Integer

Integers are an extension of the set of natural numbers ^[1] , obtained by adding zero and negative numbers ^{[2] to it} . The need to consider integers is dictated by the inability in the general case to subtract another from one natural number - you can subtract only a smaller number from a larger one. The introduction of zero and negative numbers makes subtraction the same full-fledged operation as addition ^[3] .

Integers on a number line

A real number is an integer if its decimal representation does not contain a fractional part (but may contain a sign). Examples of real numbers:

Numbers 142857; 0; −273 are integers.

Numbers 5½; 9.75 are not integers.

The set of integers is denoted by $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ $\ mathbb {Z}$ (from German. Zahlen - "numbers" ^[4] ). The study of the properties of integers deals with a branch of mathematics called number theory .

Positive and negative numbers

According to its construction, the set of integers consists of three parts:

Natural numbers (or, which is the same, integers are positive). They arise naturally when counting (1, 2, 3, 4, 5 ...) ^[5] .
Zero is the number indicated by $0$ ${\ displaystyle 0}$ $0$ . Its defining property: $0+n=n+0=n$ ${\ displaystyle 0 + n = n + 0 = n}$ $0+n=n+0=n$ for any number $n$ ${\ displaystyle n}$ $n$ .
Negative integers .

Opposite numbers (4 and –4)

Negative numbers when writing are marked with a minus sign in front: $-1,-2,-3\dots$ ${\ displaystyle -1, -2, -3 \ dots}$ $-1,-2,-3\dots$ For every integer $a$ ${\ displaystyle a}$ $a$ there is a unique number opposite to it, denoted by $-a$ ${\ displaystyle -a}$ $-a$ and possessing the property that $a+(-a)=0.$ ${\ displaystyle a + (- a) = 0.}$ $a+(-a)=0.$ If a $a$ ${\ displaystyle a}$ $a$ positively, then the opposite is negative, and vice versa. Zero is the opposite of itself ^[2] .

The absolute value of an integer $a$ ${\ displaystyle a}$ this number is called with a discarded sign ^[6] . Designation: $\left|a\right|.$ ${\ displaystyle \ left | a \ right |.}$

Examples:

\left|4\right|=4;\ \left|-5\right|=5;\ \left|0\right|=0

{\ displaystyle \ left | 4 \ right | = 4; \ \ left | -5 \ right | = 5; \ \ left | 0 \ right | = 0}

Algebraic properties

Three basic arithmetic operations are defined in a set of integers: addition , inverse to addition, subtraction and multiplication . There is also an important operation specific to natural and integer numbers: division with the remainder . Finally, for integers, an order is defined that allows you to compare numbers with each other.

Addition and Subtraction

The following table illustrates the basic properties of addition ^[7] for any integers $a, b, c$ ${\ displaystyle a, b, c}$ :

Property	Algebraic Record
Commutativity ( Reliability )	$a+b=b+a$ ${\ displaystyle a + b = b + a}$
Associativity ( combinability )	$a+\left(b+c\right)=\left(a+b\right)+c$ ${\ displaystyle a + \ left (b + c \ right) = \ left (a + b \ right) + c}$
Zero property	$a+0=a$ ${\ displaystyle a + 0 = a}$
Property of the opposite element	$a+\left(-a\right)=0$ ${\ displaystyle a + \ left (-a \ right) = 0}$

When adding and subtracting integers, the following rules of signs ^[7] ^[8] are fulfilled, which should be taken into account when opening brackets:

-\left(-a\right)=a;\ -\left(a+b\right)=-a-b;\ -\left(a-b\right)=-a+b.

{\ displaystyle - \ left (-a \ right) = a; \ - \ left (a + b \ right) = - ab; \ - \ left (ab \ right) = - a + b.}

The rules for adding integers ^[9] .

When adding integers with the same signs, we must add their absolute values and assign the sign of the terms to it. Example; $-14+\left(-28\right)=-42$ ${\ displaystyle -14+ \ left (-28 \ right) = - 42}$ .
When adding integers with different signs, we must compare their absolute values, subtract the smaller from the larger and assign the sign of the term with the absolute value greater to the result. Examples: $-4+9=9-4=5;\ -9+4=-\left(9-4\right)=-5$ ${\ displaystyle -4 + 9 = 9-4 = 5; \ -9 + 4 = - \ left (9-4 \ right) = - 5}$ .
Subtraction $a-b$ ${\ displaystyle ab}$ for integers it is always feasible, and the result can be found as $a+\left(-b\right).$ ${\ displaystyle a + \ left (-b \ right).}$ Example: $26-51=26+\left(-51\right)=-25$ ${\ displaystyle 26-51 = 26 + \ left (-51 \ right) = - 25}$ .
Geometrically, addition can be visualized as a shift of a number along the numerical axis (see the figure at the beginning of the article), and adding a positive number causes a shift to the right and a negative one to the left. For example, for a number $-3$ ${\ displaystyle -3}$ addition to it $four$ ${\ displaystyle 4}$ means its shift to the right by 4 units; clearly what turns out $+1$ ${\ displaystyle +1}$ . Similarly $-3+\left(-4\right)$ ${\ displaystyle -3+ \ left (-4 \ right)}$ by shifting $-3$ ${\ displaystyle -3}$ 4 units to the left, we get as a result $-7$ ${\ displaystyle -7}$ .
Subtraction can be visualized similarly, but in this case, on the contrary, subtracting a positive number causes a shift to the left, and a negative number - to the right. For example, $5-7$ ${\ displaystyle 5-7}$ biases $five$ ${\ displaystyle 5}$ 7 units to the number $-2$ ${\ displaystyle -2}$ , but $5-\left(-7\right)$ ${\ displaystyle 5- \ left (-7 \ right)}$ shifts it to the right to the number $12$ ${\ displaystyle 12}$ .

