> 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-tw/shu-xue/05-fractions-and-binomials.md).

# 分數與二項式

## 簡介

本文說明如何排版分數與二項式係數，以下列使用的範例開始 [`amsmath` 套件](https://ctan.org/pkg/amsmath?lang=en):

```latex
\documentclass{article}
\usepackage{amsmath}
\begin{document}
二項式係數，\(\binom{n}{k}\)，其定義如下：
\[
    \binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
\end{document}
```

[在 Overleaf 中開啟此範例。](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)

該 [`amsmath` 套件](https://ctan.org/pkg/amsmath?lang=en) 只要將下列這一行加入文件導言區即可載入：

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

以下是產生的輸出：

![顯示二項式係數的 LaTeX 排版圖片](/files/a9ef0136453a1c732ef33032f3204027c4da889d)

## 顯示分數

分數的視覺外觀會因其是行內出現、作為段落的一部分，或是排版成獨立顯示在自己一行上的內容而有所不同。下一個範例示範這些視覺外觀的變化：

```latex
\documentclass{article}
% 使用 geometry 套件來縮減
% 說明文章圖形的寬度
\usepackage[textwidth=8cm]{geometry}
\begin{document}
分數可以在段落文字中以行內方式使用，
例如 \(\frac{1}{2}\)，或獨立顯示在自己的行上，
像這樣：
\[\frac{1}{2}\]
\end{document}
```

[在 Overleaf 中開啟此範例。](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)

此範例會產生以下輸出：

![示範 LaTeX 的行內與顯示數學模式](/files/97f4e64aa8459cf106d93677f31fcaaa1bd916bf)

* **注意**：更多關於數學的行內與顯示版本的資訊可參見 Overleaf 文章 [數學模式中的顯示樣式](/latex/zh-tw/shu-xue/10-display-style-in-math-mode.md).

我們的範例分數是使用 `\frac` 命令（`\frac{1}{2}`）排版的，其一般形式為 `\frac{*分子*}{*分母*}`.

## 文字樣式分數

下列範例示範如何使用 `\text{...}` 命令來排版僅含文字的分數，該命令由 `amsmath` 套件。 `\text{...}` 提供。此命令用來避免 LaTeX 將文字排版成一般數學內容。

```latex
\documentclass{article}
% 使用 geometry 套件來縮減
% 說明文章圖形的寬度
\usepackage[textwidth=8cm]{geometry}
\usepackage{amsmath}% 用於 \text{...} 命令
\begin{document}
我們使用 \texttt{amsmath} 套件命令
\verb|\text{...}| 來建立僅含文字的分數
如下：

\[\frac{\text{分子}}{\text{分母}}\]

若沒有 \verb|\text{...}| 命令，結果
看起來會像這樣：

\[\frac{分子}{分母}\]
\end{document}
```

[在 Overleaf 中開啟此範例。](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)

此範例會產生以下輸出：

![文字分數](/files/5ee42a71a8d01c0cab9e1ff467dbb8fa974d0c99)

## 排版數學中的大小與間距

LaTeX 排版的數學內容之大小與間距是 [由演算法決定的](https://tug.org/TUGboat/tb27-1/tb86jackowski.pdf) 這些演算法會套用排版數學所用字型中所含的大小與定位資料。

有時，可能有必要或有需要覆寫 LaTeX 所選的預設數學樣式——數學元素的大小與間距——這個主題在 Overleaf 說明文章中有討論 [數學模式中的顯示樣式](/latex/zh-tw/shu-xue/10-display-style-in-math-mode.md#overriding-default-mathematical-styles).

總之，可透過下列命令變更用來排版數學的預設樣式：

* `\textstyle`：套用用於段落中排版數學的樣式；
* `\displaystyle`：套用用於單獨成行排版數學的樣式；
* `\scriptstyle`：套用用於下標或上標的樣式；
* `\scriptscriptstyle`：套用用於第二層下標或上標的樣式；

下一個範例會示範這些設定。

```latex
\documentclass{article}
% 使用 geometry 套件來縮減
% 說明文章圖形的寬度
\usepackage[textwidth=9.5cm]{geometry}
\begin{document}

段落中排版的分數通常看起來像這樣：\(\frac{3x}{2}\)。你可以強制 \LaTeX{} 使用較大的顯示樣式，例如 \( \displaystyle \frac{3x}{2} \)，這也會影響行距。段落中的數學大小也可以縮小：\(\scriptstyle \frac{3x}{2}\) 或 \(\scriptscriptstyle \frac{3x}{2}\)。在 \verb|\scriptscriptstyle| 的範例中，請注意間距的縮減：字元會更靠近 \textit{vinculum}（分子與分母之間的線）。

同樣地，你也可以變更通常以顯示樣式排版的數學樣式：

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

[在 Overleaf 中開啟這個範例](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)

此範例會產生以下輸出：

![顯示 LaTeX 預設數學樣式變化的圖片](/files/0396ef07664fe15ffc2bcb0d164db852ff06d442)

## 連分數

可以將分數嵌套以取得更複雜的式子。以下範例中顯示的第二組分數都使用了 `\cfrac` 命令，這個命令是專門設計來產生連分數的。若要使用 `\cfrac` 你必須載入 [`amsmath` 套件](https://ctan.org/pkg/amsmath?lang=en) 於文件導言區。

```latex
\documentclass{article}
% 使用 geometry 套件來縮減
% 說明文章圖形的寬度
\usepackage[textwidth=9.5cm]{geometry}
% 載入 amsmath 以存取 \cfrac{...}{...} 命令
\usepackage{amsmath}
\begin{document}
分數可以巢狀嵌套，但在這個範例中，請注意預設數學樣式（如分母所用）並未產生理想結果……

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

\noindent ……所以我們使用 \verb|\displaystyle| 來改善排版：

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

以下是使用 \texttt{amsmath} \verb|\cfrac| 命令的範例：

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

以下是另一個範例，改寫自 \texttt{amsmath} 文件，展示分子左
與右定位，分別使用 \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 中開啟此範例。](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)

此範例會產生以下輸出：

![顯示 LaTeX 中巢狀分數的圖形](/files/2a579c613b3e013a0f90fbafa76d44aa52ca40a4)

## 最後一個範例

這個範例示範更複雜的連分數：

```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 中開啟此範例。](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)

此範例會產生以下輸出：

![LaTeX 排版的進階連分數圖片](/files/99e9fea66a5d91cb867d1e0f5bf667695beeb352)

## 延伸閱讀

詳情請見：

* [數學表達式](/latex/zh-tw/shu-xue/01-mathematical-expressions.md)
* [下標與上標](/latex/zh-tw/shu-xue/02-subscripts-and-superscripts.md)
* [中括號與小括號](/latex/zh-tw/shu-xue/03-brackets-and-parentheses.md)
* [使用 amsmath 對齊方程式](/latex/zh-tw/shu-xue/06-aligning-equations-with-amsmath.md)
* [運算子](/latex/zh-tw/shu-xue/07-operators.md)
* [數學字型](/latex/zh-tw/shu-xue/12-mathematical-fonts.md)
* [LaTeX2ε 不那麼簡短的入門介紹](http://www.ctan.org/tex-archive/info/lshort/)


---

# Agent Instructions
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