Ordinal Number

Ordinal numbers (sometimes shortened to ordinals) are numbers used to show position in an ordered sequence. For example, three runners came in first, fifth, and eighth in the race. Cardinal numbers (sometimes shortened to cardinals) describe quantities. Thus, there were nine runners in the race.

The number 8 can be used in three ways: to tell how many, to tell where in a ranking, and to name someone or something. The girl with the number 8 on her baseball uniform, who is 8th in the batting order, playing on a team that scores 8 runs, is using the same number in each of these ways. When she is 8th in the batting order, she is using the number as an ordinal number. An ordinal number is one that is used to indicate where in an ordered list someone or something occurs. A number used to tell how many is called a cardinal number.

The ordinal name of a number differs somewhat from the cardinal name. In most instances, the cardinal name can be converted into the ordinal name by adding “th.” Thus, the cardinal number one thousand becomes the ordinal number one thousandth; four becomes fourth; and so on. In the case of 1, 2, and 3, however completely different names are used.

The clear distinction between cardinal and ordinal forms of 1 and 2 arises from the way in which the events or things they describe differ. A runner who comes in first comes in ahead of anyone else, and that is what is most notable about the event, not that one runner has crossed the line. Likewise, someone coming in second follows, and that, too, is something that can be noted without consciously counting the two runners. By the time the third runner comes along, counting becomes a helpful if not necessary aid in determining his or her position. The similarity between three and third (and the Latin roots from which they come) reflects this. Beyond 3, counting is almost essential, and the cardinal and ordinal forms are almost the same.

The ordinal name of a number is used in some instances where no ranking is intended or where the ranking is vestigial. The names given to the denominators of common fraction are ordinal names although they signify the number of uniform parts into which each unit is cut. Thus, three-fifths indicates that each unit has been divided into five equal parts, and the fraction represents three of them. However, when the fraction is written with numerals, 3/5, both numerator and denominator are written in the cardinal form.

On the other hand there are times when the cardinal form of a number is used in an ordinal sense. In counting a group of objects, one is putting them into one-to-one correspondence with the numbers 1, 2, 3, .... That is why the counting process works. Nevertheless, a person counts, “one, two, three, . . .,” not, “first, second, third, . . .” Mathematicians who work with infinity have extended the concept of ordinal number to apply to certain classes of infinite numbers as well.