Multiplication and

Multiplication of numbers $a, b$ ${\ displaystyle a, b}$ further designated $a\times b$ ${\ displaystyle a \ times b}$ or (only in case of lettering) simply $a b$ ${\ displaystyle ab}$ . The following table illustrates the basic properties of multiplication ^[7] for any integers $a, b, c$ ${\ displaystyle a, b, c}$ :

Property	Algebraic Record
Commutativity ( Reliability )	$a\times b=b\times a$ ${\ displaystyle a \ times b = b \ times a}$
Associativity ( combinability )	$a\times \left(b\times c\right)=\left(a\times b\right)\times c$ ${\ displaystyle a \ times \ left (b \ times c \ right) = \ left (a \ times b \ right) \ times c}$
Unit Property	$a\times 1=a$ ${\ displaystyle a \ times 1 = a}$
Zero property	$a\times 0=0$ ${\ displaystyle a \ times 0 = 0}$
The distributivity of the multiplication relative to addition	$a\times \left(b+c\right)=a\times b+a\times c$ ${\ displaystyle a \ times \ left (b + c \ right) = a \ times b + a \ times c}$

When multiplying integers, the rules of signs ^[7] ^[8] are fulfilled, which should be taken into account when opening brackets:

\left(-a\right)b=a\left(-b\right)=-ab;\ \left(-a\right)\left(-b\right)=ab

{\ displaystyle \ left (-a \ right) b = a \ left (-b \ right) = - ab; \ \ left (-a \ right) \ left (-b \ right) = ab}

Corollary : the product of numbers with the same signs is positive, with different ones it is negative.

The raising to a natural power of integers is defined in the same way as for natural numbers:

a^{n}=\underbrace {a\cdot a\cdot \ldots \cdot a} _{n}

{\ displaystyle a ^ {n} = \ underbrace {a \ cdot a \ cdot \ ldots \ cdot a} _ {n}}

The properties of raising to the power of integers are also the same as that of natural numbers:

\left(ab\right)^{n}=a^{n}b^{n};\quad a^{m}a^{n}=a^{m+n};\quad \left(a^{m}\right)^{n}=a^{mn}

{\ displaystyle \ left (ab \ right) ^ {n} = a ^ {n} b ^ {n}; \ quad a ^ {m} a ^ {n} = a ^ {m + n}; \ quad \ left (a ^ {m} \ right) ^ {n} = a ^ {mn}}

In addition to this definition, a zero degree agreement has been adopted: $a^{0}=1$ ${\ displaystyle a ^ {0} = 1}$ for any whole $a .$ ${\ displaystyle a.}$ The basis for such an agreement is the desire to maintain the above properties for a zero exponent: $a^{0}a^{n}=a^{0+n}=a^{n},$ ${\ displaystyle a ^ {0} a ^ {n} = a ^ {0 + n} = a ^ {n},}$ whence it is clear that $a^{0}=1.$ ${\ displaystyle a ^ {0} = 1.}$

Ordering

$\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ Is a linearly ordered set . The order in it is given by the relations:

\dots -2<-1<0<1<2<\dots

{\ displaystyle \ dots -2 <-1 <0 <1 <2 <\ dots}

An integer is positive if it is greater than zero, negative if it is less than zero. Positive integers are natural numbers and only they. Negative numbers are numbers opposite to positive ones. Zero is neither positive nor negative. Any negative number is less than any positive ^[2] .

For any integers $a, b, c, d$ ${\ displaystyle a, b, c, d}$ the following relations are valid ^[10] .

If a $a<b$ ${\ displaystyle a <b}$ then for any $c$ ${\ displaystyle c}$ will be $a+c<b+c$ ${\ displaystyle a + c <b + c}$ .
If a $a<b$ ${\ displaystyle a <b}$ and $c<d$ ${\ displaystyle c <d}$ then $a+c<b+d$ ${\ displaystyle a + c <b + d}$ .
If a $a<b$ ${\ displaystyle a <b}$ and $c>0$ ${\ displaystyle c> 0}$ then $ac<bc$ ${\ displaystyle ac <bc}$ .
If a $a<b$ ${\ displaystyle a <b}$ and $c<0$ ${\ displaystyle c <0}$ then $ac>bc$ ${\ displaystyle ac> bc}$ .

To compare two negative numbers, there is a rule: the larger is the number whose absolute value is less ^[10] . For example, $-6<-5$ ${\ displaystyle -6 <-5}$ .

Divisibility

Division with the remainder

The division operation, generally speaking, is not defined on a set of integers. For example, cannot be divided $3$ ${\ displaystyle 3}$ on $2$ ${\ displaystyle 2}$ - there is no integer multiplied by $2$ ${\ displaystyle 2}$ will give $3$ ${\ displaystyle 3}$ . But you can define the so-called division with the remainder ^[11] :

For any integers

a, b

{\ displaystyle a, b}

(Where

b\neq 0

{\ displaystyle b \ neq 0}

) there is a single set of integers

q, r

{\ displaystyle q, r}

such that

a=bq+r

{\ displaystyle a = bq + r}

where

0\leqslant r<\left|b\right|.

