Rational number

In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. The set of all rational numbers is denoted by Q, or in blackboard bold $\mathbb{Q}$ . Using the set-builder notation $\mathbb{Q}$ is defined as such:

$\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n \ne 0 \right\}$

Each rational number can be written in many forms, for example 3 / 6 = 2 / 4 = 1 / 2. The simplest form is when a and b have no common divisors, and every rational number has a simplest form of this type.

The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above 1. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational.

A real number that is not rational is called an irrational number.

In mathematics, the term "rational something" means that the underlying field considered is the field $\mathbb{Q}$ of rational numbers. For example, rational polynomials or rational prime ideals.

Contents

1 Arithmetic

2 History

2.1 Egyptian fractions

3 Formal construction

4 Properties

5 Real numbers

6 p-adic numbers

Arithmetic

Addition and multiplication of rational numbers are as follows:

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

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

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal iff ad = bc

Additive and multiplicative inverses exist in the rational numbers.

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

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

History

Egyptian fractions

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

For instance, $\frac{5}{7} = \frac{1}{2} + \frac{1}{6} + \frac{1}{21}$

For any positive rational number, there are infinitely many different such representations. These representations are called Egyptian fractions, because the ancient Egyptians used them. The hieroglyph used for this is the letter that looks like a mouth, which is transliterated R, so the above fraction would be written as R2R6R21. The Egyptians also had a different notation for dyadic fractions.

Formal construction

Mathematically we may define them as an ordered pair of integers $\left(a, b\right)$ , with b not equal to zero. We can define addition and multiplication of these pairs with the following rules:

$\left(a, b\right) + \left(c, d\right) = \left(ad + bc, bd\right)$

$\left(a, b\right) \times \left(c, d\right) = \left(ac, bd\right)$

To conform to our expectation that 2 / 4 = 1 / 2, we define an equivalence relation $\sim$ upon these pairs with the following rule:

$\left(a, b\right) \sim \left(c, d\right) \mbox{ iff } ad = bc$

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. (This construction can be carried out in any integral domain, see quotient field.)

We can also define a total order on Q by writing

$\left(a, b\right) \le \left(c, d\right) \mbox{ iff } ad \le bc$

Properties

The set $\mathbb{Q}$ , together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers $\mathbb{Z}$ .

The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of $\mathbb{Q}$ .

The algebraic closure of $\mathbb{Q}$ , 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.

The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones.

Real numbers

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 expressions of continued fraction.

By virtue of their order, the rationals carry an order topology. The rational numbers are a (dense) subset of the real numbers, and as such they also carry a subspace topology. The rational numbers form a metric space by using the metric $d\left(x, y\right) = |x - y|$ , and this yields a third topology on $\mathbb{Q}$ . Fortunately, 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 space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of $\mathbb{Q}$ .

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn $\mathbb{Q}$ 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 p dividing a;

in addition write | 0 | _p = 0. For any rational number $\frac{a}{b}$ , we set $\left|\frac{a}{b}\right|_p = \frac{|a|_p}{|b|_p}$ .

Then $d_p\left(x, y\right) = |x - y|_p$ defines a metric on $\mathbb{Q}$ .

The metric space $\left(\mathbb{Q}, d_p\right)$ is not complete, and its completion is the p-adic number field $\mathbb{Q}_p$ .

Topics in mathematics related to quantity

[[bg:Рационално число]] ca:Nombre racional da:Rationale tal de:Rationale Zahl et:Ratsionaalarvud es:Número racional eo:Racionalaj Nombroj eu:Zenbaki arrazional fr:Nombre rationnel [[ko:유리수]] is:Ræðar tölur it:Numero razionale nl:Rationaal getal ja:有理数 no:Rasjonalt tall pl:Liczby wymierne pt:Número racional sl:Racionalno število fi:Rationaaliluku sv:Rationella tal tr:Rasyonel Sayılar zh:有理数

Categories: Field theory | Fractions | Real numbers

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