Clever Geek Handbook
πŸ“œ ⬆️ ⬇️

Integer

Latex integers.svg

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; βˆ’273 are integers.
Numbers 5Β½; 9.75 are not integers.

The set of integers is denoted byZ {\ 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:

  1. Natural numbers (or, which is the same, integers are positive). They arise naturally when counting (1, 2, 3, 4, 5 ...) [5] .
  2. Zero is the number indicated by0 {\ displaystyle 0} {\displaystyle 0} . Its defining property:0+n=n+0=n {\ displaystyle 0 + n = n + 0 = n} {\displaystyle 0+n=n+0=n} for any numbern {\ displaystyle n} n .
  3. Negative integers .
Opposite numbers (4 and –4)

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

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

Examples:|four|=four;|-five|=five;|0|=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 integersa,b,c {\ displaystyle a, b, c}   :

PropertyAlgebraic Record
Commutativity ( Reliability )a+b=b+a{\ displaystyle a + b = b + a}  
Associativity ( combinability )a+(b+c)=(a+b)+c{\ displaystyle a + \ left (b + c \ right) = \ left (a + b \ right) + c}  
Zero propertya+0=a{\ displaystyle a + 0 = a}  
Property of the opposite elementa+(-a)=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:

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

The rules for adding integers [9] .

  1. When adding integers with the same signs, we must add their absolute values ​​and assign the sign of the terms to it. Example;-14+(-28)=-42 {\ displaystyle -14+ \ left (-28 \ right) = - 42}   .
  2. 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:-four+9=9-four=five;-9+four=-(9-four)=-five {\ displaystyle -4 + 9 = 9-4 = 5; \ -9 + 4 = - \ left (9-4 \ right) = - 5}   .
  3. Subtractiona-b {\ displaystyle ab}   for integers it is always feasible, and the result can be found asa+(-b). {\ displaystyle a + \ left (-b \ right).}   Example:26-51=26+(-51)=-25 {\ displaystyle 26-51 = 26 + \ left (-51 \ right) = - 25}   .
  4. 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 itfour {\ displaystyle 4}   means its shift to the right by 4 units; clearly what turns out+one {\ displaystyle +1}   . Similarly-3+(-four) {\ displaystyle -3+ \ left (-4 \ right)}   by shifting-3 {\ displaystyle -3}   4 units to the left, we get as a result-7 {\ displaystyle -7}   .
  5. 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,five-7 {\ displaystyle 5-7}   biasesfive {\ displaystyle 5}   7 units to the number-2 {\ displaystyle -2}   , butfive-(-7) {\ displaystyle 5- \ left (-7 \ right)}   shifts it to the right to the number12 {\ displaystyle 12}   .

Multiplication and

Multiplication of numbersa,b {\ displaystyle a, b}   further designatedaΓ—b {\ displaystyle a \ times b}   or (only in case of lettering) simplyab {\ displaystyle ab}   . The following table illustrates the basic properties of multiplication [7] for any integersa,b,c {\ displaystyle a, b, c}   :

PropertyAlgebraic Record
Commutativity ( Reliability )aΓ—b=bΓ—a{\ displaystyle a \ times b = b \ times a}  
Associativity ( combinability )aΓ—(bΓ—c)=(aΓ—b)Γ—c{\ displaystyle a \ times \ left (b \ times c \ right) = \ left (a \ times b \ right) \ times c}  
Unit PropertyaΓ—one=a{\ displaystyle a \ times 1 = a}  
Zero propertyaΓ—0=0{\ displaystyle a \ times 0 = 0}  
The distributivity of the multiplication relative to additionaΓ—(b+c)=aΓ—b+aΓ—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:

(-a)b=a(-b)=-ab;(-a)(-b)=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:

an=aβ‹…aβ‹…...β‹…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:

(ab)n=anbn;aman=am+n;(am)n=amn{\ 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:a0=one {\ displaystyle a ^ {0} = 1}   for any wholea. {\ displaystyle a.}   The basis for such an agreement is the desire to maintain the above properties for a zero exponent:a0an=a0+n=an, {\ displaystyle a ^ {0} a ^ {n} = a ^ {0 + n} = a ^ {n},}   whence it is clear thata0=one. {\ displaystyle a ^ {0} = 1.}  

Ordering

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

β‹―-2<-one<0<one<2<...{\ 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 integersa,b,c,d {\ displaystyle a, b, c, d}   the following relations are valid [10] .