{\ displaystyle 0 \ leqslant r <\ left | b \ right |.}

Here a is the dividend , b is the divisor , q is the (incomplete) quotient, r is the remainder of the division (always non-negative). If the remainder is zero, they say that division is performed entirely ^[11] .

Examples

When dividing with the remainder of a positive number $a=78$ ${\ displaystyle a = 78}$ on $b=33$ ${\ displaystyle b = 33}$ we get an incomplete quotient $q=2$ ${\ displaystyle q = 2}$ and the remainder $r=12$ ${\ displaystyle r = 12}$ . Verification: $78=33\times 2+12.$ ${\ displaystyle 78 = 33 \ times 2 + 12.}$
When dividing with the remainder of a negative number $a=-78$ ${\ displaystyle a = -78}$ on $b=33$ ${\ displaystyle b = 33}$ we get an incomplete quotient $q=-3$ ${\ displaystyle q = -3}$ and the remainder $r=21$ ${\ displaystyle r = 21}$ . Verification: $-78=33\times (-3)+21.$ ${\ displaystyle -78 = 33 \ times (-3) +21.}$
When dividing with the remainder of the number $a=78$ ${\ displaystyle a = 78}$ on $b=26$ ${\ displaystyle b = 26}$ we get the quotient $q=3$ ${\ displaystyle q = 3}$ and the remainder $r=0$ ${\ displaystyle r = 0}$ , that is, division is performed entirely. To quickly find out if a given number is divisible $a$ ${\ displaystyle a}$ by (small) number $b$ ${\ displaystyle b}$ , there are signs of divisibility .

The theory of comparisons and the Euclidean algorithm are based on the operation of division with the remainder.

Division completely. Dividers

As defined above, the number $a$ ${\ displaystyle a}$ divides (completely) by the number $b$ ${\ displaystyle b}$ if there is an integer $q$ ${\ displaystyle q}$ such that $a=bq$ ${\ displaystyle a = bq}$ . Symbolic notation: $b|a$ ${\ displaystyle b | a}$ . There are several equivalent verbal formulations of the indicated divisibility ^[12] :

$a$ ${\ displaystyle a}$ divided (wholly) into $b$ ${\ displaystyle b}$ .
$b$ ${\ displaystyle b}$ is a divider $a$ ${\ displaystyle a}$ (or: $b$ ${\ displaystyle b}$ divides $a$ ${\ displaystyle a}$ )
$a$ ${\ displaystyle a}$ multiple $b$ ${\ displaystyle b}$ .

Every integer $n$ ${\ displaystyle n}$ not equal to zero or $\pm 1$ ${\ displaystyle \ pm 1}$ has 4 trivial divisors: $1,-1,n,-n$ ${\ displaystyle 1, -1, n, -n}$ . If there are no other divisors, the number is called prime ^[13] .

The notion of the greatest common divisor of two integers, the decomposition of an integer into prime factors, and the basic arithmetic theorem for integers practically coincide (with the possible consideration of the sign) with analogues of these concepts for natural numbers ^[14] .

Integers and real numbers

There are practical tasks in which it is necessary to round the material value to the whole, that is, replace it with the nearest (in one direction or another) whole. Since rounding can be done in different ways, to clarify, you can use the " Iverson symbols " ^[15] :

\lfloor x\rfloor

{\ displaystyle \ lfloor x \ rfloor}

- closest to

x

{\ displaystyle x}

the whole down (function "floor", English floor , or "the whole part "). Gauss notation is also traditionally used.

[x]

{\ displaystyle [x]}

or Legendre designation

E\left(x\right)

{\ displaystyle E \ left (x \ right)}

.

\lceil x\rceil

{\ displaystyle \ lceil x \ rceil}

- closest to

x

{\ displaystyle x}

whole for the greater side (function "ceiling", English ceiling ).

Depending on the specifics of the problem statement, other methods may also be encountered: round to the nearest integer or cut off the fractional part (the last option for negative $x$ ${\ displaystyle x}$ differs from the “integer part” function).

Another class of problems connecting integers and real numbers is the approximation of a real number by the ratio of integers, that is, a rational number . It is proved that any real number can be approximated with any desired accuracy by rational, the best tool for such an approximation is continuous (continued) fractions ^[16] .

History

The development of mathematics began with practical counting skills (one, two, three, four ...), therefore natural numbers arose even in the prehistoric period as idealization of a finite set of homogeneous, stable and indivisible objects (people, sheep, days, etc.). Addition appeared as a mathematical model of such important events as the union of several sets (herds, bags, etc.) into one, and subtraction reflected, on the contrary, the separation of part of the set. Multiplication for natural numbers appeared as a package addition, so to speak: 3 × 4 meant the sum “ 3 times 4”, that is 4 + 4 + 4 . Properties and the relationship of operations were discovered gradually ^[17] ^[18] .

The initial step towards the expansion of natural numbers was the appearance of zero; apparently the first to use this symbol was Indian mathematicians. At first, zero was used not as a number, but as a figure in the positional record of numbers, then it gradually began to be recognized as a full-fledged number, indicating the absence of anything (for example, the complete ruin of a merchant) ^[19] .

Negative numbers were first used in ancient China and in India, where they were considered as a mathematical image of "debt." Ancient Egypt , Babylon and Ancient Greece did not use negative numbers, and if negative roots of the equations were obtained (when subtracting), they were rejected as impossible. The exception was Diophantus , who in the III century already knew the “rule of signs” and knew how to multiply negative numbers. However, he considered them only as an intermediate stage, useful for calculating the final, positive result. The usefulness and legitimacy of negative numbers was affirmed gradually. The Indian mathematician Brahmagupta (VII century) already considered them on a par with positive ones ^[20] .

