**Introduction**

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

**History of Cardinal Numbers**

George Cantor in 1874 – 1884 originated set theory and established the concept of the cardinality of a set. According to him, cardinal numbers were infinite sets and they could also be called as transfinite cardinal numbers. Cantor replaced the elementary concept of a number with a derived and abstract one based on sets (for what it is worth, the informal counting conception is closer to his ordinals than to his cardinals, the notions are equivalent in scope in the finite case but diverge dramatically for infinities). Another mathematician, Traski in 1924 proposed that every set has an association with a cardinal number. Two other mathematicians, Dana Scott and A P Morse later gave the definition of cardinal numbers by considering a set and calling the magnitude of a set as cardinality.

**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 –

- There are 6 clothes in the bag.
- 3 cars are driving in a lane.
- 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 “.

**How many Cardinal Numbers are there?**

Since we know that cardinal numbers are used as counting numbers and counting can be of less number of things or more number of things, therefore, this means that there is no end to the list of cardinal numbers. In other words, the number of cardinal numbers is infinite. But an important point to note here is that 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. These 10 digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is just the combination of these numbers which make different cardinal numbers such as 36, 43, 75, 89 and so on. The order of these numbers is called counting of numbers.

This means that the cardinal numbers, since being used as counting numbers are used in the order, 1 ,2 , 3 , 4 , 5 , 6 , 7 , 8 , 9. After 9, the numbers are formed in two digits i.e. the cardinal numbers after 9 will be 1 0, 1 1, 1 2, 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9. It can be clearly seen here that the digit in tens place remains constant while the digit in one place follows the series. Once the series in one’s place reaches the end, i.e. it reaches 9, the digit in ten’s place is changed to a number of higher-order and the pattern is followed. Therefore, after 1 9 we will have 2 0 , 2 1 , 2 2 and so on. After all the numbers in two digits are over, three-digit numbers are followed. The three-digit number starts from 100 and goes on till 999. Three-digit numbers are followed by four-digit numbers and so on. Therefore, there is no limit to the cardinal numbers which means cardinal numbers are infinite.

**Cardinal Numbers From 1 to 100**

Below is the list of cardinal numbers from 1 to 100

Nominal Number – Cardinal Number | Nominal Number – Cardinal Number | Nominal Number – Cardinal Number | Nominal Number – Cardinal Number |