  1. If aa<b {\ displaystyle a <b}   then for anyc {\ displaystyle c}   will bea+c<b+c {\ displaystyle a + c <b + c}   .
  2. If aa<b {\ displaystyle a <b}   andc<d {\ displaystyle c <d}   thena+c<b+d {\ displaystyle a + c <b + d}   .
  3. If aa<b {\ displaystyle a <b}   andc>0 {\ displaystyle c> 0}   thenac<bc {\ displaystyle ac <bc}   .
  4. If aa<b {\ displaystyle a <b}   andc<0 {\ displaystyle c <0}   thenac>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<-five {\ 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 divided3 {\ displaystyle 3}   on2 {\ displaystyle 2}   - there is no integer multiplied by2 {\ displaystyle 2}   will give3 {\ displaystyle 3}   . But you can define the so-called division with the remainder [11] :

For any integersa,b {\ displaystyle a, b}   (Wherebβ‰ 0 {\ displaystyle b \ neq 0}   ) there is a single set of integersq,r {\ displaystyle q, r}   such thata=bq+r {\ displaystyle a = bq + r}   where0β©½r<|b|. {\ 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 numbera=78 {\ displaystyle a = 78}   onb=33 {\ displaystyle b = 33}   we get an incomplete quotientq=2 {\ displaystyle q = 2}   and the remainderr=12 {\ displaystyle r = 12}   . Verification:78=33Γ—2+12. {\ displaystyle 78 = 33 \ times 2 + 12.}  
  • When dividing with the remainder of a negative numbera=-78 {\ displaystyle a = -78}   onb=33 {\ displaystyle b = 33}   we get an incomplete quotientq=-3 {\ displaystyle q = -3}   and the remainderr=21 {\ displaystyle r = 21}   . Verification:-78=33Γ—(-3)+21. {\ displaystyle -78 = 33 \ times (-3) +21.}  
  • When dividing with the remainder of the numbera=78 {\ displaystyle a = 78}   onb=26 {\ displaystyle b = 26}   we get the quotientq=3 {\ displaystyle q = 3}   and the remainderr=0 {\ displaystyle r = 0}   , that is, division is performed entirely. To quickly find out if a given number is divisiblea {\ displaystyle a}   by (small) numberb {\ 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 numbera {\ displaystyle a}   divides (completely) by the numberb {\ displaystyle b}   if there is an integerq {\ displaystyle q}   such thata=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) intob {\ displaystyle b}   .
  • b{\ displaystyle b}   is a dividera {\ displaystyle a}   (or:b {\ displaystyle b}   dividesa {\ displaystyle a}   )
  • a{\ displaystyle a}   multipleb {\ displaystyle b}   .

Every integern {\ displaystyle n}   not equal to zero orΒ±one {\ displaystyle \ pm 1}   has 4 trivial divisors:one,-one,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] :

⌊xβŒ‹{\ displaystyle \ lfloor x \ rfloor}   - closest tox {\ displaystyle x}   the whole down (function "floor", English floor , or "the whole part "). Gauss notation is also traditionally used.[x] {\ displaystyle [x]}   or Legendre designationE(x) {\ displaystyle E \ left (x \ right)}   .
⌈xβŒ‰{\ displaystyle \ lceil x \ rceil}   - closest tox {\ 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 negativex {\ displaystyle x}   differs from the β€œinteger 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 Γ— 4 meant the sum β€œ 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 β€œrule 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 SchΓΌke (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 equationx3+ax=b {\ displaystyle x ^ {3} + ax = b}   andx3=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 that0-four=0 {\ displaystyle 0-4 = 0}   , since "nothing can be less than nothing" [22] . The odd proportion was animatedly discussed.one:(-one)=(-one):one {\ 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 (β€œ 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 β€œ 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:
N{\ displaystyle \ mathbb {N}}   - natural numbers
Z{\ displaystyle \ mathbb {Z}}   - whole numbers
Q{\ displaystyle \ mathbb {Q}}   - rational numbers
R{\ displaystyle \ mathbb {R}}   - real numbers
Rβˆ–Q{\ displaystyle \ mathbb {R} \ setminus \ mathbb {Q}}   - irrational numbers

In terms of general algebra ,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 (Q {\ displaystyle \ mathbb {Q}}   ); any division is already possible in it, except division by zero [30] [31] .

