Natural numbers are numbers which arise at the account, numbering and transfer of objects. Negative and nonintegral numbers, i.e. rational, material and others do not concern them.
To definition of natural numbers allocate two approaches. First, it numbers which are used at transfer of objects or at their numbering (fifth, sixth, seventh). Secondly, at designation of quantity of objects (one, two, three). The set of natural numbers is infinite because for any natural number there will be other natural number which will be more.
Over natural numbers fundamental operations and additional are made. Operations of addition, exponentiation and multiplication belong to fundamental. Also the ring of integers is defined by binary operations of addition and multiplication. These operations are called closed, i.e. to operations which do not bring result out of a set of natural numbers. At exponentiation it is necessary to consider that if the indicator and the basis, natural numbers, then result is also natural number.
Also in addition allocate two more operations: subtraction and division. But these operations are defined not for all natural numbers. For example, it is impossible to divide into zero. At subtraction the natural number from which subtract has to be less or equally to number (if to consider zero natural number), which is subtracted. The set of natural numbers has a number of properties. First, properties of operations of addition. For any couple of natural numbers singular called their sum is defined. For it the following relations are carried out: x+y=x+y (property of commutativity), x+(y+ ß)= (x+y) + with (property of associativity). Secondly, properties of operations of multiplication. For any couple of natural numbers singular called by their work is defined. For it the following relations are carried out: x*y=y*x (property of commutativity), x * (y*c)= (x*y) of *c (property of associativity).