The rational numbers (ℚ) are included in thereal numbers(ℝ). On the other hand, they include theintegers(ℤ), which in turn include thenatural numbers(ℕ)

Inmathematics, arational numberis anynumberthat can be expressed as thequotientorfractionp/qof twointegers, anumeratorpand a non-zerodenominatorq.[1]Sinceqmay be equal to1, every integer is a rational number. Thesetof all rational numbers, often referred to as "the rationals", thefield of rationalsor thefield of rational numbersis usually denoted by a boldfaceQ(orblackboard bold{\displaystyle \mathbb {Q} }, Unicode ℚ);[2]it was thus denoted in 1895 byGiuseppe Peanoafterquoziente, Italian for "quotient".

Thedecimal expansionof a rational number always either terminates after a finite number ofdigitsor begins torepeatthe same finitesequenceof digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just forbase 10, but also for any other integerbase(e.g.binary,hexadecimal).

Areal numberthat is not rational is calledirrational. Irrational numbers include√2,π,e, andφ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers iscountable, and the set of real numbers isuncountable,almost allreal numbers are irrational.[1]

Rational numbers can beformallydefined asequivalence classesof pairs of integers(p,q)such thatq≠ 0, for theequivalence relationdefined by(p1,q1) ~ (p2,q2)if, and only ifp1q2=p2q1. With this formal definition, the fractionp/qbecomes the standard notation for the equivalence class of(p,q).

Rational numbers together withadditionandmultiplicationform afieldwhich contains theintegersand is contained in any field containing the integers. In other words, the field of rational numbers is aprime field, and a field hascharacteristic zeroif and only if it contains the rational numbers as a subfield. FiniteextensionsofQare calledalgebraic number fields, and thealgebraic closureofQis the field ofalgebraic numbers.[3]

Inmathematical analysis, the rational numbers form adense subsetof the real numbers. The real numbers can be constructed from the rational numbers bycompletion, usingCauchy sequences,Dedekind cuts, or infinitedecimals.

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