# Cardinal Numbers

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### What Are Cardinal Numbers?

Cardinal numbers are the generalization of natural numbers as it consists of all the counting numbers starting from one. Cardinal numbers are also called cardinals. The term cardinal numbers or cardinal was coined to represent the size of a set called cardinality (size) of sets. Cardinality is the number of elements present in a finite set which usually describes the size of the sets. For example, let us take two sets - Set A = {2, 4, 6, 8} and set B = {1, 2, 3, 5, 7}. The cardinality of set A is 4 as there are 4 elements present in set A whereas the number of elements present in set B is 5 so the cardinality of set B is 5. Cardinal tells us how many of something is present such as one, two, three, four, five, etc.

Apart from cardinal numbers, there are two other numbers that are ordinal and nominal. Cardinal number tells us how many of something is present, ordinal number describes the position of things and nominal number usually represents the name. For example, there were 11 players playing a game where each player has a number printed on the shirt that represents him. Suppose player number 33 comes first. Then, in this case, the cardinal number is 11 because there are a total of eleven players, the ordinal number of the player who won is 1st and the nominal number is 33 as it represents the player who one.

### Summary

• Cardinal Number (How many): 11

• Ordinal Number (Position): 1st

• Nominal Number (Name): 33

### How Many Cardinal Numbers are There?

Cardinal numbers are used as counting numbers. Counting can be of less number of things or more number of things. For example, we can count the number of fans in a house, the number of sheep on the farm, the number of hairs in our head, or the number of stars in the sky. All can be counted by using numbers like one, two, three, four, and so on. The counting can go on to infinity but there are a limited number of digits used. Basically there are 10 digits used to represent any number and those digits are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. Zero (0) alone is not a cardinal number. But zero can be used with other digits to represent numbers like 10, 20, 30, 700, 9000, etc.  The combination of numbers is used in series 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Once the number becomes double-digit, we change the ones of the number following the series. Example, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. Here, the digit in tens place remains constant while the digit in one place follows the series.

Once the series in one place reaches the end (i.e, 9), the digit in ten’s place is changed to a number of higher-order and the pattern is followed. Example, 20, 21, 22, 23, 24, 25, 26 27, 28, 29. The pattern is followed until one’s reaches 99. 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.

### Solved Examples

Example 1: Show how you can coordinate cardinal numbers with ordinal numbers?

Solution: Cardinal numbers and ordinal numbers can be coordinated as shown below:

 Cardinals Ordinals Cardinals Ordinals 1, One 1st, First 11, Eleven 11th, Eleventh 2, Two 2nd, Second 12, Twelve 12th, Twelfth 3, Three 3rd, Third 13, Thirteen 13th, Thirteenth 4, Four 4th, Fourth 14, Fourteen 14th, Fourteenth 5, Five 5th, Fifth 15, Fifteen 15th, Fifteenth 6, Six 6th, Sixth 16, Sixteen 16th, Sixteenth 7, Seven 7th, Seventh 17, Seventeen 17th, Seventeenth 8, Eight 8th, Eighth 18, Eighteen 18th, Eighteenth 9, Nine 9th, Ninth 19, Nineteen 19th, Nineteenth 10, Ten 10th, Tenth 20, Twenty 20th, Twentieth

1. How Were Cardinal Numbers Discovered?

In 1874–1884, George Cantor originated set theory and established the notion of cardinality.  The cardinality of any finite set can be found by using cardinal numbers. Note, that even unequal sets can have equal cardinality.  For example, Set A = {5,6,7,8} and Set B = {4,5,6, 7} are not equal, but they have the same cardinality because the number of elements in both the sets is the same. The equality in the cardinality of two sets can be shown by bijection, i.e. a one-to-one correspondence.  George Cantor used the same technique to find the cardinality of infinite sets that is a set of natural numbers {1, 2, 3, 4, 5, 6, 7, …} Thus, all the sets having bijection with this set are countably infinite sets and so they all have the same cardinality. He also called the cardinal numbers of these infinite sets as transfinite cardinal numbers.

In 1924, Tarski proposed that every set is associated with a cardinal number. Later, A. P. Morse and Dana Scott defined cardinal numbers by considering a set and then calling the magnitude of the set as cardinality. In order to associate cardinality with a specific set, the axiom of foundation or the axiom of choice was required. Using the axiom of choice, the cardinals can be specified through the ordinals. Therefore, the axiom of choice can be associated with an ordinal number.

2. What is the Difference Between Cardinal Numbers and Natural Numbers?

Both Cardinal numbers as well as natural numbers are counting numbers that start from 1. The only difference between cardinal numbers and natural numbers is the size. Natural numbers are finite and form the subset of cardinal numbers whereas cardinal numbers are the infinite numbers. Natural numbers include 0 but it is not the same with the cardinal numbers. In cardinal number, 0 is not used.