Infinitely cyclic decimals and numbers with infinite roots are called irrational numbers, such as π, 3. 14 1592653. ...

Rational numbers are just the opposite, and integers and fractions are collectively called rational numbers.

Include integers and fractions, and can also be expressed as finite decimals or infinite cyclic decimals.

This definition applies to decimal and other decimal (such as binary) numbers.

Mathematically, a rational number is the ratio of an integer a to a nonzero integer b, which is usually written as a/b, so it is also called a fraction. The Greek name is λ ο γ ο? The original meaning is "rational number", but the Chinese translation is not appropriate, and it has gradually become "rational number". Real numbers that are not rational numbers are called irrational numbers.

The set of all rational numbers is expressed as q, and the fractional part of rational numbers is finite or cyclic.

Rational numbers are divided into integers and fractions.

Integers are divided into positive integers, negative integers and 0.

Scores are divided into positive scores and negative scores.

Positive integers and 0 are also called natural numbers.

For example, 3, -98. 1 1, 5.272 and 7/22 are all rational numbers.

Rational numbers can also be divided into positive integers, negative integers, positive fractions, negative fractions and 0.

All rational numbers form a set, that is, rational number set, which is represented by bold letter Q, while some modern math books are represented by hollow letter Q.

The rational number set is a subset of the real number set. For related contents, see number system expansion.

A set of rational numbers is a field, that is, four operations can be performed in it (except that 0 is a divisor). For these operations, the following algorithms hold (a, b, c, etc. Represents any rational number):

① the commutative law of addition A+B = B+A;

② the associative law of addition A+(B+C) = (A+B)+C;

(3) There is a number 0, so that 0+a = a+0 = a;

(4) For any rational number A, there is an addition inverse element, which is denoted as -a, so that a+(-a) = (-a)+a = 0;

⑤ The commutative law of multiplication AB = BA

⑥ Multiplicative associative law A (BC) = (AB) C;

⑦ Distribution law A (B+C) = AB+AC;

⑧ Multiplication has a unit 1≠0, so that for any rational number A,1a = a1= a;

Pet-name ruby For rational number A which is not 0, there is a multiplication inverse 1/a, so a (1/a) = (1/a) A =1.

⑩ 0A = 0 Text explanation: A number multiplied by 0 is still equal to this number.

In addition, rational number is an ordered domain, that is, there is an ordered relation ≤

Rational number is also an Archimedes field, that is, rational numbers a and b, a≥0, B >；; 0, we can find a natural number n, which makes nb >；; Answer: It is not difficult to infer that there is no maximum rational number.

The name rational number is worth mentioning. The name "rational number" is puzzling, and rational numbers are no more "reasonable" than other numbers. However, this word comes from ancient Greece, and its English root is ratio, which means ratio (the root here is English and the Greek meaning is the same). So the meaning of this word is also very clear, that is, the "ratio" of integers. In contrast, "irrational number" is a number that cannot be accurately expressed as the ratio of two integers, but it is not unreasonable.

Rational number addition and subtraction mixed operation

1. The significance of unifying addition and subtraction of rational numbers into addition;

For subtraction in addition and subtraction mixed operation, we can convert subtraction into addition according to the rational number subtraction rule, thus unifying the mixed operation into addition operation. The unified formula is the sum of several positive numbers or negative numbers. We call this formula algebraic sum.

2. The method and steps of rational number addition and subtraction mixed operation:

(1) The subtraction in rational number mixing operation is converted into addition by using the subtraction rule.

(2) Use the law of addition, additive commutative law and the law of addition combination to perform simple operations.

The concepts of absolute value and reciprocal in rational number range have the same meaning in real number range.

Generally speaking, rational numbers are classified as follows:

Integer, fraction; Positive numbers, negative numbers and zero; Negative rational number

Integer and fraction are collectively called rational numbers, which can be expressed in the form of a/b, where a and b are both integers and prime numbers. We often use rational numbers in our daily life. Such as how much money, how many kilograms, etc.

Any real number that can't be expressed in a/b form is irrational, also called infinite acyclic decimal.