> For the complete documentation index, see [llms.txt](https://overleaf-pro.ayaka.space/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://overleaf-pro.ayaka.space/latex/zh-cn/shu-xue/05-fractions-and-binomials.md).

# 分数和二项式

## 引言

本文介绍如何排版分数和二项式系数，先从下面这个使用 [`amsmath` 宏包](https://ctan.org/pkg/amsmath?lang=en):

```latex
\documentclass{article}
\usepackage{amsmath}
\begin{document}
二项式系数，\(\binom{n}{k}\)，定义为以下表达式：
\[
    \binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
\end{document}
```

[在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Fractions+and+Binomials\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cusepackage%7Bamsmath%7D%0A%5Cbegin%7Bdocument%7D%0AThe+binomial+coefficient%2C+%5C%28%5Cbinom%7Bn%7D%7Bk%7D%5C%29%2C+is+defined+by+the+expression%3A%0A%5C%5B%0A++++%5Cbinom%7Bn%7D%7Bk%7D+%3D+%5Cfrac%7Bn%21%7D%7Bk%21%28n-k%29%21%7D%0A%5C%5D%0A%5Cend%7Bdocument%7D)

该 [`amsmath` 宏包](https://ctan.org/pkg/amsmath?lang=en) 通过在文档导言区添加以下一行来加载：

```latex
\usepackage{amsmath}
```

以下是生成的输出：

![显示二项式系数 LaTeX 排版的图片](/files/0e307ad057542fa032e579968608ca2b3f517d41)

## 显示分数

分数的外观会因其是行内显示、作为段落的一部分，还是作为独立内容单独占一行而有所不同。下面的示例演示这些外观变化：

```latex
\documentclass{article}
% 使用 geometry 宏包来减少
% 帮助文章图片的宽度
\usepackage[textwidth=8cm]{geometry}
\begin{document}
分数可以在段落文本中行内使用，例如
例如 \(\frac{1}{2}\)，或者单独显示在一行中，
像这样的：
\[\frac{1}{2}\]
\end{document}
```

[在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Inline+and+display+of+fractions+and+binomials\&snip=%5Cdocumentclass%7Barticle%7D%0A%25+Using+the+geometry+package+to+reduce%0A%25+the+width+of+help+article+graphics%0A%5Cusepackage%5Btextwidth%3D8cm%5D%7Bgeometry%7D%0A%5Cbegin%7Bdocument%7D%0AFractions+can+be+used+inline+within+the+paragraph+text%2C+for+%0Aexample+%5C%28%5Cfrac%7B1%7D%7B2%7D%5C%29%2C+or+displayed+on+their+own+line%2C+%0Asuch+as+this%3A%0A%5C%5B%5Cfrac%7B1%7D%7B2%7D%5C%5D%0A%5Cend%7Bdocument%7D)

此示例生成以下输出：

![演示 LaTeX 中的行内和独立显示数学模式](/files/0b973d463386828c71ae37af653f78a3ce5df13d)

* **注意**：关于数学的行内和显示版本的更多信息可参见 Overleaf 文章 [数学模式中的显示样式](/latex/zh-cn/shu-xue/10-display-style-in-math-mode.md).

我们的示例分数使用 `\frac` 命令（`\frac{1}{2}`）其一般形式为 `\frac{*numerator*}{*denominator*}`.

## 文本样式分数

下面的示例演示如何使用 `\text{...}` 命令，由 `amsmath` 宏包提供。 `\text{...}` 该命令用于防止 LaTeX 将文本排版为常规数学内容。

```latex
\documentclass{article}
% 使用 geometry 宏包来减少
% 帮助文章图片的宽度
\usepackage[textwidth=8cm]{geometry}
\usepackage{amsmath}% 用于 \text{...} 命令
\begin{document}
我们使用 \texttt{amsmath} 宏包中的命令
\verb|\text{...}| 创建仅文本分数
如下所示：

\[\frac{\text{numerator}}{\text{denominator}}\]

如果没有 \verb|\text{...}| 命令，结果
如下所示：

\[\frac{numerator}{denominator}\]
\end{document}
```