1 One | 2 6 Twenty Six | 5 1 Fifty One | 7 6 Seventy Six |

2 Two | 2 7 Twenty Seven | 5 2 Fifty Two | 7 7 Seventy Seven |

3 Three | 2 8 Twenty Eight | 5 3 Fifty Three | 7 8 Seventy Eight |

4 Four | 2 9 Twenty Nine | 5 4 Fifty Four | 7 9 Seventy Nine |

5 Five | 3 0 Thirty | 5 5 Fifty Five | 8 0 Eighty |

6 Six | 3 1 Thirty One | 5 6 Fifty Six | 8 1 Eighty One |

7 Seven | 3 2 Thirty two | 5 7 Fifty Seven | 8 2 Eighty Two |

8 Eight | 3 3 Thirty Three | 5 8 Fifty Eight | 8 3 Eighty Three |

9 Nine | 3 4 Thirty Four | 5 9 Fifty Nine | 8 4 Eighty Four |

1 0 Ten | 3 5 Thirty Five | 6 0 Sixty | 8 5 Eighty Five |

1 1 Eleven | 3 6 Thirty Six | 6 1 Sixty One | 8 6 Eighty Six |

1 2 Twelve | 3 7 Thirty Seven | 6 2 Sixty Two | 8 7 Eighty Seven |

1 3 Thirteen | 3 8 Thirty Eight | 6 3 Sixty Three | 8 8 Eighty Eight |

1 4 Fourteen | 3 9 Thirty Nine | 6 4 Sixty Four | 8 9 Eighty Nine |

1 5 Fifteen | 4 0 Forty | 6 5 Sixty Five | 9 0 Ninety |

1 6 Sixteen | 4 1 Forty One | 6 6 Sixty Six | 9 1 Ninety One |

1 7 Seventeen | 4 2 Forty Two | 6 7 Sixty Seven | 9 2 Ninety Two |

1 8 Eighteen | 4 3 Forty Three | 6 8 Sixty Eight | 9 3 Ninety Three |

1 9 Nineteen | 4 4 Forty Four | 6 9 Sixty Nine | 9 4 Ninety Four |

2 0 Twenty | 4 5 Forty Five | 7 0 Seventy | 9 5 Ninety Five |

2 1 Twenty One | 4 6 Forty Six | 7 1 Seventy One | 9 6 Ninety Six |

2 2 Twenty Two | 4 7 Forty Seven | 7 2 Seventy Two | 9 7 Ninety Seven |

2 3 Twenty Three | 4 8 Forty Eight | 7 3 Seventy Three | 9 8 Ninety Eight |

2 4 Twenty Four | 4 9 Forty Nine | 7 4 Seventy Four | 9 9 Ninety Nine |

2 5 Twenty Five | 5 0 Fifty | 7 5 Seventy Five | 100 One Hundred |

**Properties of Cardinal Numbers**

Following are the properties of cardinal numbers –

- Cardinal numbers help us to count the number of things or people in or around a place or a group.
- The collection of all the ordinal numbers can be denoted by the cardinal.
- Cardinal numbers can be written as words such as one, two, three, etc.
- Cardinal numbers define the measure of the size of a set but do not take account of the order.
- Cardinal numbers are also called counting numbers and natural numbers
- A group of ordinal numbers can be represented by cardinal numbers
- Cardinal numbers are always used to count and are stated by ‘how many’
- Fractions and decimals are not cardinal numbers
- Zero (0) is not a cardinal number, since it means nothing
- The cardinality of a set represents how many objects or elements are there in the set

**Cardinal Numbers vs Other Numbers**

As we know that there are other types of sets as well that possesses their own properties. For example, apart from cardinal numbers, there are two other numbers that are ordinal numbers and nominal numbers. So, how similar are cardinal numbers with ordinal numbers and nominal numbers? Or, rather how different all the three types of numbers are from each other. Let us find out.

**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.

**Ordinal number describes the position of things. **In other words, an ordinal number is a number that denotes the position or place of an object. For example, 1 ^{st}, 2 ^{nd}, 3 ^{rd} etc. are all ordinal numbers. Ordinal numbers are used for the purpose of ranking.

Let us compare cardinal numbers vs ordinal numbers

**Cardinal Numbers vs Ordinal Numbers**

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

Cardinal Numbers | Ordinal Numbers |

Cardinal Numbers are counting numbers that represent quantity. | Ordinal Numbers are based on the rank or position of an object in a given list or order. |

Cardinal numbers give us the answer of ‘how many?’ | Ordinal numbers give us the answer of ‘where’. For instance, where does the student position in the list? |

Examples of cardinal numbers are 1 , 5 , 8 , 20 , 35 etc. | Examples of ordinal numbers are 1 ^{st}, 2 ^{nd}, 3 ^{rd }etc. |

Let us now compare some cardinal numbers with their nominal numbers and ordinal counterparts

Cardinals | Ordinals |

1 One | 1^{st }First |

2 Two | 2^{nd} Second |

3 Three | 3^{rd} Third |

4 Four | 4^{th} Fourth |

5 Five | 5^{th} Fifth |

6 Six | 6^{th} Sixth |

7 Seven | 7^{th} Seventh |

8 Eight | 8^{th} Eighth |

9 Nine | 9^{th} Ninth |

1 0 Ten | 10^{th} Tenth |

1 1 Eleven | 11^{th} Eleventh |

1 2 Twelve | 12^{th} Twelfth |

1 3 Thirteen | 13^{th} Thirteenth |

1 4 Fourteen | 14^{th} Fourteenth |

1 5 Fifteen | 15^{th} Fifteenth |

1 6 Sixteen | 16^{th} Sixteenth |

1 7 Seventeen | 17^{th} Seventeenth |

1 8 Eighteen | 18^{th} Eighteenth |

1 9 Nineteen | 19^{th} Nineteenth |

2 0 Twenty | 20^{th} Twentieth |

**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.

**Solved Examples**

**Example 1** 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

- The cardinal number of set A
- 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 ).**

**Example 2** 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 element of the given set. We will have,

Element of the Set | Ordinal Number | Carinal Number |

A | 1^{st }First | 1 One |

B | 2^{nd} Second | 2 Two |

C | 3^{rd} Third | 3 Three |

D | 4^{th} Fourth | 4 Four |

E | 5^{th} Fifth | 5 Five |

F | 6^{th} Sixth | 6 Six |

G | 7^{th} Seventh | 7 Seven |

H | 8^{th} Eighth | 8 Eight |

I | 9^{th} Ninth | 9 Nine |

J | 10^{th} Tenth | 1 0 Ten |

**Key Facts and Summary**

- Cardinal numbers are the numbers that are used as counting numbers
- Cardinal numbers are the generalisation of natural numbers.
- Cardinal numbers help us to count the number of things or people in or around a place or a group.
- The collection of all the ordinal numbers can be denoted by the cardinal.
- Cardinal numbers can be written as words such as one, two, three, etc.
- Cardinal numbers define the measure of the size of a set but do not take account of the order.
- Cardinal numbers are also called counting numbers and natural numbers
- A group of ordinal numbers can be represented by cardinal numbers
- Cardinal numbers are always used to count and are stated by ‘how many’
- Fractions and decimals are not cardinal numbers
- Zero (0) is not a cardinal number, since it means nothing
- Cardinality of a set represents how many objects or elements are there in the set
- The nominal numbers and cardinal numbers are different representations of the same numbers.
- The number of cardinal numbers is infinite.
- 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.
- Ordinal number describes the position of things. In other words, an ordinal number is a number that denotes the position or place of an object. Ordinal numbers are used for the purpose of ranking.
- The cardinality of a group (set) tells how many objects or terms are there in that set or group.