Regarding the addition operationZ {\ displaystyle \ mathbb {Z}}   is an abelian group , and therefore also a cyclic group , since every nonzero elementZ {\ displaystyle \ mathbb {Z}}   can be written as the final sum 1 + 1 + ... + 1 or (βˆ’1) + (βˆ’1) + ... + (βˆ’1) . Actually,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(Z,+) {\ displaystyle (\ mathbb {Z}, +)}   . Regarding MultiplicationZ {\ 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 β€œless than or equal to” [32] , which we denote β‰Ό{\ displaystyle \ preccurlyeq}   and define as follows:

aβ‰Όb,{\ 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: 0β‰Ό-oneβ‰Όoneβ‰Ό-2β‰Ό2...{\ displaystyle 0 \ preccurlyeq -1 \ preccurlyeq 1 \ preccurlyeq -2 \ preccurlyeq 2 \ dots}   In particular,-one {\ displaystyle -1}   will be the smallest negative number. 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 oneβ‰Ό-2{\ displaystyle 1 \ preccurlyeq -2}   adding 1 to the left and right, we get the inequality 2β‰Ό-one.{\ displaystyle 2 \ preccurlyeq -1.}  

Π›ΡŽΠ±ΠΎΠ΅ упорядочСнноС ΠΊΠΎΠ»ΡŒΡ†ΠΎ с Π΅Π΄ΠΈΠ½ΠΈΡ†Π΅ΠΉ ΠΈ Π±Π΅Π· Π΄Π΅Π»ΠΈΡ‚Π΅Π»Π΅ΠΉ нуля содСрТит ΠΎΠ΄Π½ΠΎ ΠΈ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΎΠ΄Π½ΠΎ ΠΏΠΎΠ΄ΠΊΠΎΠ»ΡŒΡ†ΠΎ, ΠΈΠ·ΠΎΠΌΠΎΡ€Ρ„Π½ΠΎΠ΅ Z{\displaystyle \mathbb {Z} }   [33] .

ЛогичСскиС основания

Π Π°ΡΡˆΠΈΡ€Π΅Π½ΠΈΠ΅ Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл Π΄ΠΎ Ρ†Π΅Π»Ρ‹Ρ…, ΠΊΠ°ΠΊ ΠΈ любоС Π΄Ρ€ΡƒΠ³ΠΎΠ΅ Ρ€Π°ΡΡˆΠΈΡ€Π΅Π½ΠΈΠ΅ алгСбраичСской структуры, ставит мноТСство вопросов, основныС ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… β€” ΠΊΠ°ΠΊ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΠΎΠΏΠ΅Ρ€Π°Ρ†ΠΈΠΈ Π½Π°Π΄ Π½ΠΎΠ²Ρ‹ΠΌ Ρ‚ΠΈΠΏΠΎΠΌ чисСл (Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, ΠΊΠ°ΠΊ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ ΡƒΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΠ΅ ΠΎΡ‚Ρ€ΠΈΡ†Π°Ρ‚Π΅Π»ΡŒΠ½Ρ‹Ρ… чисСл), ΠΊΠ°ΠΊΠΈΠ΅ свойства ΠΎΠ½ΠΈ Ρ‚ΠΎΠ³Π΄Π° Π±ΡƒΠ΄ΡƒΡ‚ ΠΈΠΌΠ΅Ρ‚ΡŒ ΠΈ (Π³Π»Π°Π²Π½Ρ‹ΠΉ вопрос) допустимо Π»ΠΈ Ρ‚Π°ΠΊΠΎΠ΅ Ρ€Π°ΡΡˆΠΈΡ€Π΅Π½ΠΈΠ΅, Π½Π΅ ΠΏΡ€ΠΈΠ²Π΅Π΄Ρ‘Ρ‚ Π»ΠΈ ΠΎΠ½ΠΎ ΠΊ нСустранимым противорСчиям. Для Π°Π½Π°Π»ΠΈΠ·Π° ΠΏΠΎΠ΄ΠΎΠ±Π½Ρ‹Ρ… вопросов Π½Π°Π΄ΠΎ ΡΡ„ΠΎΡ€ΠΌΠΈΡ€ΠΎΠ²Π°Ρ‚ΡŒ Π½Π°Π±ΠΎΡ€ аксиом для Ρ†Π΅Π»Ρ‹Ρ… чисСл.