[在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Text+fractions\&snip=%5Cdocumentclass%7Barticle%7D%0A%25+Using+the+geometry+package+to+reduce%0A%25+the+width+of+help+article+graphics%0A%5Cusepackage%5Btextwidth%3D8cm%5D%7Bgeometry%7D%0A%5Cusepackage%7Bamsmath%7D%25+For+the+%5Ctext%7B...%7D+command%0A%5Cbegin%7Bdocument%7D%0AWe+use+the+%5Ctexttt%7Bamsmath%7D+package+command%0A%5Cverb%7C%5Ctext%7B...%7D%7C+to+create+text-only+fractions%0Alike+this%3A%0A%0A%5C%5B%5Cfrac%7B%5Ctext%7Bnumerator%7D%7D%7B%5Ctext%7Bdenominator%7D%7D%5C%5D%0A%0AWithout+the+%5Cverb%7C%5Ctext%7B...%7D%7C+command+the+result+%0Alooks+like+this%3A%0A%0A%5C%5B%5Cfrac%7Bnumerator%7D%7Bdenominator%7D%5C%5D%0A%5Cend%7Bdocument%7D)

此示例生成以下输出：

![文字分数](/files/66ee29dee0323c365d2320f7bc32c0e5a9e76662)

## 排版数学中的大小和间距

LaTeX 排版的数学内容的大小和间距是 [由算法决定的](https://tug.org/TUGboat/tb27-1/tb86jackowski.pdf) 这些算法会应用用于排版数学的字体中包含的大小和位置信息。

偶尔，可能需要或希望覆盖 LaTeX 选择的默认数学样式——数学元素的大小和间距——这是 Overleaf 帮助文章中讨论的一个主题 [数学模式中的显示样式](/latex/zh-cn/shu-xue/10-display-style-in-math-mode.md#overriding-default-mathematical-styles).

总之，用于排版数学的默认样式可以通过以下命令更改：

* `\textstyle`：应用用于在段落中排版数学的样式；
* `\displaystyle`：应用用于单独成行排版数学的样式；
* `\scriptstyle`：应用用于下标或上标的样式；
* `\scriptscriptstyle`：应用用于二级下标或上标的样式；

这些在下一个示例中演示。

```latex
\documentclass{article}
% 使用 geometry 宏包来减少
% 帮助文章图片的宽度
\usepackage[textwidth=9.5cm]{geometry}
\begin{document}

段落中排版的分数通常看起来像这样：\(\frac{3x}{2}\)。你可以强制 \LaTeX{} 使用更大的显示样式，例如 \( \displaystyle \frac{3x}{2} \)，这也会影响行距。段落中的数学内容大小也可以减小：\(\scriptstyle \frac{3x}{2}\) 或 \(\scriptscriptstyle \frac{3x}{2}\)。对于 \verb|\scriptscriptstyle| 示例，请注意间距的缩小：字符更靠近 \textit{vinculum}（分子和分母之间的分隔线）。

同样，你也可以更改通常以显示样式排版的数学的样式：

\[f(x)=\frac{P(x)}{Q(x)}\quad \textrm{and}\quad \textstyle f(x)=\frac{P(x)}{Q(x)}\quad \textrm{and}\quad \scriptstyle f(x)=\frac{P(x)}{Q(x)}\]
\end{document}
```

