Integers can also be regarded as fractions with a denominator of 1.Nonrationalreal numberIt is called irrational number, that is, the decimal part of irrational number is infinite and non cyclic.It is one of the important contents in the field of "number and algebra" and has wide application in real lifeAlgebraic expression、equation、Inequality、Rectangular coordinate system、function, statistics and other mathematical contents, as well as the basis of related discipline knowledge.
The set of rational numbers can use capital black regular symbolsQRepresentative.butQIt does not mean rational number. Rational number set and rational number are two different concepts.The set of rational numbers is composed of all rational numbersaggregateAnd rational numbers are allelement。
The name "rational number" is difficult to understand. Rational number is no more "reasonable" than other numbers.In fact, this seems to be a mistake in translation.The word rational number comes from the West. In English, it is rational number, and rational usually means "rational".In modern times, China translated western scientific works according to Japanese translation methods,circulate erroneous reports, which is translated as "rational number".But the word comes fromancient Greek, whose English root is ratio, that isratioThe root here is in English, and the Greek meaning is the same.So the meaning of this word is also very obvious, that is, the "ratio" of integers.In contrast, "irrational numbers" are numbers that cannot be accurately expressed as the ratio of two integers, rather than being unreasonable.[1]
Understanding of rational numbers
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Rational number isinteger(positive integer, 0, negative integer) and fraction[2]。Positive Integer SumPositive fractionCollectively referred to asPositive rational number, negative integer sumNegative fractionCollectively referred to asNegative rational number。Therefore, the number of rational number set can be divided into positive rational number, negative rational number andFatal Frame。Because any integer or fraction can be converted to decimalRecurring decimalOn the contrary, every decimal recurring decimal can also be converted into an integer or fraction, so rational numbers can also be defined as decimal recurring decimal.
Rule on the order of magnitude of rational numbers a and b: if a-b is a positive rational number, it is said that when a is greater than b or b is less than a, it is recorded as a>b or b<a.Any two unequal rational numbers can be compared in size.
An important difference between rational number set and integer set is that rational number set is dense while integer set is dense.After rational numbers are arranged in order of size, there must be other rational numbers between any two rational numbers, which is density.Integer sets do not have this feature, and there are no other integers between two adjacent integers.
Rational numbers are compact subsets of real numbers: every real number has any rational number close to it.A related property is that only rational numbers can be reduced to finite continued fractions.According to their sequence, rational numbers have aOrdered topology。Rational number is real (dense)subset, so it also has a subspace topology.[1]
Classification of rational numbers
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Classification of rational numbers
According to different standards, rational numbers can be classified according to definition and symbols;By definition, rational numbers are divided intointeger、fraction；Classified by sign Rational numbers are divided intoPositive rational number、0、Negative rational number。[2][3]Decimals can be changed into fractions, so they are regarded as fractions.[3]
Basic algorithm
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Addition operation
1. Add two numbers with the same sign, take the same sign as the addend, and add the absolute value.
2. Add two numbers with different signs, ifabsolute valueEqual is mutualInverse numberThe sum of two numbers of is 0;If the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the smaller absolute value from the larger absolute value.
5. Two numbers that are opposite to each other can be added first.
6. Numbers with the same sign can be added first.
7. Numbers with the same denominator can be added first.
8. If you can add several numbers to get an integer, you can add them first.[1]
Subtraction
Subtracting a number is equal to adding the opposite number of the number, that is, subtracting a rational number by using the opposite number of the number to add.[1]
Multiplication
1、Same numberGet positive, get negative, and multiply the absolute value.
2. Any number multiplied by 0 will get 0.
3. Multiplying several numbers that are not equal to 0, the sign of the product is determined by the number of negative factors. When there are odd numbers of negative factors, the product is negative, and when there are even numbers of negative factors, the product is positive.
4. When several numbers are multiplied, one factor is 0, and the product is 0.
5. When multiplying several numbers that are not equal to 0, first determine the sign of the product, and then multiply the absolute value.[1]
Division operation
1. Dividing by a number that is not equal to zero is equal to multiplying the reciprocal of the number.
2. When two numbers are divided, the same sign is positive, the different sign is negative, and the absolute value is divided.0 is divided by any number that is not equal to 0.
The division and multiplication of rational numbers are reciprocal operations.
In the division operation, the sign is determined first according to the rule that the same sign is positive and the different sign is negative, and then the absolute value is divided.If there isWith fraction, usually formed firstFalse fractionCalculate.If it is impossible to divide, the division operation is converted to multiplication operation.[1]
Real Number Classification Chart
Power operation
1. Odd times of negative numberspowerIs a negative number, and the even power of a negative number is a positive number.For example: (- 2)³(3rd power of - 2)=- 8, (- 2)²(power 2 of - 2)=4.
2. Any power of a positive number is positive, and any power of 0 is 0.For example: 2two=4，2three=8，zerothree=0。
3. The power of 0 is meaningless.
4. Since the power is a special case of multiplication, the power operation of rational numbers can be completed by the multiplication operation of rational numbers.
5. Any power of 1 is 1, the even power of - 1 is 1, and the odd power is - 1.[1]
Operation law of rational number
Addition law:
1、Additive commutative law: Add two numbers to exchange the position of the addend, and the sum remains unchanged, that is.
2、Associative law of additionTo add three numbers, first add the first two numbers or add the last two numbers, and the sum remains the same.
Subtraction law:
Subtraction operation law: subtracting a number is equal to addingInverse number。That is:
。
Multiplication law:
1、Commutative law of multiplication: Multiply two numbers, exchange the position of the factor, and the product remains unchanged, that is.
2、Associative law of multiplication: Multiply three numbers. Multiply the first two numbers first, or multiply the last two numbers first. The product remains unchanged.
3、distributive law Multiplying a number by the sum of two numbers is equal to multiplying the number by these two numbers respectively, and then adding the products, namely:
。
Hybrid algorithm
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For the addition, subtraction, multiplication and division of rational numbers, if there is no parenthesis to indicate what operation to do first, it shall be carried out in the order of "multiplication and division first, then addition and subtraction". If it isSibling operation, the calculation is carried out from left to right.
Related issues
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The fallacy of dividing by zero
Improper use of dividing by zero in algebraic operationsInvalid proof：
。premise
Not equal to
。
From: 0a=0, 0b=0, 0a=0b.
Divide two sides by zero to get 0a/0=0b/0.
To simplify, a=b.
The above fallacy assumes that it is permissible to divide a number by 0, and[1]
Algebraic processing
If a mathematical system complies withfieldIn this mathematical system, dividing by zero must be meaningless.This is because division is defined asmultiplicationThe reverse operation of
Value is an equation
in
The solution of (if any).If set
, Equation
Can be written as
Or directly
。Therefore, equation
No solution (when
However, any numerical value can also solve this equation (when
When).[1]
integer
integer, is the general name of all numbers in the sequence {..., - 3, - 2, - 1, 0, 1, 2, 3,...}, includingnegtive integer, zero (0) andpositive integer。Like natural numbers, integers are countableInfinite set。thisaggregateIt is usually expressed mathematically asboldZOr, derived from German wordsZahlen(meaning "number").
ZIt's an additionCyclic group, because any integer is the sum of several 1 or - 1.1 and - 1 YesZThe only two generators of.A cyclic group with an infinite number of elements is associated with（Z，+）isomorphism。[1]
