Rational numbers

Ancient Greek mathematicians of the classical era used only integer and fractional positive numbers. These are the so-called rational numbers. Fractions and operations on them were used, for example, by Sumerians, ancient Egyptians and Greeks. In ancient Greece, rational numbers in general were a symbol of the harmony of the surrounding world and a manifestation of the divine principle, and all the segments, until some time, were considered commensurable, ie, the ratio of their lengths had to be expressed by a rational number (another god could not allow!). The term rational is derived from the Latin "ratio" - an attitude that is similar in meaning to the Greek word "logos". Thus, historically, the first expansion of the notion of number is joining to the set natural numbers of the set of all fractional numbers. Rational numbers are also called relative, because any of them can be represented by the ratio of two integers. Using rational numbers, you can different dimensions (for example , the length of the segment for the selected unit of scale) with any accuracy, ie, rational numbers are sufficient to satisfy most practical needs.

Ancient Greek mathematicians of the classical era used only integer and fractional positive numbers

The set of rational numbers is everywhere dense on the numerical axis: at least one rational is located between any two different rational numbers. Since the time of the ancient Greeks, it is known about the existence of numbers that are not representable as fractions: they have proved that there is no rational number whose square is equal to two. The lack of rational numbers to express all quantities led to the concept of a real number.

The term fractional number is sometimes used as a synonym for the term rational number, and sometimes a synonym for any non-integer number. In the latter case, fractional and rational numbers are different things, since then non-integral rational numbers are just a special case of fractional numbers.