Cardinal & Ordinal Numbers

Three Ways of Using Numbers

In daily life, numbers are used in 2 senses - as answers to "How Many?" as well as "What is the numerical order in which this item occurs?".

When they answer the first question (I have 3 pencils), they are called Cardinal numbers.

When they form part of the second question (It is the 4thseat on 5throw) they are called Ordinal numbers.

With increasing use of technology, there is a third way in which numbers are used. They are called Nominal Numbers. These are cases where a number is used as a way of naming something.

Your telephone numbers, postal codes and personal identity/ security numbers are of this type. 603103 is the code for Kelambakkam in Kanchipuram District in the Indian Postal System.

Only Cardinal Numbers can be Arithmetically Operated with

Cardinal numbers tell the quantity. Cardinality of a given set does not change. But the order in which things are arranged within a set can change depending on the path of counting. Cardinal numbers are called One, Two, Three ..... etc.

Ordinal numbers inform us about the order or position. Ordinal numbers are called First, Second, Third etc.

The 4 arithmetic operations can be done on numbers, only when they are used in the cardinal sense. In a running race the first and second cannot be added to give third!

The Words First & Second

It is interesting to see that from number 3 onwards, the ordinal number is in some ways related to the cardinal name - Three > Third, Four> Fourth etc.

But words First and Second seem to be independent of words One & Two. We do not say Oneth and Twoth. This seems to be true of most languages. This is possibly an indication that concepts & words denoting order like first & second were used much before or independent of the concept of numbers.

The Use of Cardinal Names While Counting

In daily life transactions, many a time, Ordinal numbers are also referred to as One, Two, Three. This can lead to confusions in the minds of children. Though this confusion will go away when they grow up, it could delay their understanding of number sense and other related concepts. Let us see few such instances.

Teachers introduce the idea of Number as quantity (Number Sense) to children through counting. A child when asked to find the number of fingers in one of the hands is taught to recite the sequence One, Two, Three ..... at the same time touching each of the fingers. Hence at some point in the counting process, a child touches one finger and calls out "Two". At another point, it touches another (single) finger and calls out "Three". If the counting is done in a different order, the child could be touching the same finger and calling out different number names.

This assignment of different numbers, like Two or Three, to a single finger could confuse the child, unless the teachers point out that when it calls out "two" it is actually implying "second".

All schools have sports days and a podium on which the first three medal winners stand and receive their medals. The numbers written on the podium are 1, 2 and 3 which are familiar to children as quantities. In the above case they actually represent first, second and third, though they are represented as 1, 2, 3.

It is very clear that First + Second is definitely not Third and the operation has no meaning.

While counting, numbers are used both in ordinal and cardinal senses!

The numbers we call out, one, two, three etc are actually ordinal numbers indicating their order.

But the last number thus called out, say twenty eight, has both an ordinal and cardinal interpretation. On the one hand it means that the last object counted is the twenty eighth. It also means that the total number of objects in that set was twenty eight.

Teaching Ordinal Numbers

The difference between the cardinal and ordinal use of numbers can be brought out by using the terminology "First", "Second", "Third" etc.

We can do the following activity for teaching the difference between cardinal and ordinal forms. Make a group of children to stand in a line and give instructions like "Two boys" to step out or "the Second" boy to step out.

In this activity, the directional nature of ordinal numbers will also come out clearly. A student who may be "second" from one end of the line could be "fifth" from the other end of the line.

Hence it may be necessary to say "second from the left".

Children should be trained in calling out numbers in Ordinal form as well as Cardinal form. When they reach higher classes, it can be clarified to them that actually when they are counting a collection, though they call out One, Two and Three, they actually mean "First", "Second" and "Third".

Use of Roman Numerals as Ordinal Numbers

In common usage, we use Roman numerals to indicate Ordinal Numbers. Kings are called James I or James III or different sectors in modern cities are denoted using Roman Numerals.