Rational number


Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold \mathbb{Q}, Unicode, which stands for quotient.)

The decimal expansion of a rational number always either terminates after finitely many 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 binary, hexadecimal, or any other integer base.

A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

The rational numbers can be formally defined as the equivalence classes of the quotient set (Z × (Z ∖ {0})) / ~, where the cartesian product Z × (Z ∖ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not zero (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2m2n1 = 0.

In abstract algebra, the rational numbers together with certain operations of addition and multiplication form a field. This is the archetypical field of characteristic zero, and is the field of fractions for the ring of integers. 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 either Cauchy sequences, Dedekind cuts, or infinite decimals.

Zero divided by any other integer equals zero, therefore zero is a rational number (but division by zero is undefined).

Contents

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, the adjective rational often means that the underlying field considered is the field Q of rational numbers. For example, a rational integer is an algebraic integer which is also a rational number, which is to say, an ordinary integer, and a rational matrix is a matrix whose coefficients are rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a “polynomial over the rationals”. However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.

Arithmetic

Two rational numbers a/b and c/d are equal if, and only if, ad = bc.

Two fractions are added as follows

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

The rule for multiplication is

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

Additive and multiplicative inverses exist in the rational numbers

 - \left( \frac{a}{b} \right) = \frac{-a}{b} = \frac{a}{-b} \quad\mbox{and}\quad 
        \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \mbox{ if } a \neq 0.

It follows that the quotient of two fractions is given by

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

Egyptian fractions

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers, such as

\frac{5}{7} = \frac{1}{2} + \frac{1}{6} + \frac{1}{21}.

For any positive rational number, there are infinitely many different such representations, called Egyptian fractions, as they were used by the ancient Egyptians. The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems.

Formal construction

a diagram showing a representation of the equivalent classes of pairs of integers

Mathematically we may construct the rational numbers as equivalence classes of ordered pairs of integers (m,n), with n ≠ 0. This space of equivalence classes is the quotient space (Z × (Z ∖ {0})) / ∼, where (m1,n1) ∼ (m2,n2) if, and only if, m1n2m2n1 = 0. 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) := \left(m_1n_2 + n_1m_2, n_1n_2\right)
\left(m_1, n_1\right) \times \left(m_2, n_2\right) := \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)} := \left(m_1n_2, n_1m_2\right).

The equivalence relation (m1,n1) ~ (m2,n2) if, and only if, m1n2m2n1 = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set (Z × (Z ∖ {0})) / ∼, 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 integral domain: 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 [(m1,n1)] = [(m2,n2)]. 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 gcd(m,n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write [(1,2)] instead of [(2,4)] or [(−12,−24)], even though [(1,2)] = [(2,4)] = [(−12,−24)].

We can also define a total order on Q. Let ∧ be the and-symbol and ∨ be the or-symbol. We say that [(m1,n1)] ≤ [(m2,n2)] 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 embedding that maps m to [(m, 1)].

Properties

a diagram illustrating the countability of the rationals

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

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. 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) = |xy|, 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) = |xy|, 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 prime number and for any non-zero integer a, let |a|p = pn, where pn is the highest power of p dividing a.

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) = |xy|p defines a metric on Q.

The metric space (Q,dp) is not complete, and its completion is the p-adic number field Qp. 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 p-adic absolute value.

See also

External links


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