With natural numbers you can create infinite sets.
Natural numbers are those that allow the elements of a set to be counted. This is the first set of numbers that was used by humans to count objects. One (1), two (2), five (5) and nine (9), for example, are natural numbers.
Originating in the Latin numĕrus , the concept of numbers is used in mathematics to refer to the signs or set of signs that serve to express a quantity in relation to its unit . There are different groups of numbers, such as integers , real numbers and others.
There is controversy regarding considering zero (0) as a natural number. Generally, Set Theory includes zero within this group, while Number Theory prefers to exclude it.
Examples of natural numbers
The natural numbers are infinite, so the examples of them are also infinite. 3, 25, 698, 3450, 187790 and 826898123 are just a few examples of natural numbers.
An even number , an odd number , or a prime number can be natural numbers. Negative numbers , on the other hand, are not, since they do not make it possible to count the elements of a set.
It could be said, as we already indicated, that natural numbers have two main uses: they are used to specify the size of finite sets and to describe what position an element occupies within an ordered sequence.
However, in addition to these two major functions mentioned, with natural numbers we can also carry out both the identification and differentiation of the various elements that are part of the same group or set. Thus, for example, within a football club each member has a number that distinguishes them from the rest. As an example of this, the following phrases would serve: “Manuel is member number 3,250 of the Barcelona Football Club” , “ My grandfather Ricardo was member number 4 of the club in my neighborhood” .
A clothing store can also number its items in order to organize its stock and facilitate its operations. In this way, it identifies blue pants as item 104 and a white cotton t-shirt as product 278, for example, by appealing to different natural numbers.
It is possible to arrange the figures of natural numbers on an endless number line.
Main features
In addition to the above, we cannot ignore the fact that the natural numbers are ordered . In this way, thanks to this order the numbers can be compared with each other. So, for example, we could emphasize in this sense that 8 is greater than 3 or that 1 is less than 6.
Likewise, another of the qualities that differentiate natural numbers is that they are unlimited , as we noted above. That means that whenever 1 is added to one of them it will give rise to another different natural number.
For all these reasons, these numbers can be represented in a straight line, always from smallest to largest. Thus, once we mark 0 on the line, we will proceed to establish the rest of the number (1, 2, 3...) to the right of that.
An irrational number is not part of the natural numbers.
Other peculiarities of natural numbers
Natural numbers belong to the set of positive integers : they do not have decimals, they are not fractional and they are found to the right of zero on the real line. They are infinite, since they include all the elements of a sequence (1, 2, 3, 4, 5...), but from what has been said they do not include any decimal number or a fraction .
Due to their characteristics, natural numbers constitute a closed set for addition and multiplication operations since, when operating with any of their elements, the result will always be a natural number: 5+4=9, 8×4=32. The same does not happen, however, with subtraction (5-12= -7) or with division (4/3=1.33).
Suppose a child has a bag with five candies. It can be said that the bag represents a set with 5 elements . It is not possible to subtract 8 from that set (the result would be a negative number: there is no way to count those candies negatively) or divide its components by 6 (a decimal number would be obtained, which would imply that the candies would no longer exist as wholes as such. ). On the other hand, you can add candy to the bag or eventually multiply its quantity, also incorporating more.
Peano's axioms
A series of postulates developed by the Italian mathematician Giuseppe Peano (1858-1932) for the definition of natural numbers is known as Peano's axioms . Each Peano axiom defines a characteristic or property of these numbers.
Peano's five axioms are as follows:
- 1 is a natural number.
- Every natural number has a successor number.
- 1 has the peculiarity of not being the successor of another natural number (this idea leaves 0 out of this set).
- If there are two natural numbers with the same successor, those natural numbers are the same number.
- When 1 is part of a set and given any natural number, the successor also integrates the set, all the natural numbers belong to the set in question.