Natural number

Natural numbers is special set, class of equivalence on the sets, that indicates the amount of their elements, with special properties postulated. Each element of this class of equivalence is called 'natural number. This set also has name \(\mathbb N\).

In order to include the character \(\mathbb N\) in the Latex or in the TORI format, the following sequence of asscii-characters can be typed: \(\mathbb N\).

Some obsolete typesetting systems imply the searching of such a character in the labyrinth of menu.

The basic properties of elements of \(\mathbb N\) are postulated as axioms; other are deduced on the base of logic. This article supposes that the logic is already constructed; so, the axioms of the Boolean algebra are assumed.

Definitions
Unity as class of equivalence of sets that have huts one unique element. This class of equivalence is denoted with character 1.

Then, "two" can be defined as class of equivalence of combination (conjunction) of a set that has one element to another set that also has one element. Such class of equivalence is denoted with symbol 2.

In the similar way one can define 3 and so on.

However, such a construction implies that the expression "has just a single element" already has some certain sense. Practically, the definition of such expressions is neither easier, not shorter than the explicit declaration of the existence of natural numbers, satisfying the axioms below; then the equivalence of these numbers to certain sets appears as a scientific concept. The goal of TORI is the application, id est, the construction of applicable tools. Therefore, ASSUME, that there exist set of objects, called "natural numbers", denoted also as \(\mathbb N\), satisfying the axioms below.

Operations and notations
Operations with these objects are called arithmetical operations. The unary arithmetic operation is expressed in the following way: name of operation, open parenthesis, natural number, closer parenthesis. The most elementary operations is unitary increment; its name is ++. Such an operation is defined by its properties, any stuff satisfying such properties can be called "unity increment". Example: \(2= +\!+(1)\).

The so-called "binary" operations are those that are expressed in the following way: open parenthesis, natural number, name of the operation, natural number, closed parenthesis. Usually, the the binary operation returns either Boolean value (true or false), or, again, natural number. (However, for construction of other numbers, the the operation with results in some other set can be constructed.)

The numbers are usually denoted with Italics characters of the Latin alphabet, the same alphabet used to write this article. The advanced typesetting systems, such as MathJax, used at TORI, or in Latex (accepted in the scientific journals, the text between the dollar sigs is interpreted in mathematical mode, to any ascii letters appear with Italics. For example, in order to denote some number with first character of the Latin alphabet, id set, \(a\), it is sufficient to type \(s\).

The statement that some \(a\) is natural number, can be expressed with formula \(a\in\mathbb N\). Below, it is assumed that all letters denote natural numbers.

Axioms
Operation of relation are "smaller than", "equal" and "larger than". Then have short names \(<\), \(=\) and \(>\). They are binary operations, and the result of each of such operation is logic value true or false.

Then important operations are unity increment, summation and multiplication. They have short names ++, + and *. The inverse operations are unity decrement, subtraction and division; they have short names --, - and /.

The following axioms are postulated for the basic operations

For any \(a\) и \(b\), one and the only one of the relations
 * \(a < b\),
 * \(a = b\),
 * \(a > b\)

has value true, other two have value false. (in order to use word "two", this notation have been defined above). Then,
 * if \(a\!=\!b\) then \(b\!=\!a\),
 * if \(a\!<\!b\) then \(b\!>\!a\),
 * if \(a\!>\!b\) then \(b\!<\!a\).

It is assumed that in any mathematical expression, one may replace any natural number to any other natural number, considered as equal and such a replacement does not change the value of the expression. Usually, in mathematics, the operation of equality (equivalence) for any objects is defined in such a way, that such a property holds.

For natural numbers \(a\), \(b\) and \(c\), the following relations hold:
 * if \(a=b\) and \(b=c\), then \(a=c\);
 * if \(ab\) and \(b>c\), then \(a>c\);

Note, that, according to the Boolean definition of logic, operation "or" is not exclusive; and the additional operations "more or equal" and less than equal" with short names \(\ge\) and \(\le\) are defined in the following way:
 * \((a \le b) \) \(=\) \(\Big( (a < b)  {~\mathrm{or}~}  (a=b) \Big)\)
 * \((a \ge b)\) \(=\) \(\Big( (a > b)  {~\mathrm{or}~}  (a=b) \Big)\)

Then, for example,
 * \(\Big ( a \le b) {~\mathrm {and}~} ( a \ge b) \Big)\) \(=\)  \((a\!=\!b) \)

Note that in the up to last case, the symbol = is used to indicate the equivalence of the logic variables, while in the last case such symbol indicates the equivalence (equality) of natural numbers.

Such an overload is typical for the mathematical notations; sense of the operation depends on the set the arguments belong to. In the similar way, the same sign is used to indicate equivalence of integer numbers, rational numbers and other (usually, more complicated) objects.

Axioms of unity increment
At set \(\mathbb N\) , the unity increment operation is defined with result in \(\mathbb N\). This operation is called \(+\!+\). With respect to number \(a\), the number \(+\!+(a)\) is called sequent. With respect to number \(+\!+(a)\), number \(a\) is called precedent.

For each number from \(\mathbb N\) there exist, and only unique sequent element.. If for \(a \in\mathbb N\) the number \(b\) is precedent, and number \(c\) is precedent, then \(b\!=\!c\), id est,
 * if \(a=++(b)\) and \(a=++( c )\) then \(b=c\)

If for \(a\in \mathbb N\) there is no precedent \(b\in \mathbb N\), then \(a\) is called unity. For this case, the special symbol is used, 1. It is presumed that all unities are equal.