[在 Overleaf 中打开此示例](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Changing+default+math+styles\&snip=%5Cdocumentclass%7Barticle%7D%0A%25+Using+the+geometry+package+to+reduce%0A%25+the+width+of+help+article+graphics%0A%5Cusepackage%5Btextwidth%3D9.5cm%5D%7Bgeometry%7D%0A%5Cbegin%7Bdocument%7D%0A%0AFractions+typeset+within+a+paragraph+typically+look+like+this%3A+%5C%28%5Cfrac%7B3x%7D%7B2%7D%5C%29.+You+can+force+%5CLaTeX%7B%7D+to+use+the+larger+display+style%2C+such+as+%5C%28+%5Cdisplaystyle+%5Cfrac%7B3x%7D%7B2%7D+%5C%29%2C+which+also+has+an+effect+on+line+spacing.+The+size+of+maths+in+a+paragraph+can+also+be+reduced%3A+%5C%28%5Cscriptstyle+%5Cfrac%7B3x%7D%7B2%7D%5C%29+or+%5C%28%5Cscriptscriptstyle+%5Cfrac%7B3x%7D%7B2%7D%5C%29.+For+the+%5Cverb%7C%5Cscriptscriptstyle%7C+example+note+the+reduction+in+spacing%3A+characters+are+moved+closer+to+the+%5Ctextit%7Bvinculum%7D+%28the+line+separating+numerator+and+denominator%29.%0A%0AEqually%2C+you+can+change+the+style+of+mathematics+normally+typeset+in+display+style%3A%0A%0A%5C%5Bf%28x%29%3D%5Cfrac%7BP%28x%29%7D%7BQ%28x%29%7D%5Cquad+%5Ctextrm%7Band%7D%5Cquad+%5Ctextstyle+f%28x%29%3D%5Cfrac%7BP%28x%29%7D%7BQ%28x%29%7D%5Cquad+%5Ctextrm%7Band%7D%5Cquad+%5Cscriptstyle+f%28x%29%3D%5Cfrac%7BP%28x%29%7D%7BQ%28x%29%7D%5C%5D%0A%5Cend%7Bdocument%7D)

此示例生成以下输出：

![显示 LaTeX 默认数学样式变化的图片](/files/7550d96983e28a9e0c6ec406617fb868de3d2df3)

## 连分数

分数可以嵌套以得到更复杂的表达式。下面示例中显示的第二组分数都使用了 `\cfrac` 命令，它专门用于生成连分数。要使用 `\cfrac` 你必须加载 [`amsmath` 宏包](https://ctan.org/pkg/amsmath?lang=en) 到文档导言区。

```latex
\documentclass{article}
% 使用 geometry 宏包来减少
% 帮助文章图片的宽度
\usepackage[textwidth=9.5cm]{geometry}
% 加载 amsmath 以使用 \cfrac{...}{...} 命令
\usepackage{amsmath}
\begin{document}
分数可以嵌套，但在这个示例中请注意，分母中使用的默认数学样式并不能产生理想效果……

\[ \frac{1+\frac{a}{b}}{1+\frac{1}{1+\frac{1}{a}}} \]

\noindent ……因此我们使用 \verb|\displaystyle| 来改进排版：

\[ \frac{1+\frac{a}{b}} {\displaystyle 1+\frac{1}{1+\frac{1}{a}}} \]

下面是一个使用 \texttt{amsmath} 的 \verb|\cfrac| 命令的示例：

\[
  a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}
\]

下面是另一个示例，源自 \texttt{amsmath} 文档，演示了左
和右对齐放置分子，分别使用 \verb|\cfrac[l]| 和 \verb|\cfrac[r]|：
\[
\cfrac[l]{1}{\sqrt{2}+
\cfrac[r]{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+\dotsb}}}
\]
\end{document}
```

