Arithmetic/Introduction to Natural Numbers
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The ability to count things has been essential throughout the ages. Over time, several systems for counting things were developed; the first of which was the natural numbers. As a set, the natural numbers can be written like so: {1, 2, 3, . . .}. If we also include the number zero (0) in the set, it becomes the whole numbers: {0, 1, 2, 3, . . .}.
[edit] Formulation
The whole numbers can be formed in many ways. The easiest way is to use what is called an inductive definition. This is when we define the first of a series of numbers, and then make it possible to derive any given number's successor so that given any number we can always find the next. The first of the whole numbers is 0. The way we can derive the next is to simply add one to the previous number. This is easily demonstrated: 0 + 1 = 1, so zero's successor is one; 1 + 1 = 2, so one's successor is two; 2 + 1 = 3, so two's successor is three; and this can be continued "ad infinitum," which is just a Latin phrase meaning "to infinity".
[edit] Uses
The natural numbers are used for three main purposes: For counting, ordering, and defining other concepts. To count something using the natural numbers, you must simply assign one and only one natural number to one element of a group of objects, starting with one. To the next object, selected arbitrarily, that has not been assigned a number, you would assign the next number in the group of natural numbers and then proceed to move on to the next, until all of the items have been assigned a number or in rare cases ad infinitum. The attentive will notice that this is once again, an inductive definition: we define the first term and come up with a way of deriving any given term's successor.
Ordering is where you assign natural numbers to members of a group not arbitrarily, but with some property in mind. To do this, you select the object that has the greatest value of that property (i.e. the biggest, the smartest, the fattest, etc. . .) and assign it the natural number one, then you move on to the next object with the next greatest value of that property and assign it the next natural number, in this case, two. You then proceed to the next object with the next greatest value of that property and assign it, once again, the next natural number, repeating this step until all objects have been ordered or, in rare cases, ad infinitum. Once again, we use an inductive definition.
It should be noted that in all of the above cases zero does not come into play. Zero is a unique case where, in the case of counting, you do not have any objects to assign the first natural number to. For example, if you are attempting to count the amount of apples you own, and you own no apples, then the amount you count is zero. With ordering, the number zero is never used because if you have nothing to order, you can never assign the value of one to anything, and there is no 0th place.
The natural numbers also play an integral part in the definition of many other mathematical concepts such as the very concept of mathematical induction we have used to define three concepts within this page. It should be noted that since the definition used on this page uses mathematical induction, we cannot use it in a more formal situation to define the natural numbers, as the concept of induction is dependant on the numbers already being defined. Students who have an understanding of set theory can find the formal definition of natural numbers on Natural Numbers.
[edit] Properties
One notices that the natural numbers go on forever, with any singular one of them having an infinite number of successors, as any successor has a successor, and that successor has a successor onwards to infinity. Yet in spite of the infinite size, we can still count the numbers. This makes the set countably infinite. Mathematicians have created a whole set of special numbers called the cardinal numbers to describe the different sized infinities; in this case, the set of natural numbers is aleph-null sized. This is important to remember for further studies in mathematics.
