What is the difference between ordinal and cardinal numbers

Again the first set of numbers can be arranged in ascending order considering the size of each element and comparing them. In the process of ordering, the numbers are considered as cardinals. Likewise, the set of all nonnegative integers can be ordered in a set; i. But in this case, the size of the set becomes infinite, and giving it in terms of ordinals is not possible. No matter how large a number you pick to give the size of the set, still there will be numbers left out of the set you pick and which are nonnegative integers.

Formally the ordinal number is the order type of a well ordered set. Therefore, the ordinal number of the finite sets can be given by cardinal numbers, but for infinite sets ordinal is given by transfinite numbers such as Aleph Home » Cardinal Numbers vs. It seems difficult to comprehend, but numbers can be used for other purposes than signifying quantity.

Sometimes, for instance, you might want to outline a sequence of events in the correct order. To simplify these pursuits, English has both cardinal and ordinal numbers.

If you care about concepts such as dimension or measure , and demand that they be preserved by any functions you consider, then you can never bring a short line segment into one-to-one correspondance with a long one, or with a square. However, if you allow arbitrary functions, which may completely ignore the structural notions that you cherish, then you may obtain results which you find surprising, or even revolting to your intuition.

This would ultimately be because there is a conflict between the ideas that you wish to consider, and the way in which you are considering it. Actually, the difference cal already be seen for finite numbers, although they get really manifest only in the infinite numbers. Cardinals are about the question "how many". For example, there are ten athletes at the competition. Ordinals are about the order. There's the winner, then there is the second one, then the third one, and so on.

Now for finite sets like the ten athletes above , there is essentially only one way to arrange them ignoring the selection who gets first etc. However, as soon as we get infinite sets, this changes dramatically. Consider the natural numbers.

This amount of course doesn't depend on how we arrange them. But now there are many substantially different ways to arrange them; indeed, even more such ways than there are natural numbers.

However not all possible ways to arrange them correspond to an ordinal number; the ordinal numbers correspond to so-called well-orders, that is, orders where from any subset you can still say which one came first. This is for example not the case for the integers ordered by size; if you look at the negative numbers, there is no first one, as there's always one preceding it. For the natural numbers, the most obvious well-ordering is the usual ordering: You can easily say e.

Note that this is the same order type you get when you e. While the exact ordering is different, you can get the original back by just renaming the individual numbers to the one appropriate for its position. But now consider the alternative arrangement of the natural numbers where you first take all odd numbers, and then all even numbers. That is, your ordering now looks like. This is substantially different from the usual ordering: While in the usual ordering, starting from 0 you can reach any specific natural number in a finite number of steps, now this is only true for the odd numbers; to reach an even number, you first have to go past the infinitely many even numbers, and then possibly a finite number of further steps.

And you cannot remove that difference by renaming the numbers; the fact remains that there are numbers that are preceded by infinitely many other numbers. However this is still a well order. You can still ask what is, in this order, about the first prime number 3 , the first common multiple of 12 and 15 still 0 , and the first three-digit number Well, just put the 0 on the left of the 1,2,3, …, instead of the right.

What do you get? Well, exactly the usual order of the natural numbers! So you have the unusual property that addition of ordinal numbers is not commutative. Note that all those examples used the natural numbers, therefore all of them have the same number of elements, that is the same cardinality. The alphabetical ordering isn't important.

Although you can count the elements in each set - they both have three - this isn't what should be done. First take any element from the first set, say 'b' and match it to one of the second set, say 'B'; carry on doing this until either set is empty or both. If the first set is empty before the second then it is smaller in cardinality etc.

However, in mathematics cardinal numbers have a slight different meaning. Cardinal refers to the measuring the cardinality the number of elements present of a set or between two sets. Even in this sense, cardinal numbers must have numerals or whole numbers. The cardinality of a finite set is a natural number, while the transfinite cardinal numbers describe the sizes of infinite sets.

Cardinality is defined in terms of bijective functions, which requires an on-to-one function. For every element in one set, an element must be present in the pairing set.

Ordinal numbers are words that represent rank and order in a set.