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 $\backslash \backslash mathbb\{Q\}$. Using the set-builder notation $\backslash \backslash mathbb\{Q\}$ is defined as such: :$\backslash \backslash mathbb\{Q\}\; =\; \backslash \backslash left\backslash \backslash \{\backslash \backslash frac\{m\}\{n\}\; :\; m\; \backslash \backslash in\; \backslash \backslash mathbb\{Z\},\; n\; \backslash \backslash in\; \backslash \backslash mathbb\{Z\},\; n\; \backslash \; e\; 0\; \backslash ight\backslash \backslash \}$ Each rational number can be written in infinitely 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 $\backslash \backslash mathbb\{Q\}$ of rational numbers. For example, rational polynomials or rational prime ideals.

Arithmetic

Addition and multiplication of rational numbers are as follows: :$\backslash \backslash frac\{a\}\{b\}\; +\; \backslash \backslash frac\{c\}\{d\}\; =\; \backslash \backslash frac\{ad+bc\}\{bd\}$   :$\backslash \backslash frac\{a\}\{b\}\; \backslash \backslash cdot\; \backslash \backslash frac\{c\}\{d\}\; =\; \backslash \backslash frac\{ac\}\{bd\}$   Two rational numbers $\backslash \backslash frac\{a\}\{b\}$ and $\backslash \backslash frac\{c\}\{d\}$ are equal iff $ad\; =\; bc$ Additive and multiplicative inverses exist in the rational numbers. :$-\; \backslash \backslash left(\; \backslash \backslash frac\{a\}\{b\}\; \backslash ight)\; =\; \backslash \backslash frac\{-a\}\{b\}$   :$\backslash \backslash left(\backslash \backslash frac\{a\}\{b\}\backslash ight)^\{-1\}\; =\; \backslash \backslash frac\{b\}\{a\}\; \backslash \backslash mbox\{\; if\; \}\; a\; \backslash \; eq\; 0$

History

Egyptian fractions

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance, $\backslash \backslash frac\{5\}\{7\}\; =\; \backslash \backslash frac\{1\}\{2\}\; +\; \backslash \backslash frac\{1\}\{6\}\; +\; \backslash \backslash frac\{1\}\{21\}$ 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, or, using the hieroglyphs and writing left to right: |Aa13 |style="padding-left:1em; padding-right:1em;" |D21:Z1

• Z1*Z1*Z1*Z1*Z1

|D21:V20

• V20*Z1

½ is one of exactly three exceptions: it is written as shown in the first hieroglyph above. The two other exceptions were the two only non-unit fractions for which there were symbols: |D22 |$=\; \backslash \backslash frac\{2\}\{3\}$ | style="padding-left:1em;" |D23 |$=\; \backslash \backslash frac\{3\}\{4\}$ The Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals.

Formal construction

Mathematically we may define them as an ordered pair of integers $\backslash \backslash left(a,\; b\backslash ight)$, with $b$ not equal to zero. We can define addition and multiplication of these pairs with the following rules: : $\backslash \backslash left(a,\; b\backslash ight)\; +\; \backslash \backslash left(c,\; d\backslash ight)\; =\; \backslash \backslash left(ad\; +\; bc,\; bd\backslash ight)$ : $\backslash \backslash left(a,\; b\backslash ight)\; \backslash \backslash times\; \backslash \backslash left(c,\; d\backslash ight)\; =\; \backslash \backslash left(ac,\; bd\backslash ight)$ To conform to our expectation that $2/4\; =\; 1/2$, we define an equivalence relation $\backslash \backslash sim$ upon these pairs with the following rule: : $\backslash \backslash left(a,\; b\backslash ight)\; \backslash \backslash sim\; \backslash \backslash left(c,\; d\backslash ight)\; \backslash \backslash mbox\{\; iff\; \}\; ad\; =\; bc$ 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 : $\backslash \backslash left(a,\; b\backslash ight)\; \backslash \backslash le\; \backslash \backslash left(c,\; d\backslash ight)\; \backslash \backslash mbox\{\; iff\; \}\; ad\; \backslash \backslash le\; bc$

Properties

The set $\backslash \backslash mathbb\{Q\}$, together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers $\backslash \backslash mathbb\{Z\}$. The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of $\backslash \backslash mathbb\{Q\}$. The algebraic closure of $\backslash \backslash mathbb\{Q\}$, 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 $d\backslash \backslash left(x,\; y\backslash ight)\; =\; |x\; -\; y|$, and this yields a third topology on $\backslash \backslash mathbb\{Q\}$. 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 $\backslash \backslash mathbb\{Q\}$.

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn $\backslash \backslash mathbb\{Q\}$ into a topological field: let $p$ be a prime number and for any non-zero integer $a$ let $|a|\_p\; =\; p^\{-n\}$, where $p^n$ is the highest power of $p$ dividing $a$; in addition write $|0|\_p\; =\; 0$. For any rational number $\backslash \backslash frac\{a\}\{b\}$, we set $\backslash \backslash left|\backslash \backslash frac\{a\}\{b\}\backslash ight|\_p\; =\; \backslash \backslash frac$. > Then $d\_p\backslash \backslash left(x,\; y\backslash ight)\; =\; |x\; -\; y|\_p$ defines a metric on $\backslash \backslash mathbb\{Q\}$. The metric space $\backslash \backslash left(\backslash \backslash mathbb\{Q\},\; d\_p\backslash ight)$ is not complete, and its completion is the p-adic number field$\backslash \backslash mathbb\{Q\}\_p$. 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 Category:Field theory Category:Fractions Category:Real numbers