# 有理数

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

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

## 运算

${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}\,\ \ \ \ \ \ {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}}$

${\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}\,\ \ \ \ \ \ \ \ a\neq 0}$时，${\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}}$

## 古埃及分数

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

## 形式构建

${\displaystyle \left(a,b\right)+\left(c,d\right)=\left(ad+bc,bd\right)}$
${\displaystyle \left(a,b\right)\times \left(c,d\right)=\left(ac,bd\right)}$

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

Q上的全序关系可以定义为：

${\displaystyle \left(a,b\right)\leq \left(c,d\right)}$当且仅当
1. ${\displaystyle bd>0}$并且${\displaystyle ad\leq bc}$
2. ${\displaystyle bd<0}$并且${\displaystyle ad\geq bc}$

## 性质

${\displaystyle \mathbb {Q} }$代数闭包，例如有理数多项式的根的域，是代数数域

## p进数

${\displaystyle p}$素数，对任何非零整数${\displaystyle a}$${\displaystyle |a|_{p}=p^{-n}}$，这里${\displaystyle p^{n}}$整除${\displaystyle a}$${\displaystyle p}$的最高次幂；

${\displaystyle d_{p}\left(x,y\right)=|x-y|_{p}}$${\displaystyle \mathbb {Q} }$上定义了一个度量

## 参考文献

1. ^ 三平方の定理 (ピタゴラスの定理) の歴史 - 何ゆえ有理数と呼ぶか ? - 名前の由来 -. asait.world.coocan.jp.
2. ^ Oxford English Dictionary 2nd. Oxford University Press. 1989. Entry ratio, n., sense 2.a.
3. ^ Oxford English Dictionary 2nd. Oxford University Press. 1989. Entry rational, a. (adv.) and n.1, sense 5.a.