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 which is used to indicate where in an ordered list someone or something occurs. A number used to tell how many is a "cardinal number." A number which is used to name something is neither a cardinal nor an ordinal 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 which 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
|two||second||2nd or 2d|
|three||third||3rd or 3d|
|one hundred||one hundredth||100th|
|one hundred one||one hundred first||101st|
a group of objects one is putting them into one-to-one correspondence with the numbers 1, 2, 3,... in order. That is why the counting process works. Nevertheless one 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.