Аксиоматика Ρ†Π΅Π»Ρ‹Ρ… чисСл

ΠŸΡ€ΠΎΡ‰Π΅ всСго ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚ΡŒ аксиоматику мноТСства Ρ†Π΅Π»Ρ‹Ρ… чисСл Z{\displaystyle \mathbb {Z} }   , Ссли ΠΎΠΏΠΈΡ€Π°Ρ‚ΡŒΡΡ Π½Π° ΡƒΠΆΠ΅ построСнноС мноТСство Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл N{\displaystyle \mathbb {N} }   (ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠ΅ прСдполагаСтся Π½Π΅ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΠ²Ρ‹ΠΌ, Π° свойства Π΅Π³ΠΎ β€” извСстными). ИмСнно, ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΠΌ Z{\displaystyle \mathbb {Z} }   ΠΊΠ°ΠΊ минимальноС ΠΊΠΎΠ»ΡŒΡ†ΠΎ , содСрТащСС мноТСство Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл. Π‘ΠΎΠ»Π΅Π΅ строго, аксиомы Ρ†Π΅Π»Ρ‹Ρ… чисСл ΡΠ»Π΅Π΄ΡƒΡŽΡ‰ΠΈΠ΅ [34] [35] .

Z1 : Для всяких Ρ†Π΅Π»Ρ‹Ρ… чисСл a,b{\displaystyle a,b}   ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π° ΠΈΡ… сумма a+b{\displaystyle a+b}   .
Z2 : Π‘Π»ΠΎΠΆΠ΅Π½ΠΈΠ΅ ΠΊΠΎΠΌΠΌΡƒΡ‚Π°Ρ‚ΠΈΠ²Π½ΠΎ : a+b=b+a{\displaystyle a+b=b+a}   . Для краткости ΠΎΠ³ΠΎΠ²ΠΎΡ€ΠΊΡƒ «для всяких a,b...{\displaystyle a,b\dots }   Β» Π΄Π°Π»Π΅Π΅, ΠΊΠ°ΠΊ ΠΏΡ€Π°Π²ΠΈΠ»ΠΎ, опускаСм.
Z3 : Π‘Π»ΠΎΠΆΠ΅Π½ΠΈΠ΅ ассоциативно : (a+b)+c=a+(b+c).{\displaystyle \left(a+b\right)+c=a+\left(b+c\right).}  
Z4 : БущСствуСт элСмСнт 0 (ноль) Ρ‚Π°ΠΊΠΎΠΉ, Ρ‡Ρ‚ΠΎ a+0=a{\displaystyle a+0=a}   .
Z5 : Для всякого Ρ†Π΅Π»ΠΎΠ³ΠΎ числаa {\ displaystyle a}   сущСствуСт ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΠΏΠΎΠ»ΠΎΠΆΠ½Ρ‹ΠΉ Π΅ΠΌΡƒ элСмСнт -a{\displaystyle -a}   Ρ‚Π°ΠΊΠΎΠΉ, Ρ‡Ρ‚ΠΎ a+(-a)=0.{\displaystyle a+\left(-a\right)=0.}  
Z6 : Для всяких Ρ†Π΅Π»Ρ‹Ρ… чисСл a,b{\displaystyle a,b}   ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΎ ΠΈΡ… ΠΏΡ€ΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ΠΈΠ΅ ab{\displaystyle ab}   .
Z7 : Π£ΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΠ΅ ассоциативно : (ab)c=a(bc).{\displaystyle \left(ab\right)c=a\left(bc\right).}  
Z8 : Π£ΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΠ΅ связано со слоТСниСм Ρ€Π°ΡΠΏΡ€Π΅Π΄Π΅Π»ΠΈΡ‚Π΅Π»ΡŒΠ½Ρ‹ΠΌΠΈ (дистрибутивными) Π·Π°ΠΊΠΎΠ½Π°ΠΌΠΈ: (a+b)c=ac+bc;c(a+b)=ca+cb.{\displaystyle \left(a+b\right)c=ac+bc;\ c\left(a+b\right)=ca+cb.}  
Z9 : ΠœΠ½ΠΎΠΆΠ΅ΡΡ‚Π²ΠΎ Ρ†Π΅Π»Ρ‹Ρ… чисСл Z{\displaystyle \mathbb {Z} }   содСрТит подмноТСство, ΠΈΠ·ΠΎΠΌΠΎΡ€Ρ„Π½ΠΎΠ΅ мноТСству Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл N{\displaystyle \mathbb {N} }   . Для простоты Π΄Π°Π»Π΅Π΅ это подмноТСство обозначаСтся Ρ‚ΠΎΠΉ ΠΆΠ΅ Π±ΡƒΠΊΠ²ΠΎΠΉ N{\displaystyle \mathbb {N} }   .
Z10 ( аксиома ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡŒΠ½ΠΎΡΡ‚ΠΈ ): ΠŸΡƒΡΡ‚ΡŒM {\ displaystyle M}   β€” подмноТСство Z{\displaystyle \mathbb {Z} }   , Π²ΠΊΠ»ΡŽΡ‡Π°ΡŽΡ‰Π΅Π΅ N{\displaystyle \mathbb {N} }   ΠΈ Ρ‚Π°ΠΊΠΎΠ΅, Ρ‡Ρ‚ΠΎ опСрация вычитания Π½Π΅ Π²Ρ‹Π²ΠΎΠ΄ΠΈΡ‚ Π·Π° ΠΏΡ€Π΅Π΄Π΅Π»Ρ‹M {\ displaystyle M}   . ThenM {\ displaystyle M}   совпадаСт со всСм Z{\displaystyle \mathbb {Z} }   .