[在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Example+of+continued+fractions\&snip=%5Cdocumentclass%7Barticle%7D%0A%25+Using+the+geometry+package+to+reduce%0A%25+the+width+of+help+article+graphics%0A%5Cusepackage%5Btextwidth%3D9.5cm%5D%7Bgeometry%7D%0A%25+Load+amsmath+to+access+the+%5Ccfrac%7B...%7D%7B...%7D+command%0A%5Cusepackage%7Bamsmath%7D%0A%5Cbegin%7Bdocument%7D%0AFractions+can+be+nested+but%2C+in+this+example%2C+note+how+the+default+math+styles%2C+as+used+in+the+denominator%2C+don%27t+produce+ideal+results...%0A%0A%5C%5B+%5Cfrac%7B1%2B%5Cfrac%7Ba%7D%7Bb%7D%7D%7B1%2B%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B1%7D%7Ba%7D%7D%7D+%5C%5D%0A%0A%5Cnoindent+...so+we+use+%5Cverb%7C%5Cdisplaystyle%7C+to+improve+typesetting%3A%0A%0A%5C%5B+%5Cfrac%7B1%2B%5Cfrac%7Ba%7D%7Bb%7D%7D+%7B%5Cdisplaystyle+1%2B%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B1%7D%7Ba%7D%7D%7D+%5C%5D%0A%0AHere+is+an+example+which+uses+the+%5Ctexttt%7Bamsmath%7D+%5Cverb%7C%5Ccfrac%7C+command%3A%0A%0A%5C%5B%0A++a_0%2B%5Ccfrac%7B1%7D%7Ba_1%2B%5Ccfrac%7B1%7D%7Ba_2%2B%5Ccfrac%7B1%7D%7Ba_3%2B%5Ccdots%7D%7D%7D%0A%5C%5D%0A%0AHere+is+another+example%2C+derived+from+the+%5Ctexttt%7Bamsmath%7D+documentation%2C+which+demonstrates+left%0Aand+right+placement+of+the+numerator+using+%5Cverb%7C%5Ccfrac%5Bl%5D%7C+and+%5Cverb%7C%5Ccfrac%5Br%5D%7C+respectively%3A%0A%5C%5B%0A%5Ccfrac%5Bl%5D%7B1%7D%7B%5Csqrt%7B2%7D%2B%0A%5Ccfrac%5Br%5D%7B1%7D%7B%5Csqrt%7B2%7D%2B%0A%5Ccfrac%7B1%7D%7B%5Csqrt%7B2%7D%2B%5Cdotsb%7D%7D%7D%0A%5C%5D%0A%5Cend%7Bdocument%7D)

此示例生成以下输出：

![演示 LaTeX 中嵌套分数排版的图示](/files/66ff5049d954d0bcc54ea2d6708df2410281d810)

## 最后一个示例

这个示例演示一个更复杂的连分数：

```latex
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\newcommand*{\contfrac}[2]{%
{
  \rlap{$\dfrac{1}{\phantom{#1}}$}%
  \genfrac{}{}{0pt}{0}{}{#1+#2}%
}
}
\[
  a_0 +
  \contfrac{a_1}{
  \contfrac{a_2}{
  \contfrac{a_3}{
  \genfrac{}{}{0pt}{0}{}{\ddots}
  }}}
\]

\end{document}
```

[在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=A+more+complex+continued+fraction\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cusepackage%7Bamsmath%7D%0A%5Cbegin%7Bdocument%7D%0A%5Cnewcommand%2A%7B%5Ccontfrac%7D%5B2%5D%7B%25%0A%7B%0A++%5Crlap%7B%24%5Cdfrac%7B1%7D%7B%5Cphantom%7B%231%7D%7D%24%7D%25%0A++%5Cgenfrac%7B%7D%7B%7D%7B0pt%7D%7B0%7D%7B%7D%7B%231%2B%232%7D%25%0A%7D%0A%7D%0A%5C%5B%0A++a_0+%2B%0A++%5Ccontfrac%7Ba_1%7D%7B%0A++%5Ccontfrac%7Ba_2%7D%7B%0A++%5Ccontfrac%7Ba_3%7D%7B%0A++%5Cgenfrac%7B%7D%7B%7D%7B0pt%7D%7B0%7D%7B%7D%7B%5Cddots%7D%0A++%7D%7D%7D%0A%5C%5D%0A%0A%5Cend%7Bdocument%7D)

此示例生成以下输出：

![LaTeX 排版的高级连分数图片](/files/74be7718de467e4791f381efc25c0878f6e60bd9)

## 进一步阅读

更多信息请参见：

* [数学表达式](/latex/zh-cn/shu-xue/01-mathematical-expressions.md)
* [下标和上标](/latex/zh-cn/shu-xue/02-subscripts-and-superscripts.md)
* [括号](/latex/zh-cn/shu-xue/03-brackets-and-parentheses.md)
* [使用 amsmath 对齐方程](/latex/zh-cn/shu-xue/06-aligning-equations-with-amsmath.md)
* [运算符](/latex/zh-cn/shu-xue/07-operators.md)
* [数学字体](/latex/zh-cn/shu-xue/12-mathematical-fonts.md)
* [LaTeX2ε 不那么简短的介绍](http://www.ctan.org/tex-archive/info/lshort/)


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