Integer is a collection of positive integers, zero, and negative integers.All components of integersInteger set, integer set is aNumber ring。In the integer system, zero and positive integers are collectively referred to as natural numbers-1、-2、-3、…、-n、…（nIs a non-zero natural number) isnegtive integer。bepositive integer, zero, and negative integers form an integer system.
Integer does not includedecimal、fraction。Unless otherwise specified, the numbers involved are integersletterIt also represents an integer.Integer can be regarded asdenominatorA fraction of 1.
Note: Zero and positive integers are collectively referred to as natural numbers.
Integers can also be divided intoOdd numberAnd even numbers.
positive integer
It is a tool for human counting since ancient times.It can be said that the process of abstracting from "one cow, two cows" or "five people, six people" to a positive integer is quite natural.
Fatal Frame
Zero not only means "none" ("none"), but also means the symbol of vacancy.In ancient China, when calculating numbers and performing operations with arithmetic chips, no arithmetic chips were left in the empty space. Although there was no empty mark, it was still possible to record numbers andFour arithmetic operationsCreate good conditions.India ArabiaThe zero in the numerology comes from the Indian word Sunya, which also means "empty" or "blank".
negtive integer
China was the first to introduce negative numbers.The "positive and negative numbers" discussed in "Nine Chapters of Arithmetic. Equations" are the addition and subtraction of integers.The need for subtraction also promotesnegtive integerIntroduction of.Subtraction can be regarded as solving equation
, if
、bIf it is a natural number, the given equation may not have a natural number solution.In order to make it always have a solution, it is necessary to expand the natural number system to the integer system.[1]
Odd even number
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In an integer, the number that can be divided by 2 is called an even number.Numbers that cannot be divided by 2 are calledOdd number。That is, whennWhen it is an integer, the even number can be expressed as 2n (n is an integer);Odd numbers can be expressed as 2n+1 (or 2n-1).
Even numbers include positive even numbers (also calledeven numbers）, negative even and 0.All integers are either odd or even.
staydecimal systemYou can judge whether the number is odd or even by looking at the single digit: the number whose single digit is 1, 3, 5, 7, 9 is odd;The number of bits 0, 2, 4, 6, 8 is even.
Algebraic property
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The following table shows the basic properties of addition and multiplication of integers.(that is, for any integer a, b and c)
Addition and multiplication of integers
nature
addition
multiplication
Closure
Is an integer
Is an integer
Associative law
Commutative law
Unit yuan exists
Existence of inverse element
In the set of integers, only 1 or - 1 has integer inverses with respect to multiplication
Distributive law
Properties of 1 and 0
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1 is the divisor of any number, that is, for any integer, there is always 1|
。
0 is a multiple of any non-zero number,
, is an integer, then
|0。
Divisible feature
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1. If the last digit of a number is a single even number, the number can be divided by 2.
2. If the sum of all digits of a number can be divided by 3, the integer can be divided by 3.
3. If the last two digits of a number can be divided by 4, the number can be divided by 4.
4. If the last digit of a number is 0 or 5, the number can be divided by 5.
5. If a number can be divided by 2 and 3, it can be divided by 6.
6. If the single digit of a number is cut off, then subtract 2 times of the single digit from the remaining number. If the difference is a multiple of 7, the original number can be divided by 7.If the difference is too large or it is difficult to see whether it is a multiple of 7 by mental calculation, it is necessary to continue the above process of "truncation, multiple, subtraction, and error check" until a clear judgment can be made.For example, the process of determining whether 133 is a multiple of 7 is as follows:
, so 133 is a multiple of 7;For example, the process of determining whether 6139 is a multiple of 7 is as follows:
，
So 6139 is a multiple of 7, and so on.
7. If the last three digits of a number can be divided by 8, the number can be divided by 8.
8. If the sum of all digits of a number can be divided by 9, the integer can be divided by 9.
9. If the last digit of a number is 0, the number can be divided by 10.
10. If the difference between the sum of odd digits and the sum of even digits of a number can be divided by 11, the number can be divided by 11.The multiple test method of 11 can also be used“Tail cutting method”Processing.The only difference in the process is that the multiple is not 2 but 1.
11. If a number can be divided by 3 and 4, it can be divided by 12.
12. If the single digit of a number is truncated, then add 4 times of the single digit from the remaining number. If the sum is a multiple of 13, the original number can be divided by 13.If the difference is too large or it is difficult to see whether the multiple of 13 is calculated in mind, repeat the process of "truncation, multiple, addition, and summation" until a clear judgment can be made.
13. If the single digit of a number is cut off, and then 5 times of the single digit is subtracted from the remaining number, if the difference is a multiple of 17, the original number can be divided by 17.If the difference is too large or it is difficult to see whether it is a multiple of 17 by mental calculation, repeat the previous process until a clear judgment can be made.
14. If the single digit of a number is cut off, then add 2 times of the single digit from the remaining number. If the difference is a multiple of 19, the original number can be divided by 19.If the difference is too large or it is difficult to see whether it is a multiple of 19 by mental calculation, repeat the previous calculation idea until a clear judgment can be made.
15. If the difference between the last three digits of a number and the three times of the previous separated number can be divided by 17, then the number can be divided by 17.
16. If the difference between the last three digits of a number and the seven times of the previous separated number can be divided by 19, then the number can be divided by 19.
17. If the difference between the last four digits of a number and the previous five times of the separated number can be divided by 23 (or 29), then the number can be divided by 23
Parity
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oneOdd number± odd number=even number, even number ± even number=even number, odd number ± even number=odd number, even number × even number=even number, odd number × even number=even number, odd number × odd number=odd number;That is, the sum, difference and product of any number of even numbers are still even numbers, the sum and difference of odd numbers are odd numbers, and the sum and difference of even numbers and odd numbers are even numbers;
2. The square of odd numbers can be expressed as
The square of an even number can be expressed as
or
The form of;
3. If the product of a finite number of integers is odd, then each integer is odd;If the product of finite integers is even, then at least one of these integers is even;The sum and difference of two integers have the same parity;If the square root of an integer is an integer, they have the same parity.
Representation of Integer Set
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Why
What about representing an integer set?This involves the contribution of a German female mathematician to the ring theory. Her name isNott。
In 1920, she had introduced the concepts of "left model" and "right model".Written in 1921《Integral ringThe theory of ideal is a milestone in the development of commutative algebra.Among them, when Nott introduced the concept of integer ring (integer set itself is also aNumber ring）She is German, and the integer in German is called Zahlen, so she recorded the integer ring asZSince then, integer sets have been used