If \(a = ++(b)\) and \(a = ++(c)\) then \(b=c\).

Axioms of summation
For any \(a \in \mathbb N\) and \(b \in \mathbb N\) the operation of summation is defined, and the result of the operation is also element of \(\mathbb N\). This is expressed as и \((a+b)\). For any \(a\), \(b\), \(c\) \(\in\) \(\mathbb N\), the following relations hold:
 * \(+\!+(a)=a\!+\!1\)
 * \(a\!+\!b>a\)
 * \((a\!+\!b)=(b\!+\!a)\)
 * \(((a\!+\!b)+c)= (a+(b\!+\!c))\)
 * if \(a = (b+c)\) and \(a = (d+c)\), then \(b = d\).

Similar axioms usually take place also for more complicated mathematical objects. Often, some of the parenthesis are omitted; sometimes, this causes confusions.

Axioms of multiplivation
The axioms of multiplication of the natural numbers are very similar to those of summation. For all \(a\), \(b\) \(\in\) \(\mathbb N\), the multiplication \({\rm product}(a, b)\) which is also \((a*b)\) exists such that the following relations hold:
 * \(a*1=a\)
 * \(a*b =b*a\)
 * \((a+b)+c= a+(b+c)\)

if \(a = b*c\) and \(a = d*c\) then \(b = d\).

Axiom of distributivity
For all \(a\), \(b\), \(c\) \(\in\) \(\mathbb N\), the following relation holds:
 * \(a*(b+c) = a*b+a*c\)

Analogy with logics
In the axioms above, the + looks very similar to the boolean "or", while * looks pretty similar to the Boolean "and". However, the distributivity appears only in the unilateral manner; the relation above at the swapping of signs + and * gives the wrong
 * \(a+(b*c) = (a+b)*(a+c)\)

Such a relation would mean that
 * \(a=(a*a)+ (a*b)+(a*c)\),

Giving contraction with the axiom that for any \(a\) and \(d\), \(a<(a+d)\) = "true".

Names
Some natural number have simple names. In prticular,
 * 2 = ++(1)
 * 3 = ++(3)
 * 4 = ++(4)
 * 5 = ++(5)
 * 6 = ++(6)
 * 7 = ++(7)
 * 8 = ++(8)
 * 9 = ++(9)

There are special rules for generation of names of natural numbers larger than 9. In the human civilization of 17–21 centuries, the most usual is the Aramean (arabian) positional system of numeration.

Inverse operations
Operations "--" "-" and "/" are defined as follows: For \(a\in math N\), the precedent \(c = --(a)\) is natural number such that \(++b = a\). For \(a, b \in math N\), the difference \(c= a-b\) is natural number such that \(c+b=a\). For \(a, b \in math N\), the ratio \(c= a/b\) is natural number such that \(c*b=a\).

The inverse operations above are defined not for all natural numbers. In order to extend the applicabilities of the inverse operations, the more complicated objects are introduced: integer numbers, rational numbers, real numbers, and so on.

Theorems
There are many theorems about natural numbers; some of them are very beautiful. Few simple examples are suggested below.

Theorem. The difference of natural numbers \(a\) and \(b\) is natural numbers if and only if \(a > b\).

(to prove, use the axiom that sum is larger than any of summands.)

Theorem. \(2+2=4\)
 * Proof. Using the definition of symbol 2 and axioms above, transform the left hand side of the requested relation:
 * \(2 = ++1 = 1+1\)
 * \(2+2 = 2 + (1+1) = (2+1)+1 = 3+1 = ++(3) = 4\)

(end of proof)

Theorem. \(2*2=4\)
 * Proof. Using the definition of symbol 2 and axioms above, transform the left hand side of the requested relation:
 * \(2*2 = (1+1)*2 = 1*2 + 1*2 = 2+2\)
 * Then, from the previous theorem, \(2*2=4\).

The tables of summation and multiplication can be presented as theorems. In centuries 18–20, the children had to remember such tables instead of to prove them, in the similar way, as they still have to memorize the Euclid's axioms instead of to deduce them.

The special branch of mathematics is dedicated to the problem of division of the natural numbers, id set, the analysis of the condition of the existence of the ratio of two natural numbers in the set of natural numbers.

More concepts
As an exercise, one may consider the concepts that contradict the postulates above. One of such intents is considered in the article Mizugadro's number . Some economics or technological concepts require the revisions of the rules of arithmetics above.

The financial piramides usually pretend that each of participants gets back more money than the inscription fee. At the law of conservation of the number of money bills (if the organizes do not print banknotes by themselves), the sum should be dependent on the order of summands.

In the similar way, for the inertial propulsion

, developed in the Russian Khrunichev's institute, the sum of components of the momentum of each particle is declared to be dependent on the order of summation; the special tornado–like movement is declared to alliterate the order of summation of the momenta in such a way that the device generates the support-less force, violating the law of conservation of Energy–Momentum. Such a violation implies the break of the axioms of arithmetics.

The paragraph above refers to the fraud, quite analogous to the report about the spending of some liquid for the washing of the optical axes. As a more realistic concept, one may try to reduce the amount of axioms above, replacing them to theorems. (id set, nerving of them on the base of remaining axioms.) Such a reduction worth as an exercise, but for the application in the Mathematical Analysis and physics the postulates suggested seem to be sufficient.

Conclusions
The rules of summation multiplication established for the natural numbers, hold also for more complicated objects - integer numbers, rational numbers, real numbers, etc. Such objects can be either postulated with their properties, or constricted from relatively simpler objects.