Из этих аксиом Π²Ρ‹Ρ‚Π΅ΠΊΠ°ΡŽΡ‚ ΠΊΠ°ΠΊ слСдствия всС ΠΏΡ€ΠΎΡ‡ΠΈΠ΅ свойства Ρ†Π΅Π»Ρ‹Ρ… чисСл, Π² Ρ‚ΠΎΠΌ числС ΠΊΠΎΠΌΠΌΡƒΡ‚Π°Ρ‚ΠΈΠ²Π½ΠΎΡΡ‚ΡŒ умноТСния, ΡƒΠΏΠΎΡ€ΡΠ΄ΠΎΡ‡Π΅Π½Π½ΠΎΡΡ‚ΡŒ, ΠΏΡ€Π°Π²ΠΈΠ»Π° дСлСния Π½Π°Ρ†Π΅Π»ΠΎ ΠΈ дСлСния с остатком [36] . ПокаТСм, Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, ΠΊΠ°ΠΊ вводится порядок Ρ†Π΅Π»Ρ‹Ρ… чисСл. Π‘ΡƒΠ΄Π΅ΠΌ Π³ΠΎΠ²ΠΎΡ€ΠΈΡ‚ΡŒ, Ρ‡Ρ‚ΠΎa<b {\ displaystyle a <b}   , Ссли b-a{\displaystyle ba}   Π΅ΡΡ‚ΡŒ Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½ΠΎΠ΅ число. Аксиомы порядка Π»Π΅Π³ΠΊΠΎ ΠΏΡ€ΠΎΠ²Π΅Ρ€ΡΡŽΡ‚ΡΡ. Из опрСдСлСния сразу слСдуСт, Ρ‡Ρ‚ΠΎ всС Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Π΅ числа большС нуля ( ΠΏΠΎΠ»ΠΎΠΆΠΈΡ‚Π΅Π»ΡŒΠ½Ρ‹ ), Π° всС ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΠΏΠΎΠ»ΠΎΠΆΠ½Ρ‹Π΅ ΠΈΠΌ мСньшС нуля ( ΠΎΡ‚Ρ€ΠΈΡ†Π°Ρ‚Π΅Π»ΡŒΠ½Ρ‹ ). Для Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл Π½ΠΎΠ²Ρ‹ΠΉ порядок совпадаСт со старым [37] .

ΠŸΡ€ΠΈΠ²Π΅Π΄Ρ‘Π½Π½Π°Ρ аксиоматика Ρ†Π΅Π»Ρ‹Ρ… чисСл ΠΊΠ°Ρ‚Π΅Π³ΠΎΡ€ΠΈΡ‡Π½Π° , Ρ‚ΠΎ Π΅ΡΡ‚ΡŒ Π»ΡŽΠ±Ρ‹Π΅ Π΅Ρ‘ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΈΠ·ΠΎΠΌΠΎΡ€Ρ„Π½Ρ‹ ΠΊΠ°ΠΊ ΠΊΠΎΠ»ΡŒΡ†Π° [38] .