In Europe, recognition came a thousand years later, and even then for a long time negative numbers were called "false", "imaginary" or "absurd". The first description of them in European literature appeared in Leonard Pisansky 's Book of the Abacus (1202), which also interpreted negative numbers as debt. Bombelli and Girard in their writings considered negative numbers quite acceptable and useful, in particular, to indicate a lack of something. Negative numbers were freely used by Nicolas Schüke (1484) and Michael Stiefel (1544) ^[20] .

In the XVII century, with the advent of analytical geometry , negative numbers received a clear geometric representation on the numerical axis . From this moment comes their full equality. The legalization of negative numbers has led to numerous conveniences - for example, the transfer of the terms of an equation to another part of it became possible regardless of the sign of this term (earlier, say, the equation $x^{3}+ax=b$ ${\ displaystyle x ^ {3} + ax = b}$ and $x^{3}=ax+b$ ${\ displaystyle x ^ {3} = ax + b}$ were considered fundamentally different) ^[21] .

Nevertheless, the theory of negative numbers has long been in its infancy. Pascal , for example, believed that $0-4=0$ ${\ displaystyle 0-4 = 0}$ , since "nothing can be less than nothing" ^[22] . The odd proportion was animatedly discussed. $1:\left(-1\right)=\left(-1\right):1$ ${\ displaystyle 1: \ left (-1 \ right) = \ left (-1 \ right): 1}$ - in it, the first term on the left is greater than the second, and on the right - on the contrary, and it turns out that more is equal to less (“ Arno paradox”). Wallis believed that negative numbers are less than zero, but at the same time more than infinity ^[23] . It was also unclear what the meaning of the multiplication of negative numbers was, and why the product of negative numbers was positive; heated discussions took place on this subject. An echo of those times is the fact that in modern arithmetic the subtraction operation and the sign of negative numbers are denoted by the same symbol ( minus ), although algebraically these are completely different concepts. In 1831, Gauss considered it necessary to clarify that negative numbers basically have the same rights as positive ones, and the fact that they are not applicable to all things does not mean anything, because fractions also do not apply to all things (for example, they are not applicable to counting people) ^[24] .

A complete and completely rigorous theory of negative numbers was created only in the 19th century ( William Hamilton and Hermann Gunter Grassman ) ^[25] .

Application

In Applied Sciences

Marks of integer temperature values on the thermometer scale

Integers are widely used in the study of objects that, by their nature or by the characteristics of the problem statement, are indivisible (for example, people, ships, buildings, sometimes days, etc.). Negative numbers can also be used in such models - say, when planning trade deals, you can indicate sales with positive numbers and purchases with negative numbers. An example from physics is quantum numbers , which play a fundamental role in the microworld; all of them are integer (or half-integer ) numbers with a sign ^[26] .

To solve the problems arising from this, special mathematical methods have been developed that take into account the specifics of the problems. In particular, the solution in integers of algebraic equations (of different degrees) is considered by the theory of “ Diophantine equations ” ^[27] . Integer optimization issues are studied by integer programming ^[28] .

In Computer Science

The integer type is often one of the main data types in programming languages . Integer data types are typically implemented as a fixed set of bits , one of which encodes the sign of a number, and the rest encodes binary digits. Modern computers have a rich set of instructions for arithmetic operations with integers ^[29] .

Place in General Algebra

Hierarchy of number sets:

\mathbb {N}

{\ displaystyle \ mathbb {N}}

- natural numbers

\mathbb {Z}

{\ displaystyle \ mathbb {Z}}

- whole numbers

\mathbb {Q}

{\ displaystyle \ mathbb {Q}}

- rational numbers

\mathbb {R}

{\ displaystyle \ mathbb {R}}

- real numbers

\mathbb {R} \setminus \mathbb {Q}

{\ displaystyle \ mathbb {R} \ setminus \ mathbb {Q}}

- irrational numbers

In terms of general algebra , $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ with respect to addition and multiplication, it is an infinite commutative ring with unity, without zero divisors ( region of integrity ). The ring of integers is Euclidean (and therefore factorial ) and Noetherian , but it is not Artinian . If we expand this ring by adding all kinds of fractions to it (see the field of quotients ), we obtain a field of rational numbers ( $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ ); any division is already possible in it, except division by zero ^[30] ^[31] .

Regarding the addition operation $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ is an abelian group , and therefore also a cyclic group , since every nonzero element $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ can be written as the final sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1) . Actually, $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ is the only infinite cyclic addition group by virtue of the fact that any infinite cyclic group is isomorphic to the group $(\mathbb {Z} ,+)$ ${\ displaystyle (\ mathbb {Z}, +)}$ . Regarding Multiplication $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ does not form a group, since division in a set of integers is, generally speaking, impossible ^[30] .

The set of integers with the usual order is an ordered ring , but it is not completely ordered , since, for example, there is no smallest among negative numbers. However, it can be made quite ordered by defining the non-standard relation “less than or equal to” ^[32] , which we denote $\preccurlyeq$ ${\ displaystyle \ preccurlyeq}$ and define as follows:

a\preccurlyeq b,

{\ displaystyle a \ preccurlyeq b,}

if either

a=b,

{\ displaystyle a = b,}

or

|a|<|b|,

{\ displaystyle | a | <| b |,}

or

|a|=|b|

{\ displaystyle | a | = | b |}

and

a<0<b.

