Cardinal & Ordinal Numbers

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In daily life, numbers are used in 2 senses – as answers to &ldquo;How Many?&rdquo; as well as &ldquo;Which item or What is the order of this item?&rdquo; When they answer the first question (I have 3 pencils), they are called Cardinal numbers. When they form part of the second question (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 Kelambakkam in Kanchipuram District for the Indian Postal System.

Cardinal numbers tell the quantity. Ordinal numbers inform us about the order or position. Cardinality of a given set does not change. But its order can change depending on the direction of counting.

The idea of ordinal numbers arises out of the cardinal property of numbers which allows them to be arranged in an order. Cardinal numbers are One, Two, Three &hellip;&hellip;. etc and Ordinal numbers are First, Second, Third etc. The 4 arithmetic operations can be done on numbers, only when they are used in the cardinal sense.

It is interesting to see that from number 3 onwards, the ordinal number is in some ways related to the cardinal name – Three &gt; Third, Four&gt; Fourth etc. But words First and Second seem to be independent of words One &amp; Two. We do not say Oneth and Twoth. This seems to be true of most languages. This is possibly an indication that concepts of order like first &amp; second were formed much before or independent of the concept of numbers.

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 &hellip;. 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 &ldquo;Two&rdquo;. At another point, it touches another (single) finger and calls out &ldquo;Three&rdquo;. 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 numbers, like Two or Three, to a single finger could confuse the child, unless the teacher point out that when it calls out &ldquo;Two&rdquo; it is actually meaning &ldquo;Second&rdquo;.

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.

The difference between the cardinal and ordinal use of numbers can be brought out by using the terminology &ldquo;First&rdquo;, &ldquo;Second&rdquo;, Third&rdquo; etc. It is very clear that First + Second is definitely not Third and has no meaning.

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 &ldquo;Two boys&rdquo; to step out or &ldquo;Second&rdquo; boy to step out.

In this activity, the directional nature of ordinal numbers will also come out clearly. A student who may be &ldquo;Second&rdquo; from one end of the line could be &ldquo;Fifth&rdquo; from the other end of the line.

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 &ldquo;First&rdquo;, &ldquo;Second&rdquo; and &ldquo;Third&rdquo;.

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.

While counting, one particular number is both ordinal and cardinal in nature!

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.

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