# Rational number

# Rational number

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).

A real number that is not rational is called irrational. Irrational numbers include √2, π, *e**φ* rational 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*such that *q* ≠ 0, for the equivalence relation defined by (*p*1, *q*1) ~ (*p*2, *q*2) if, and only if *p*1*q*2 = *p*2*q*1. With this formal definition, the fraction *p*/*q*becomes the standard notation for the equivalence class of (*p*, *q*).

Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of **Q** are called algebraic number fields, and the algebraic closure of **Q** is the field of algebraic numbers.^{[3]}

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 (see Construction of the real numbers).

Terminology

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 (that is 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, for avoiding confusion with "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.

Arithmetic

Irreducible fraction

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

Starting from a rational number *a*/*bgreatest common divisor, and, if

Embedding of integers

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

Equality

- if and only if

If both fractions are in canonical form then

- if and only ifand

Ordering

If both denominators are positive, and, in particular, if both fractions are in canonical form,

- if and only if

If either denominator is negative, 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.

Addition

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.

Subtraction

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

Multiplication

The rule for multiplication is:

Even if both fractions are in canonical form, the result may be a reducible fraction.

Inverse

Every rational number *a*/*badditive inverse, often called its

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

A nonzero rational number *a*/*bmultiplicative inverse, also called its

Division

If both *b* and *c* are nonzero, the division rule is

Thus, dividing *a/ by* c*/*d*is equivalent to multiplying* a*/*breciprocal of

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

Continued fraction representation

A **finite continued fraction** is an expression such as

where *an* are integers. Every rational number *a*/*bcoefficients anEuclidean algorithm to ( a*,*b*).

Other representations

common fraction:

mixed numeral:

repeating decimal using a vinculum:

repeating decimal using parentheses:

continued fraction using traditional typography:

continued fraction in abbreviated notation: [2; 1, 2]

egyptian fraction:

prime power decomposition:

quote notation:

**3!6**

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

- if and only if.

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.)

It is often convenient to choose, once for all, in each equivalence class a specific element called the *canonical representative element*. This 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.

Properties

The set **Q**, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers **Z**.

The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of **Q**. The rational numbers are therefore the prime field for characteristic zero.

The algebraic closure of **Q**, i.e. the field of roots of rational polynomials, is the 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.

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that

Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

Real numbers and topological properties

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.

By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric *d*(*x*,*y*) = |x − |, and this yields a third topology on* **Q** All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of **Q** under the metric *d* (*x*,*y*) = |*x*−* y*|, above.

*p*-adic numbers

*p*-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn **Q** into a topological field:

Let *p* be a prime number and for any non-zero integer *a*, let |*a*|*p*=* p*−*n*, where* pn*is the highest power of* pdividing

In addition set |0|*p*= 0. For any rational number* a*/*b*, we set |*a*/*b*|*p*= |*a*|*p*/ |*b*|*p*.

Then *dp* (*x*,*y*) = |*x*−* y*|*pmetric on Q.

The metric space (**Q**,*dp*p*-adic number field*-adic number field]]* **Q** Q lue or a *p*-adic absolute value.

See also

Floating point

Ford circles

Niven's theorem

Rational data type

height of a rational number in lowest term = naive height

Gaussian rational

## References

*Discrete Mathematics and its Applications*(6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. ISBN 978-0-07-288008-3.

*Elements of Modern Algebra*(6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN 0-534-40264-X.