{\ displaystyle a <0 <b.}

Then the order of the integers will be like this: $0\preccurlyeq -1\preccurlyeq 1\preccurlyeq -2\preccurlyeq 2\dots$ ${\ displaystyle 0 \ preccurlyeq -1 \ preccurlyeq 1 \ preccurlyeq -2 \ preccurlyeq 2 \ dots}$ In particular, $-1$ ${\ displaystyle -1}$ will be the smallest negative number. $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ with the new order will be a completely ordered set, but will no longer be an ordered ring, since this order is not consistent with the operations of the ring: for example, from $1\preccurlyeq -2$ ${\ displaystyle 1 \ preccurlyeq -2}$ adding 1 to the left and right, we get the inequality $2\preccurlyeq -1.$ ${\ displaystyle 2 \ preccurlyeq -1.}$

Любое упорядоченное кольцо с единицей и без делителей нуля содержит одно и только одно подкольцо, изоморфное $\mathbb {Z}$ $\mathbb {Z}$ ^[33] .

Логические основания

Расширение натуральных чисел до целых, как и любое другое расширение алгебраической структуры, ставит множество вопросов, основные из которых — как определить операции над новым типом чисел (например, как определить умножение отрицательных чисел), какие свойства они тогда будут иметь и (главный вопрос) допустимо ли такое расширение, не приведёт ли оно к неустранимым противоречиям. Для анализа подобных вопросов надо сформировать набор аксиом для целых чисел.

Аксиоматика целых чисел

Проще всего определить аксиоматику множества целых чисел $\mathbb {Z}$ $\mathbb {Z}$ , если опираться на уже построенное множество натуральных чисел $\mathbb {N}$ $\mathbb {N}$ (которое предполагается непротиворечивым, а свойства его — известными). Именно, определим $\mathbb {Z}$ $\mathbb {Z}$ как минимальное кольцо , содержащее множество натуральных чисел. Более строго, аксиомы целых чисел следующие ^[34] ^[35] .

Z1 : Для всяких целых чисел

a, b

{\displaystyle a,b}

определена их сумма

a+b

{\displaystyle a+b}

.

Z2 : Сложение коммутативно :

a+b=b+a

{\displaystyle a+b=b+a}

. Для краткости оговорку «для всяких

a,b\dots

a,b\dots

» далее, как правило, опускаем.

Z3 : Сложение ассоциативно :

\left(a+b\right)+c=a+\left(b+c\right).

\left(a+b\right)+c=a+\left(b+c\right).

Z4 : Существует элемент 0 (ноль) такой, что

a+0=a

{\displaystyle a+0=a}

.

Z5 : Для всякого целого числа

a

{\ displaystyle a}

существует противоположный ему элемент

-a

{\displaystyle -a}

такой, что

a+\left(-a\right)=0.

a+\left(-a\right)=0.

Z6 : Для всяких целых чисел

a, b

{\displaystyle a,b}

определено их произведение

a b

{\displaystyle ab}

.

Z7 : Умножение ассоциативно :

\left(ab\right)c=a\left(bc\right).

\left(ab\right)c=a\left(bc\right).

Z8 : Умножение связано со сложением распределительными (дистрибутивными) законами:

\left(a+b\right)c=ac+bc;\ c\left(a+b\right)=ca+cb.

\left(a+b\right)c=ac+bc;\ c\left(a+b\right)=ca+cb.

Z9 : Множество целых чисел

\mathbb {Z}

\mathbb {Z}

содержит подмножество, изоморфное множеству натуральных чисел

\mathbb {N}

\mathbb {N}

. Для простоты далее это подмножество обозначается той же буквой

\mathbb {N}

\mathbb {N}

.

Z10 ( аксиома минимальности ): Пусть

M

{\ displaystyle M}

— подмножество

\mathbb {Z}

\mathbb {Z}

, включающее

\mathbb {N}

\mathbb {N}

и такое, что операция вычитания не выводит за пределы

M

{\ displaystyle M}

. Then

M

{\ displaystyle M}

совпадает со всем

\mathbb {Z}

\mathbb {Z}

.

Из этих аксиом вытекают как следствия все прочие свойства целых чисел, в том числе коммутативность умножения, упорядоченность, правила деления нацело и деления с остатком ^[36] . Покажем, например, как вводится порядок целых чисел. Будем говорить, что $a<b$ ${\ displaystyle a <b}$ , если $b-a$ ${\displaystyle ba}$ есть натуральное число. Аксиомы порядка легко проверяются. Из определения сразу следует, что все натуральные числа больше нуля ( положительны ), а все противоположные им меньше нуля ( отрицательны ). Для натуральных чисел новый порядок совпадает со старым ^[37] .

Приведённая аксиоматика целых чисел категорична , то есть любые её модели изоморфны как кольца ^[38] .

Непротиворечивость

Стандартный способ доказать непротиворечивость новой структуры — смоделировать ( интерпретировать ) её аксиомы с помощью объектов другой структуры, чья непротиворечивость сомнений не вызывает. В нашем случае мы должны реализовать эти аксиомы на базе пар натуральных чисел ^[39] .

