Numbers

## Introduction

In mathematics, we come across different types of sets of numbers that have their own characteristics. One such type of number is the ordinal number. So, what are these numbers and how are they used? Let us find out.

## Definition

Ordinal numbers are the numbers that talk about the position of objects. The numbers 1st, 2nd, 3rd and so on are examples of ordinal numbers.

## What are Ordinal Numbers?

We have now leant that Ordinal numbers are the numbers that talk about the position of objects. In other words, ordinal numbers are the numbers that indicate the relation or order of one object in relation to another object. For instance, if we say that Henry stood 3rd in the examination, in this case, the number 3rd is an ordinal number that indicates the position of Henry with respect to other candidates who appeared in an examination. Some other examples of ordinal numbers are –

1. Peter stood first in the class.
2. Nancy was the fifth girl standing in the row.
3. Simona stays on the seventh floor of the apartment.

Let us now learn about ordinal numbers from 1 to 100

## Ordinal Numbers from 1 to 100

It is very important to learn how to write ordinal numbers from 1 to 100 as they are useful in representing the position of an object. The below table indicates the ordinal numbers from 1 to 100.

When we talk of ordinal numbers, another type of numbers is important to understand are cardinal numbers. What are cardinal numbers and how are they ordinal numbers different from cardinal numbers? Let us find out.

## What are Cardinal Numbers?

Cardinal numbers are the numbers that are used as counting numbers. In other words, the numbers that we use for counting are called cardinal numbers. Another name by which cardinal numbers are known as is natural numbers. In fact, cardinal numbers are the generalisation of natural numbers. The word “ Cardinal” means “how many” of anything is existing in a group. Like if we want to count the number of oranges that are present in the basket, we will have to make use of these numbers, such as 1, 2, 3, 4, 5….and so on.  Let us understand the cardinal numbers by an example.

Suppose we want to tell how many students are present in the class. The answer could be any number such as 8, 20, 45 and so on. These numbers that tell the count of the number of students present in the class are cardinal numbers. Some other examples of cardinal numbers are –

1. There are 6 clothes in the bag.
2. 3 cars are driving in a lane.
3. Peter has 2 dogs and 1 cat as pets in his house.

In the above three examples, the numbers 6, 3, 2 and 1 are the cardinal numbers. So basically it denotes the quantity of something, irrespective of their order. It defines the measure of the size of a set but does not take account of the order.

### How are Cardinal Numbers Represented in English?

We now know that Cardinal numbers define how many things or people are there, for example, five women standing under a tree. In this sentence, the word “five “ represents the cardinal number “ 5 “.

### What is Cardinality of a Set?

The cardinality of a group (set) tells how many objects or terms are there in that set or group. In other words, the cardinality of a finite set is a natural number: the number of elements in the set. For example, the set A  = { 1 , 3 , 5 , 7 , 9 } has a cardinality of 3 as there are 3 elements in the set.

## Difference between Ordinal Numbers, Nominal Numbers and Cardinal Numbers

What is the difference between ordinal numbers, nominal numbers and cardinal numbers ?

Nominal numbers are numbers in numeric form. For instance, 25 is the nominal number for the number “ Twenty Five “. Hence, the nominal numbers and cardinal numbers are different representations of the same numbers. We can also say that the nominal numbers are another type of number, different from cardinals and ordinals, used to name an object or a thing in a set of groups. They are used for the identification of something. It is not for representing the quantity or the position of an object.

Now, let us consider an example. Suppose in a race there were 10 athletes who participated. The athlete who came first was awarded a gold medal, while a silver medal was given to the candidate who stood second and a bronze medal was given to the athlete who came third. In this case, the number 10 which represents the number of athletes that participated in the race is the cardinal number. On the other hand, the positions first ( 1 st ), second (2 nd ) and third (3 rd ) are ordinal numbers as they represent the position.

Let us summarise the difference between cardinal numbers and ordinal numbers

Let us now compare some ordinal numbers with their cardinal counterparts.

## Solved Examples

Example 1 These are the first 10 English letters given in order. Express them in ordinal numbers as well as cardinal numbers where D is the fourth letter at the number 4 in the set.

{ A,   B,   C,   D,   E,   F,   G,   H,   I,   J }

Solution We have been given the set { A,   B,   C,   D,   E,   F,   G,   H,   I,   J }. It has also been given that D is the fourth letter at the number 4 in the set.

Let us write the ordinal and the cardinal numbers for the element of the given set. We will have,

Example 2 In the word, “MAGNIFICENT”, which is the third, fifth and the seventh letter?

Solution We have been given the word, “MAGNIFICENT” and we need to find the third, fifth and the seventh letter.

Let us write the position of all the letter of the given word. We will have,

We can see from the above table that there are 12 letters in the word “ MAGNIFICIENT” . Also, the third, fifth and the seventh letters are –

Third letter – G

Fifth Letter – I

Seventh Letter – I

Hence, the third, fifth and the seventh letters are G, I and I respectively.

Example 3 What is the cardinal number of set A = { 3, 5, 7, 9, 10, 11, 4, 19 } ? What is the ordinal number of the number 7 in the set?

Solution We have been given the set A = { 3, 5, 7, 9, 10, 11, 4, 19 }. We need to find

1. The cardinal number of set A
2. The ordinal number of the number 7 in the set

Let us find these one by one.

First, let us count the number of elements in the set A.  The total number of elements in the given set is 8.

Therefore, the cardinal number of the set A is 8.

Now, let us check the position of the element 7 in the given set. It is at the third position in the set.

Therefore, the ordinal number of the number 7 in the set A is 3 rd ( Third ).

## Key Facts and Summary

1. Ordinal numbers are the numbers that talk about the position of objects.
2. Ordinal numbers give us the answer of ‘where’. For instance, where does the student position in the list?
3. Nominal numbers are numbers in numeric form.
4. Cardinal numbers define the measure of the size of a set but do not take account of the order.
5. The nominal numbers and cardinal numbers are different representations of the same numbers.
6. The number of cardinal numbers is infinite.
7. Although the counting of numbers can go on to infinity, but the digits that are used to count the numbers are fixed. In fact, there are only 10 digits that are used for counting of numbers or we can say to represent cardinal numbers.