rational number

Decimals with infinite cycles and numbers with infinite roots are called irrational numbers, such as π, while rational numbers are just the opposite. Integers and fractions are collectively called rational numbers, including 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 non-zero integer b, which is usually written as a/b, so it is also called a fraction. However, due to improper Chinese translation, it has gradually become a "reasonable number". Real numbers that are not rational numbers are called irrational numbers.

The set of all rational numbers is expressed as q, 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 fractions, and positive fractions, negative fractions, positive integers and 0 are also called natural numbers. For example, 3,7/22 are rational numbers. As follows:

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 rational number set 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. In fact, this seems to be a mistake in translation.

The word rationalnumber comes from the west and is a rational number in English. Rational usually means "rational". China translated western scientific works in modern times into "rational numbers" according to Japanese translation methods. 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.