Рассмотрим всевозможные упорядоченные пары натуральных чисел $\left(a,b\right)$ $\left(a,b\right)$ . Чтобы смысл дальнейших определений стал понятен, сразу поясним, что мы намерены в дальнейшем каждую такую пару рассматривать как целое число $a-b,$ ${\displaystyle ab,}$ например, пары $\left(3,2\right)$ $\left(3,2\right)$ or $\left(6,5\right)$ $\left(6,5\right)$ будут изображать единицу, а пары $\left(1,4\right)$ $\left(1,4\right)$ or $\left(8,11\right)$ $\left(8,11\right)$ будут изображать $-3.$ ${\displaystyle -3.}$

Далее определим ^[40] :

Couples $\left(a,b\right)$ $\left(a,b\right)$ and $\left(c,d\right)$ $\left(c,d\right)$ считаются равными, если $a+d=b+c$ ${\displaystyle a+d=b+c}$ . Это связано с тем, что, как показано в примерах, любое целое число можно представить бесконечным числом пар.
Сложение : сумма пар $\left(a,b\right)$ $\left(a,b\right)$ and $\left(c,d\right)$ $\left(c,d\right)$ определяется как пара $\left(a+c,b+d\right)$ $\left(a+c,b+d\right)$ .
Умножение : произведение пар $\left(a,b\right)$ $\left(a,b\right)$ and $\left(c,d\right)$ $\left(c,d\right)$ определяется как пара $\left(ac+bd,ad+bc\right)$ $\left(ac+bd,ad+bc\right)$ .

It is easy to verify that the results of addition and multiplication do not change if we replace any pair by its equal, that is, a new pair-result will be equal to the previous one (in the sense of Definition 1, the equality). It is also easy to verify that the described structure of pairs satisfies the entire list of axioms of integers. Positive numbers are modeled in pairs. $\left(a,b\right)$ ${\ displaystyle \ left (a, b \ right)}$ , in which $a>b$ ${\ displaystyle a> b}$ , zero represent pairs of the form $\left(a,a\right)$ ${\ displaystyle \ left (a, a \ right)}$ , and couples $\left(a,b\right)$ ${\ displaystyle \ left (a, b \ right)}$ with $a<b$ ${\ displaystyle a <b}$ correspond to negative numbers ^[40] .

This model allows us to clarify how their properties uniquely follow from the axioms of integers; We show this for the "rule of signs." For example, multiplying two “negative numbers” $\left(a,b\right)$ ${\ displaystyle \ left (a, b \ right)}$ and $\left(c,d\right)$ ${\ displaystyle \ left (c, d \ right)}$ , which $a<b,\ c<d$ ${\ displaystyle a <b, \ c <d}$ , by definition we get a pair $\left(ac+bd,ad+bc\right)$ ${\ displaystyle \ left (ac + bd, ad + bc \ right)}$ . Difference $ac+bd-\left(ad+bc\right)$ ${\ displaystyle ac + bd- \ left (ad + bc \ right)}$ is equal to $\left(b-a\right)\left(d-c\right)$ ${\ displaystyle \ left (ba \ right) \ left (dc \ right)}$ , this number is positive, therefore the para-product represents a positive integer, therefore, the product of negative numbers is positive. Any other rule (say, “the product of negative numbers is negative”) would make the theory of integers inconsistent.

The described model proves that the reduced axiomatics of integers is consistent. Because if there was a contradiction in it, it would mean a contradiction in the arithmetic of natural numbers, which is basic for this model, which we previously assumed to be consistent ^[39] .

Cardinality

The set of integers is infinite. Although natural numbers are only part of the set of integers, integers are the same as natural numbers, in the sense that the power of the set of integers is the same as the set of natural numbers - they are both countable ^[41] .

Variations and generalizations

Some algebraic structures are similar in their properties to the ring of integers $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ . Among them:

Gaussian integers . These are complex numbers. $a+bi$ ${\ displaystyle a + bi}$ where $a, b$ ${\ displaystyle a, b}$ - whole numbers. For Gaussian numbers, as well as for ordinary integers, one can define the concepts of divisors , prime and modulo comparisons . An analogue of the basic theorem of arithmetic is valid ^[42] .
Eisenstein integers ^[43] .

