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In , a rational number is any that can be expressed as the or fraction of two , a and a non-zero .

(2018). 9780072880083, McGraw-Hill.
Since may be equal to 1, every integer is a rational number. 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 (or \mathbb{Q}, Unicode ℚ); it was thus denoted in 1895 by after , Italian for "".

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

A that is not rational is called irrational. Irrational numbers include , , , and . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is , and the set of real numbers is , real numbers are irrational.

Rational numbers can be formally defined as equivalence classes of pairs of integers such that , for the equivalence relation defined by . With this formal definition, the fraction becomes the standard notation for the equivalence class of .

Rational numbers together with and form a field which contains the and is contained in any field containing the integers. In other words, the field of rational numbers is a , and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite of are called algebraic number fields, and the algebraic closure of is the field of .

(2018). 053440264X, Thomson Brooks/Cole. 053440264X

In mathematical analysis, the rational numbers form a of the real numbers. The real numbers can be constructed from the rational numbers by completion, using , , or infinite .

The term rational in reference to the set Q refers to the fact that a rational number represents a of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the are rational numbers. For example, a is a point with rational (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 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 , where and are , and . This is often called the .

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

Embedding of integers
Any integer can be expressed as the rational number , which is its canonical form as a rational number.

\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, and, in particular, if both fractions are in canonical form,
\frac{a}{b} < \frac{c}{d} if and only if ad < bc.

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.

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 and are .

\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 and are .

The rule for multiplication is:
\frac{a}{b} \cdot\frac{c}{d} = \frac{ac}{bd}.

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

Every rational number has an , often called its opposite,
- \left( \frac{a}{b} \right) = \frac{-a}{b}.
If is in canonical form, the same is true for its opposite.

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

\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}.
If is in canonical form, then the canonical form of its reciprocal is either or , depending on the sign of .

If both and are nonzero, the division rule is
\frac{\frac{a}{b}} {\frac{c}{d}} = \frac{ad}{bc}.

Thus, dividing by is equivalent to multiplying by the reciprocal of :

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

Exponentiation to integer power
If 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 . In particular,
\left(\frac{a}{b}\right)^0 = 1.

If , then

\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}.
If is in canonical form, the canonical form of the result is \frac{b^n}{a^n} if either or is even. Otherwise, the canonical form of the result is \frac{-b^n}{-a^n}.

Continued fraction representation
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 an are integers. Every rational number a/ b can be represented as a finite continued fraction, whose an can be determined by applying the Euclidean algorithm to ( a, b).

Other representations
  • : \frac{8}{3}
  • : 2\tfrac{2}{3}
  • repeating decimal using a vinculum: 2.\overline{6}
  • repeating decimal using : 2.(6)
  • continued fraction using traditional typography: 2 + \cfrac{1}{1 + \cfrac{1}{2} }
  • continued fraction in abbreviated notation: 2;
  • egyptian fraction: 2 + \frac{1}{2} + \frac{1}{6}
  • prime power decomposition: 2^3\times3^{-1}
  • : 3!6

are different ways to represent the same rational value.

Formal construction
Mathematically we may construct the rational numbers as equivalence classes of of ( m, n), with . This space of equivalence classes is the quotient space where if, and only if, We can define addition and multiplication of these pairs with the following rules:
\left(m_1, n_1\right) + \left(m_2, n_2\right) \equiv \left(m_1n_2 + n_1m_2, n_1n_2\right)
\left(m_1, n_1\right) \times \left(m_2, n_2\right) \equiv \left(m_1m_2, n_1n_2\right)
and, if m2 ≠ 0, division by
\frac{\left(m_1, n_1\right)} {\left(m_2, n_2\right)} \equiv \left(m_1n_2, n_1m_2\right).

The equivalence relation ( m1, n1) ~ ( m2, n2) if, and only if, is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the i.e. we identify two pairs ( m1, n1) and ( m2, n2) if they are equivalent in the above sense. (This construction can be carried out in any : see field of fractions.) We denote by ( m1, n1) the equivalence class containing ( m1, n1). If ( m1, n1) ~ ( m2, n2) then, by definition, ( m1, n1) belongs to ( m2, n2) and ( m2, n2) belongs to ( m1, n1); in this case we can write . Given any equivalence class ( m, n) there are a countably infinite number of representation, since

\cdots = (-2m,-2n) = (-m,-n) = (m,n) = (2m,2n) = \cdots.

The canonical choice for ( m, n) is chosen so that n is positive and , i.e. m and n share no common factors, i.e. m and n are . For example, we would write (1,2) instead of (2,4) or (−12,−24), even though .

We can also define a on Q. Let ∧ be the and-symbol and ∨ be the or-symbol. We say that if:

(n_1n_2 > 0 \ \and \ m_1n_2 \le n_1m_2) \ \or \ (n_1n_2 < 0 \ \and \ m_1n_2 \ge n_1m_2).

The integers may be considered to be rational numbers by the that maps m to ( m,1).

The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the 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 for characteristic zero.

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

The set of all rational numbers is . Since the set of all real numbers is uncountable, we say that real numbers are irrational, in the sense of , i.e. the set of rational numbers is a .

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

\frac{a}{b} < \frac{c}{d}
(where b,d are positive), we have
\frac{a}{b} < \frac{ad + bc}{2bd} < \frac{c}{d}.

Any 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 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 expansions as regular continued fractions.

By virtue of their order, the rationals carry an . The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a by using the absolute difference metric 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 . The rationals are characterized topologically as the unique metrizable space without . The space is also totally disconnected. The rational numbers do not form a complete metric space; the are the completion of Q under the metric above.

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 and for any non-zero integer a, let , where pn is the highest power of p a.

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

Then defines a on Q.

The metric space ( Q, dp) is not complete, and its completion is the Q p. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a absolute value.

See also

External links

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