Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠\(\tfrac p q\)⁠ of two integers, a numerator p and a nonzero denominator q. For example, ⁠\(\tfrac{3}{7}\)⁠ is a rational number, as is every integer (for example, \(-5 = \tfrac{-5}{1}\)). The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface Q, or blackboard bold ⁠\(\Q.\)⁠

A rational number is a real number. The real numbers that are rational are those whose decimal expansion 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...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).

A real number that is not rational is called irrational. Irrational numbers include the square root of 2 (⁠\(\sqrt 2\)⁠), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

The field of rational numbers is the unique field that contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field. 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.

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).

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, though it sometimes also refers to a matrix whose entries are rational functions; 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.

Although nowadays rational numbers are defined in terms of ratios, the term rational is not a derivation of ratio. On the contrary, it is ratio that is derived from rational: the first use of ratio with its modern meaning was attested in English about 1660, while the use of rational for qualifying numbers appeared almost a century earlier, in 1570. This meaning of rational came from the mathematical meaning of irrational, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of ἄλογος)".

This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers". So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).

Every rational number may be expressed in a unique way as an irreducible fraction ⁠\(\tfrac a b,\)⁠ 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 ⁠\(\tfrac a b,\)⁠ its canonical form may be obtained by dividing both a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.

\(\frac{a}{b} = \frac{c}{d}\) if and only if \(ad = bc\)

If both fractions are in canonical form, then:

\(\frac{a}{b} = \frac{c}{d}\) if and only if \(a = c\) and \(b = d\)

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

\(\frac{a}{b} < \frac{c}{d}\) if and only if \(ad < bc.\)

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:

\(\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.\)

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

\(\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}.\)

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

The rule for multiplication is:

\(\frac{a}{b} \cdot\frac{c}{d} = \frac{ac}{bd}.\)

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

Every rational number ⁠\(\tfrac a b\)⁠ has an additive inverse, often called its opposite,

\(- \left( \frac{a}{b} \right) = \frac{-a}{b}.\)

If ⁠\(\tfrac a b\)⁠ is in canonical form, the same is true for its opposite.

A nonzero rational number ⁠\(\tfrac a b\)⁠ has a multiplicative inverse, also called its reciprocal,

\(\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}.\)

If ⁠\(\tfrac a b\)⁠ is in canonical form, then the canonical form of its reciprocal is either ⁠\(\tfrac b a\)⁠ or ⁠\(\tfrac{-b}{-a},\)⁠ depending on the sign of a.

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

\(\frac{\,\dfrac{a}{b}\,} {\dfrac{c}{d}} = \frac{ad}{bc}.\)

Thus, dividing ⁠\(\tfrac a b\)⁠ by ⁠\(\tfrac c d\)⁠ is equivalent to multiplying ⁠\(\tfrac a b\)⁠ by the reciprocal of ⁠\(\tfrac c d:\)⁠

\(\frac{ad}{bc} = \frac{a}{b} \cdot \frac{d}{c}.\)

If n is a non-negative integer, then

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.\)

The result is in canonical form if the same is true for ⁠\(\tfrac a b.\)⁠ In particular,

\(\left(\frac{a}{b}\right)^0 = 1.\)

If a ≠ 0, then

\(\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}.\)

If ⁠\(\tfrac a b\)⁠ is in canonical form, the canonical form of the result is ⁠\(\tfrac{b^n}{a^n}\)⁠ if a > 0 or n is even. Otherwise, the canonical form of the result is ⁠\(\tfrac{-b^n}{-a^n}.\)⁠

A finite continued fraction is an expression such as

\(a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}},\)

where a_n are integers. Every rational number ⁠\(\tfrac a b\)⁠ can be represented as a finite continued fraction, whose coefficients a_n can be determined by applying the Euclidean algorithm to (a, b).

• common fraction: ⁠\(\tfrac 8 3\)⁠
• mixed numeral: ⁠\(2\tfrac 2 3\)⁠
• repeating decimal using a vinculum: \(2.\overline 6\)
• repeating decimal using parentheses: \(2.(6)\)
• continued fraction using traditional typography: \(2 + \tfrac 1 {1 + \tfrac 1 2}\)
• continued fraction in abbreviated notation: \([2; 1, 2]\)
• Egyptian fraction: \(2 + \tfrac 1 2 + \tfrac 1 6\)
• prime power decomposition: \(2^3 \times 3^{-1}\)
• quote notation: \(3'6\)

are different ways to represent the same rational value.