ΠΠ΅ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΠ²ΠΎΡΡ‚ΡŒ

Π‘Ρ‚Π°Π½Π΄Π°Ρ€Ρ‚Π½Ρ‹ΠΉ способ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚ΡŒ Π½Π΅ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΠ²ΠΎΡΡ‚ΡŒ Π½ΠΎΠ²ΠΎΠΉ структуры β€” ΡΠΌΠΎΠ΄Π΅Π»ΠΈΡ€ΠΎΠ²Π°Ρ‚ΡŒ ( ΠΈΠ½Ρ‚Π΅Ρ€ΠΏΡ€Π΅Ρ‚ΠΈΡ€ΠΎΠ²Π°Ρ‚ΡŒ ) Π΅Ρ‘ аксиомы с ΠΏΠΎΠΌΠΎΡ‰ΡŒΡŽ ΠΎΠ±ΡŠΠ΅ΠΊΡ‚ΠΎΠ² Π΄Ρ€ΡƒΠ³ΠΎΠΉ структуры, Ρ‡ΡŒΡ Π½Π΅ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΠ²ΠΎΡΡ‚ΡŒ сомнСний Π½Π΅ Π²Ρ‹Π·Ρ‹Π²Π°Π΅Ρ‚. Π’ нашСм случаС ΠΌΡ‹ Π΄ΠΎΠ»ΠΆΠ½Ρ‹ Ρ€Π΅Π°Π»ΠΈΠ·ΠΎΠ²Π°Ρ‚ΡŒ эти аксиомы Π½Π° Π±Π°Π·Π΅ ΠΏΠ°Ρ€ Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл [39] .

Рассмотрим всСвозмоТныС упорядочСнныС ΠΏΠ°Ρ€Ρ‹ Π½Π°Ρ‚ΡƒΡ€Π°Π»ΡŒΠ½Ρ‹Ρ… чисСл (a,b){\displaystyle \left(a,b\right)}   . Π§Ρ‚ΠΎΠ±Ρ‹ смысл Π΄Π°Π»ΡŒΠ½Π΅ΠΉΡˆΠΈΡ… ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΈΠΉ стал понятСн, сразу поясним, Ρ‡Ρ‚ΠΎ ΠΌΡ‹ Π½Π°ΠΌΠ΅Ρ€Π΅Π½Ρ‹ Π² дальнСйшСм ΠΊΠ°ΠΆΠ΄ΡƒΡŽ Ρ‚Π°ΠΊΡƒΡŽ ΠΏΠ°Ρ€Ρƒ Ρ€Π°ΡΡΠΌΠ°Ρ‚Ρ€ΠΈΠ²Π°Ρ‚ΡŒ ΠΊΠ°ΠΊ Ρ†Π΅Π»ΠΎΠ΅ число a-b,{\displaystyle ab,}   Π½Π°ΠΏΡ€ΠΈΠΌΠ΅Ρ€, ΠΏΠ°Ρ€Ρ‹ (3,2){\displaystyle \left(3,2\right)}   or (6,five){\displaystyle \left(6,5\right)}   Π±ΡƒΠ΄ΡƒΡ‚ ΠΈΠ·ΠΎΠ±Ρ€Π°ΠΆΠ°Ρ‚ΡŒ Π΅Π΄ΠΈΠ½ΠΈΡ†Ρƒ, Π° ΠΏΠ°Ρ€Ρ‹ (one,four){\displaystyle \left(1,4\right)}   or (eight,eleven){\displaystyle \left(8,11\right)}   Π±ΡƒΠ΄ΡƒΡ‚ ΠΈΠ·ΠΎΠ±Ρ€Π°ΠΆΠ°Ρ‚ΡŒ -3.{\displaystyle -3.}  

Π”Π°Π»Π΅Π΅ ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΠΈΠΌ [40] :

