What is a rational number, what is a rational number

A rational number is the ratio of an integer a to a non-zero integer b, usually written a b.

This includes integers and what is commonly referred to as a fraction, which can also be expressed as a finite decimal or an infinite loop decimal.

Numbers that can be expressed in the form of fractions are often referred to as rational numbers.

What is the concept of a rational number.

Rational numbers are integers, fractions.

01 Rational numbers are a collective term for integers (positive integers, 0s, negative integers) and fractions, and are a collection of integers and fractions. Positive integers and positive fractions are collectively called positive rational numbers, and negative integers and negative fractions are collectively referred to as negative rational numbers. Therefore, the number of rational numbers in a set of rational numbers can also be divided into positive rational numbers, negative rational numbers, and zeros.

02 The name "rational number" is incomprehensible, and rational numbers are no more "reasonable" than other numbers. In fact, it seems to be a translation error. The term "rational number" came from the West and is rational number in English, and rational usually means "rational".

In modern times, when China translated Western scientific works, it translated them as "rational numbers" according to the translation method in Japanese. However, in ancient Greece, the English root of this word ** is ratio, which means ratio (the root here is in English, and the Greek meaning is the same). So the meaning of this word is also obvious, which is the "ratio" of integers.

In contrast, an "irrational number" is a number that cannot be accurately expressed as the ratio of two integers, and it is not unreasonable.

According to their sequence, rational numbers have an ordinal topology. A rational number is a (dense) subset of real numbers, so it also has a subspace topology.

Rational numbers are a collective term for positive integers, negative integers, positive fractions, negative fractions, and zeros. Mathematically, a number that can be expressed as a ratio of two integers is defined as a rational number.

Rational numbers are one of the important contents in the field of "number and algebra", which have a wide range of applications in real life, and are the basis for continuing to learn real numbers, algebraic formulas, equations, inequalities, Cartesian coordinate systems, functions, statistics and other mathematical content and related subject knowledge.

Mathematically, a rational number is the ratio of an integer a to a positive integer b, e.g. 3 8, and the general rule a is also a rational number. A rational number is a set of integers and fractions, and an integer can also be thought of as a fraction with a denominator of one. The decimal part of a rational number is a finite or infinitely looping number.

Real numbers that are not rational numbers are called irrational numbers, i.e., the fractional part of an irrational number is an infinite number that is not cyclical.

The set of rational numbers can be represented by the uppercase black orthography symbol q. But q does not denote a rational number, and a set of rational numbers and a rational number are two different concepts. A rational number set is a set of elements that are all rational numbers, while a rational number is a set of all the elements in a rational number set.

Etymology. Rational number originally meant "proportional number" in Greek, and the English took its meaning, taking ratio as the root word, and adding -nal at the end of the word to form an adjective, the full name is rational number, which is literally translated into Chinese as "comparable number". Correspondingly, irrational numbers are "incomparable".

At the end of the Ming Dynasty, the mathematician Xu Guangqi and the scholar Matteo Ricci translated the first six volumes of the Geometry Originals in Latin. They translated the word as "li", which refers to "ratio". Before the Meiji Restoration, most of the translations of European and American mathematical classics were based on Chinese Chinese.

Japanese scholars have translated the word "li" in classical Chinese directly into "li", rather than the "ratio" as explained in classical Chinese. Later, Japanese scholars directly translated "rational numbers" and "irrational numbers" with a wrong understanding. (There is no meaning of ratio in classical Chinese).

When rational numbers were transmitted back to China from Japan, the error continued. At the end of the Qing Dynasty, China sent students to Japan to bring the term back to China, so much so that both China and Japan now use the terms "rational numbers" and "irrational numbers".

It can be seen that due to the lack of understanding of Chinese classical Chinese by Japanese scholars at that time, today's mistranslation has occurred.