Rational Number
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Rational Number
The rational numbers (Q) are included in the real numbers (R), while themselves including the integers (Z), which in turn include the natural numbers (N)

In mathematics, a rational number is a number such as -3/7 that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Every integer is a rational number: for example, 5 = 5/1. The set of all rational numbers, often referred to as "the rationals"[], the field of rationals[] or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode Q/Q);[2][3] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".

The decimal expansion of a rational number either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...).[4] Conversely, any repeating or terminating decimal represents a rational number. These statements are true not just in base 10, but also in any other integer base (for example, binary or hexadecimal).

A real number that is not rational is called irrational.[5] Irrational numbers include , ?, e, and ?. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1]

Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ? 0, using the equivalence relation defined as follows:

The fraction p/q then denotes the equivalence class of (p, q).


In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (for more, see Construction of the real numbers).


The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.


Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers and b > 0. This is often called the canonical form of the rational number.

Starting from a rational number a/b, its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.

Embedding of integers

Any integer n can be expressed as the rational number n/1, which is its canonical form as a rational number.


if and only if

If both fractions are in canonical form, then:

if and only if and


If both denominators are positive (particularly if both fractions are in canonical form):

if and only if

On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator--by changing the signs of both its numerator and denominator.


Two fractions are added as follows:

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.




The rule for multiplication is:

where the result may be a reducible fraction--even if both original fractions are in canonical form, .


Every rational number a/b has an additive inverse, often called its opposite,

If a/b is in canonical form, the same is true for its opposite.

A nonzero rational number a/b has a multiplicative inverse, also called its reciprocal,

If a/b is in canonical form, then the canonical form of its reciprocal is either or , depending on the sign of a.


If b, c, and d are nonzero, the division rule is

Thus, dividing a/b by c/d is equivalent to multiplying a/b by the reciprocal of c/d:

Exponentiation to integer power

If n is a non-negative integer, then

The result is in canonical form if the same is true for a/b. In particular,

If a ? 0, then

If a/b is in canonical form, the canonical form of the result is if either a > 0 or n is even. Otherwise, the canonical form of the result is

Continued fraction representation

A finite continued fraction is an expression such as


Other representations

are different ways to represent the same rational value.

Formal construction


The rational numbers may be built as equivalence classes of ordered pairs of integers.

More precisely, let (Z × (Z \ {0})) be the set of the pairs (m, n) of integers such n ? 0. An equivalence relation is defined on this set by

Addition and multiplication can be defined by the following rules:

This equivalence relation is a congruence relation, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers Q is the defined as the quotient set by this equivalence relation, (Z × (Z \ {0})) / ~, equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any integral domain and produces its field of fractions.)

The equivalence class of a pair (m, n) is denoted Two pairs (m1, n1) and (m2, n2) belong to the same equivalence class (that is are equivalent) if and only if this means that if and only

Every equivalence class may be represented by infinitely many pairs, since

Each equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (m, n) in the equivalence class such that m and n are coprime, and n > 0. It is called the representation in lowest terms of the rational number.

The integers may be considered to be rational numbers identifying the integer n with the rational number

A total order may be defined on the rational numbers, that extends the natural order of the integers. One has if


A diagram illustrating the countability of the positive rationals

The set Q of all rational numbers, together with the addition and multiplication operations shown above, forms a field.

Q has no field automorphism other than the identity.

With the order defined above, Q is an ordered field that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield isomorphic to Q.

Q is a prime field, which is a field that has no subfield other than itself.[7] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to Q.

Q is the field of fractions of the integers Z.[8] The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the field of algebraic numbers.

The set of all rational numbers is countable, while the set of all real numbers (as well as the set of irrational numbers) is uncountable. Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.


(where are positive), we have


Real numbers and topological properties



p-adic numbers


Let p be a prime number and for any non-zero integer a, let , where pn is the highest power of p dividing a.

In addition set For any rational number a/b, we set

Then defines a metric on Q.


See also


  1. ^ a b Rosen, Kenneth. Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158-160. ISBN 978-0-07-288008-3.
  3. ^ Rouse, Margaret. "Mathematical Symbols". Retrieved 2015.
  4. ^ "Rational number". Encyclopedia Britannica. Retrieved .
  5. ^ Weisstein, Eric W. "Rational Number". mathworld.wolfram.com. Retrieved .
  6. ^ Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243-244. ISBN 0-534-40264-X.
  7. ^ S?gakkai, Nihon (1993). Encyclopedic Dictionary of Mathematics, Volume 1. London, England: MIT Press. p. 578. ISBN 0-2625-9020-4.
  8. ^ Bourbaki, N. (2003). Algebra II: Chapters 4 - 7. Springer Science & Business Media. p. A.VII.5.

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