  1. Couples (a,b){\displaystyle \left(a,b\right)}   and (c,d){\displaystyle \left(c,d\right)}   ΡΡ‡ΠΈΡ‚Π°ΡŽΡ‚ΡΡ Ρ€Π°Π²Π½Ρ‹ΠΌΠΈ, Ссли a+d=b+c{\displaystyle a+d=b+c}   . Π­Ρ‚ΠΎ связано с Ρ‚Π΅ΠΌ, Ρ‡Ρ‚ΠΎ, ΠΊΠ°ΠΊ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ Π² ΠΏΡ€ΠΈΠΌΠ΅Ρ€Π°Ρ…, любоС Ρ†Π΅Π»ΠΎΠ΅ число ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡ€Π΅Π΄ΡΡ‚Π°Π²ΠΈΡ‚ΡŒ бСсконСчным числом ΠΏΠ°Ρ€.
  2. Π‘Π»ΠΎΠΆΠ΅Π½ΠΈΠ΅ : сумма ΠΏΠ°Ρ€ (a,b){\displaystyle \left(a,b\right)}   and (c,d){\displaystyle \left(c,d\right)}   опрСдСляСтся ΠΊΠ°ΠΊ ΠΏΠ°Ρ€Π° (a+c,b+d){\displaystyle \left(a+c,b+d\right)}   .
  3. Π£ΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΠ΅ : ΠΏΡ€ΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½ΠΈΠ΅ ΠΏΠ°Ρ€ (a,b){\displaystyle \left(a,b\right)}   and (c,d){\displaystyle \left(c,d\right)}   опрСдСляСтся ΠΊΠ°ΠΊ ΠΏΠ°Ρ€Π° (ac+bd,ad+bc){\displaystyle \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.(a,b) {\ displaystyle \ left (a, b \ right)}   , in whicha>b {\ displaystyle a> b}   , zero represent pairs of the form(a,a) {\ displaystyle \ left (a, a \ right)}   , and couples(a,b) {\ displaystyle \ left (a, b \ right)}   witha<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 β€œnegative numbers”(a,b) {\ displaystyle \ left (a, b \ right)}   and(c,d) {\ displaystyle \ left (c, d \ right)}   , whicha<b,c<d {\ displaystyle a <b, \ c <d}   , by definition we get a pair(ac+bd,ad+bc) {\ displaystyle \ left (ac + bd, ad + bc \ right)}   . Differenceac+bd-(ad+bc) {\ displaystyle ac + bd- \ left (ad + bc \ right)}   is equal to(b-a)(d-c) {\ 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, β€œthe 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 integersZ {\ displaystyle \ mathbb {Z}}   . Among them:

  • Gaussian integers . These are complex numbers.a+bi {\ displaystyle a + bi}   wherea,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

  1. ↑ Here we have in mind the oldest understanding of natural numbers with the first unit element:one,2,3,four,five... {\ displaystyle 1,2,3,4,5 \ dots}  
  2. ↑ 1 2 3 Handbook of Elementary Mathematics, 1978 , p. 111-113.
  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.
  5. ↑ Elementary mathematics, 1976 , p. 18.
  6. ↑ Handbook of Elementary Mathematics, 1978 , p. 114.
  7. ↑ 1 2 3 4 Elementary Mathematics, 1976 , p. 24-28.
  8. ↑ 1 2 Elementary mathematics from the point of view of higher education, 1987 , p. 39.
  9. ↑ Handbook of Elementary Mathematics, 1978 , p. 114-115.
  10. ↑ 1 2 Handbook of Elementary Mathematics, 1978 , p. 172-173.
  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.
  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 .
  20. ↑ 1 2 Gleizer G.I. History of mathematics at school. - M .: Enlightenment, 1964. - S. 132-135. - 376 p.
  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. Β§ 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 .
  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.
  33. ↑ Numerical Systems, 1975 , p. 100.
  34. ↑ Numerical Systems, 1975 , p. 95-96.
  35. ↑ Encyclopedia of Elementary Mathematics, 1951 , p. 160-162.
  36. ↑ Numerical Systems, 1975 , p. 96-98.
  37. ↑ Encyclopedia of Elementary Mathematics, 1951 , p. 170-171.
  38. ↑ Numerical Systems, 1975 , p. 98.
  39. ↑ 1 2 Numerical systems, 1975 , p. 100-102.
  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.

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.
Source - https://ru.wikipedia.org/w/index.php?title= Integer&oldid = 98750649


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