Notes

↑ Here we have in mind the oldest understanding of natural numbers with the first unit element: $1,2,3,4,5\dots$ ${\ displaystyle 1,2,3,4,5 \ dots}$
↑ ¹ ² ³ Handbook of Elementary Mathematics, 1978 , p. 111-113.
↑ Elementary mathematics from the point of view of higher education, 1987 , p. 37.
↑ Paul Pollack. Earliest Uses of Symbols of Number Theory (Neopr.) . Date of treatment October 22, 2017.
↑ Elementary mathematics, 1976 , p. 18.
↑ Handbook of Elementary Mathematics, 1978 , p. 114.
↑ ¹ ² ³ ⁴ Elementary Mathematics, 1976 , p. 24-28.
↑ ¹ ² Elementary mathematics from the point of view of higher education, 1987 , p. 39.
↑ Handbook of Elementary Mathematics, 1978 , p. 114-115.
↑ ¹ ² Handbook of Elementary Mathematics, 1978 , p. 172-173.
↑ ¹ ² Division // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2.
↑ Sushkevich A.K. Number Theory. Elementary course. - Kh .: Publishing house of Kharkov University, 1954. - S. 5.
↑ Elementary mathematics, 1976 , p. 20.
↑ The concept of divisibility // Elements of the theory of divisibility: Methodological recommendations for students of the faculty of pedagogy and psychology of childhood / comp. S.V. Pomortseva, O.V. Ivanova. - Omsk: Omsk state. ped University, 2008 .-- 37 p.
↑ Knut D. The art of computer programming. T. 1. The main algorithms. - M .: Mir , 1976 .-- S. 68 .-- 735 p.
↑ Khinchin A. Ya. Chain fractions . - M .: GIFFL, 1960.
↑ Mach E. Cognition and error // Albert Einstein and the theory of gravity. - M .: Mir, 1979. - S. 74 (footnote). - 592 p. : "Before the concept of number arises, experience must exist that, in a sense, equivalent objects exist multiple and invariable ."
↑ Kline M. Mathematics. Loss of certainty. - M .: Mir, 1984. - S. 109-112. - 446 p.
↑ Lamberto Garcia del Cid. Special numbers of other cultures // Wonderful numbers. Zero, 666 and other beasts. - DeAgostini, 2014 .-- T. 21. - P. 115. - 159 p. - (World of mathematics). - ISBN 978-5-9774-0716-8 .
↑ ¹ ² Gleizer G.I. History of mathematics at school. - M .: Enlightenment, 1964. - S. 132-135. - 376 p.
↑ Handbook of Elementary Mathematics, 1978 , p. 113-114.
↑ Sukhotin A.K. The vicissitudes of scientific ideas. M .: Mol. guard. 1991, p. 34.
↑ Panov V.F. Negative numbers // Ancient and young mathematics. - Ed. 2nd, corrected. - M .: MSTU im. Bauman , 2006 .-- S. 399 .-- 648 p. - ISBN 5-7038-2890-2 .
↑ Alexandrova N.V. Mathematical terms. (Reference). M.: Higher School, 1978, p. 164.
↑ Mathematics of the 18th Century // History of Mathematics / Edited by A.P. Yushkevich , in three volumes. - M .: Nauka, 1972 .-- T. III. - S. 48-49.
↑ Sivukhin D.V. § 38. Four quantum numbers of an electron and the fine structure of spectral terms // General course of physics. - M. , 2005. - T. V. Atomic and nuclear physics. - S. 226.
↑ Gelfond A.O. Solution of equations in integers . - M .: Nauka, 1978. - ( Popular lectures on mathematics ).
↑ Karmanov V.G. Mathematical programming. - M .: Nauka , 1986 .-- 288 p.
↑ M. Ben-Ari. Chapter 4. Elementary data types // Programming languages. Practical Benchmarking = Understanding Programming Language. - M .: Mir, 2000 .-- S. 53-74. - 366 p. - ISBN 5-03-003314-9 .
↑ ¹ ² Vinberg E. B. The course of algebra. 2nd ed. - M .: Publishing House of the Moscow Center for Scientific and Technical Education, 2013. - S. 15-16, 113-114. - 590 s. - ISBN 978-5-4439-0209-8 .
↑ Atia M., MacDonald I. Introduction to commutative algebra. - M .: Mir, 1972. - S. 94. - 160 p.
↑ Donald Knut . The Art of Programming, Volume I. Basic Algorithms. - M .: Mir , 1976 .-- S. 571 (15b). - 736 p.
↑ Numerical Systems, 1975 , p. 100.
↑ Numerical Systems, 1975 , p. 95-96.
↑ Encyclopedia of Elementary Mathematics, 1951 , p. 160-162.
↑ Numerical Systems, 1975 , p. 96-98.
↑ Encyclopedia of Elementary Mathematics, 1951 , p. 170-171.
↑ Numerical Systems, 1975 , p. 98.
↑ ¹ ² Numerical systems, 1975 , p. 100-102.
↑ ¹ ² Encyclopedia of Elementary Mathematics, 1951 , p. 162-168.
↑ N. Ya. Vilenkin . Stories about sets . - 3rd ed. - M .: ICMMO , 2005 .-- S. 65-66. - 150 s. - ISBN 5-94057-036-4 .
↑ Okunev L. Ya. Integer complex numbers. - M .: State. student-ped. Publishing House of the People's Commissariat of the RSFSR, 1941. - 56 p.
↑ Eric W. Weisstein. Eisenstein Integer (Neopr.) . Date accessed August 19, 2017.

Literature

Vygodsky M. Ya. Handbook of Elementary Mathematics . - M .: Science, 1978.
- Reprint: M .: AST, 2006, ISBN 5-17-009554-6 , 509 pp.
Zaitsev V.V., Ryzhkov V.V., Skanavi M.I. Elementary mathematics. Repeat course. - Third edition, stereotyped. - M .: Nauka, 1976 .-- 591 p.
Klein F. Elementary mathematics from the point of view of higher. - M .: Nauka, 1987. - T. I. Arithmetic. Algebra. Analysis. - 432 s.
Nechaev V.I. Numerical systems. - M .: Education, 1975 .-- 199 p.
Encyclopedia of Elementary Mathematics (in 5 volumes). - M .: Fizmatgiz, 1951. - T. 1. - S. 160-168. - 448 p.

[1] Here we have in mind the oldest understanding of natural numbers with the first unit element: $1,2,3,4,5\dots$ ${\ displaystyle 1,2,3,4,5 \ dots}$

[VYG111-2] ¹ ² ³ Handbook of Elementary Mathematics, 1978 , p. 111-113.

[_9e6a90d265d73639-3] Elementary mathematics from the point of view of higher education, 1987 , p. 37.

[4] Paul Pollack. Earliest Uses of Symbols of Number Theory (Neopr.) . Date of treatment October 22, 2017.

