Rational number

A number expressible as a fraction of integers. The formal theory of rational numbers is developed using pairs of integers. One considers ordered pairs of integers and for which . Two such pairs, and , are called equivalent (equal) if and only if . This is an equivalence relation, being reflexive, symmetric and transitive, and so partitions the set of all such pairs into equivalence classes. A rational number is defined as an equivalence class of pairs. A pair is also called a rational fraction (or fraction of integers). Distinct classes define distinct rational numbers. The set of all rational numbers is countable. The rational number containing a pair of the form is called zero. If is a rational number and , then the rational number containing is called the (additive) inverse of , and is denoted by . A rational number is called positive (negative) if it contains a rational fraction with and of the same sign (of different signs). If a rational number is positive (negative), then its (additive) inverse is negative (positive). The set of rational numbers can be ordered in the following way: Every negative rational number is less then every positive one, and a positive rational number is less than another positive rational number (written ) if there exist rational fractions and , , such that ; every negative (positive) rational number is smaller (greater) then zero: (); a negative rational number is less than another negative rational number if the positive rational number is greater than the positive rational number : . The absolute value of a rational number is defined in the usual way: if and if .

The sum of two rational fractions and is defined as the rational fraction and the product as . The sum and product of two rational numbers and are defined as the equivalence classes of rational fractions containing the sum and product of two rational fractions and belonging to and , respectively. The order, sum and product of rational numbers and do not depend on the choice of representatives of the corresponding equivalence classes, that is, they are uniquely determined by and themselves. The rational numbers form an ordered field, denoted by .

A rational number is denoted by any rational fraction from its equivalence class, i.e. . Thus, one and the same rational number can be written as distinct, but equivalent, rational fractions.

If every rational number containing a rational fraction of the form is associated with the integer , then one obtains an isomorphism from the set of such rational numbers onto the ring of integers. Therefore, the rational number containing a rational fraction of the form is denoted by .

Each function of the form

(1)

is a norm on the field of rational numbers , that is, it satisfies the conditions:

1) for any , ;

2) ;

3) for all . The field of rational numbers is not complete with respect to the norm (1). The completion of with respect to the norm (1) yields the field of real numbers.

Consider the function

(2)

where is a prime number, is a rational number and is determined by:

where is an integer, is an irreducible rational fraction such that and are not divisible by , and is a fixed number, . Then is a norm on . It induces the so-called -adic metric. is not complete with respect to this metric. By completing with respect to the norm (2), one obtains the field of -adic numbers (cf. -adic number). The metrics induced by (1) and (2) (for all prime numbers) exhaust all non-trivial metrics on .

In decimal notation, only rational numbers are representable as periodic decimals fractions.

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2]	C. Pisot, M. Zamansky, "Mathématiques générales: algèbre-analyse" , Dunod (1966)

Comments

Another property characterizing rational numbers is that their continued fraction is finite. A very important theme in number theory is to find only the rational solutions of equations such as , , etc. (see Diophantine equations). Finally, the subject of "rational numbers" is intimately connected with that of irrational numbers (cf. Irrational number). It is not known, for example, whether or not , or the Euler constant are rational.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII
[a2]	G. Bachman, "Introduction to -adic numbers and valuation theory" , Acad. Press (1964)
[a3]	B.L. van der Waerden, "Algebra" , 2 , Springer (1971) (Translated from German)