Cardinal numbers
A cardinal number is a natural number that is used to represent how many of something there are in a group. Cardinality is studied as a part of set theory.
Given a set of objects, A, the cardinal number of the set, n(A), is the number of elements in the set. Given the set A = {1, 2, 3}, there are 3 elements, so the cardinal number is n(A) = 3. Another way to say this is that the cardinality of the set is 3.
Example
Given the following sets of numbers: {1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, determine the cardinality of each set.
Although the numbers in each set of numbers above are different, they all have the same cardinality: 3.
We can also count that there are 4 sets of 3 numbers; 4 is another cardinal number.
One real life example of a cardinal number is a team. A soccer team has 11 players on the field at any one time, a baseball team has 9, and so on. Players on the team may have the jersey numbers 11 and 9, but those numbers wouldn't be considered cardinal numbers, because they aren't actually counting anything, and are instead used as identification; these types of numbers are known as nominal numbers.
Properties of cardinal numbers
Cardinal numbers are used for counting, so fractions, decimals, negative numbers, etc., are not considered cardinal numbers, only natural numbers. Also, the cardinality of a set can only be determined if the set is finite, since the cardinality is dependent on the number of elements in the set; a set with infinite elements cannot be counted, so we cannot find the cardinality of an infinite set.
0 is also not included in the cardinal numbers. If a given set is empty, or A = {}, the cardinality of the set is 0.
Cardinal arithmetic follows many of the same rules and shares a number of properties of arithmetic operations with natural numbers. For example, cardinal addition is also associative and commutative. Cardinal multiplication also follows the respective rules for associativity and commutativity in ordinary multiplication.
There are various other properties of cardinal numbers, these are just a few.
Cardinal, ordinal, and nominal numbers
These three types of numbers are often discussed together since it is necessary to understand them in order to effectively communicate using numbers. Briefly, a nominal number differs from a cardinal number in that nominal numbers are only used for identification purposes, typically to distinguish elements within a set of similar objects. They do not represent a count of something as cardinal numbers do.
Ordinal numbers, unlike cardinal or nominal numbers, tell us something about order, rank, or position. They can be identified by using terms such as first, second, third, and fourth to represent the cardinal numbers 1, 2, 3, and 4.
As an example that includes the use of all three types of numbers, imagine that there are 15 people running in a race. 15 is a cardinal number. The participants in the race are assigned the numbers 1-15 depending on when they registered for the race. Their respective numbers are nominal numbers. Only the three people who finish the race fastest get a prize based on whether they ranked first, second, or third. First, second, and third are ordinal numbers.