[_4950b67f0a6dd5cc-5] Elementary mathematics, 1976 , p. 18.

[_88eba4517a55aab3-6] Handbook of Elementary Mathematics, 1978 , p. 114.

[ELEM-7] ¹ ² ³ ⁴ Elementary Mathematics, 1976 , p. 24-28.

[_9e6a90d265d73637-8] ¹ ² Elementary mathematics from the point of view of higher education, 1987 , p. 39.

[_1c531b20f049eef2-9] Handbook of Elementary Mathematics, 1978 , p. 114-115.

[VYG172-10] ¹ ² Handbook of Elementary Mathematics, 1978 , p. 172-173.

[BSE-11] ¹ ² Division // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2.

[12] Sushkevich A.K. Number Theory. Elementary course. - Kh .: Publishing house of Kharkov University, 1954. - S. 5.

[_4950b97f0a6ddb3d-13] Elementary mathematics, 1976 , p. 20.

[14] The concept of divisibility // Elements of the theory of divisibility: Methodological recommendations for students of the faculty of pedagogy and psychology of childhood / comp. S.V. Pomortseva, O.V. Ivanova. - Omsk: Omsk state. ped University, 2008 .-- 37 p.

[15] Knut D. The art of computer programming. T. 1. The main algorithms. - M .: Mir , 1976 .-- S. 68 .-- 735 p.

[16] Khinchin A. Ya. Chain fractions . - M .: GIFFL, 1960.

[17] Mach E. Cognition and error // Albert Einstein and the theory of gravity. - M .: Mir, 1979. - S. 74 (footnote). - 592 p. : "Before the concept of number arises, experience must exist that, in a sense, equivalent objects exist multiple and invariable ."

[18] Kline M. Mathematics. Loss of certainty. - M .: Mir, 1984. - S. 109-112. - 446 p.

[19] Lamberto Garcia del Cid. Special numbers of other cultures // Wonderful numbers. Zero, 666 and other beasts. - DeAgostini, 2014 .-- T. 21. - P. 115. - 159 p. - (World of mathematics). - ISBN 978-5-9774-0716-8 .

[GLE1964-20] ¹ ² Gleizer G.I. History of mathematics at school. - M .: Enlightenment, 1964. - S. 132-135. - 376 p.

[_d421848d49d5e980-21] Handbook of Elementary Mathematics, 1978 , p. 113-114.

[22] Sukhotin A.K. The vicissitudes of scientific ideas. M .: Mol. guard. 1991, p. 34.

[23] Panov V.F. Negative numbers // Ancient and young mathematics. - Ed. 2nd, corrected. - M .: MSTU im. Bauman , 2006 .-- S. 399 .-- 648 p. - ISBN 5-7038-2890-2 .

[24] Alexandrova N.V. Mathematical terms. (Reference). M.: Higher School, 1978, p. 164.

[25] Mathematics of the 18th Century // History of Mathematics / Edited by A.P. Yushkevich , in three volumes. - M .: Nauka, 1972 .-- T. III. - S. 48-49.

[26] Sivukhin D.V. § 38. Four quantum numbers of an electron and the fine structure of spectral terms // General course of physics. - M. , 2005. - T. V. Atomic and nuclear physics. - S. 226.

[27] Gelfond A.O. Solution of equations in integers . - M .: Nauka, 1978. - ( Popular lectures on mathematics ).

[28] Karmanov V.G. Mathematical programming. - M .: Nauka , 1986 .-- 288 p.

[29] M. Ben-Ari. Chapter 4. Elementary data types // Programming languages. Practical Benchmarking = Understanding Programming Language. - M .: Mir, 2000 .-- S. 53-74. - 366 p. - ISBN 5-03-003314-9 .

[VIN-30] ¹ ² Vinberg E. B. The course of algebra. 2nd ed. - M .: Publishing House of the Moscow Center for Scientific and Technical Education, 2013. - S. 15-16, 113-114. - 590 s. - ISBN 978-5-4439-0209-8 .

[31] Atia M., MacDonald I. Introduction to commutative algebra. - M .: Mir, 1972. - S. 94. - 160 p.

[32] Donald Knut . The Art of Programming, Volume I. Basic Algorithms. - M .: Mir , 1976 .-- S. 571 (15b). - 736 p.

[_991400c961cefe8b-33] Numerical Systems, 1975 , p. 100.

[_0f0c92bd73c71983-34] Numerical Systems, 1975 , p. 95-96.

[_4ee1b3449ab312cc-35] Encyclopedia of Elementary Mathematics, 1951 , p. 160-162.

[_366e48275e464c50-36] Numerical Systems, 1975 , p. 96-98.

[_5def59dfd8fabe0b-37] Encyclopedia of Elementary Mathematics, 1951 , p. 170-171.

[_44f71d47156948b9-38] Numerical Systems, 1975 , p. 98.

[NECH100-39] ¹ ² Numerical systems, 1975 , p. 100-102.

[EEM162-40] ¹ ² Encyclopedia of Elementary Mathematics, 1951 , p. 162-168.

[41] N. Ya. Vilenkin . Stories about sets . - 3rd ed. - M .: ICMMO , 2005 .-- S. 65-66. - 150 s. - ISBN 5-94057-036-4 .

[42] Okunev L. Ya. Integer complex numbers. - M .: State. student-ped. Publishing House of the People's Commissariat of the RSFSR, 1941. - 56 p.

[43] Eric W. Weisstein. Eisenstein Integer (Neopr.) . Date accessed August 19, 2017.