> 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/02-subscripts-and-superscripts.md). # 下標與上標 ## 簡介 在數學表達式中，指數和索引，以及某些特殊運算子，使用上標和下標非常常見。本文將說明如何在簡單表達式、積分、求和等等情況下撰寫上標和下標。 定積分是最常見的數學表達式之一，所以讓我們來看一個範例： ```latex \[ \int\limits_0^1 x^2 + y^2 \ dx \] ``` [在 Overleaf 中開啟此範例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%5Cint%5Climits_0%5E1+x%5E2+%2B+y%5E2+%5C+dx+%5C%5D%0A%5Cend%7Bdocument%7D) 這段 LaTeX 程式碼會產生： $$\int\limits\_0^1 x^2 + y^2 \ dx$$ 按照慣例，在 LaTeX 中，上標和下標分別使用字元 `^` 和 `_` ，例如，套用於 $$x$$ 和 $$y$$ 的指數。在上面的程式碼片段中。這些字元也可以與數學符號一起使用，例如積分（`\int`）就包含在上面的範例中，其中 `_` 用來設定下限，而 `^` 用來設定上限。 命令 `\limits` 會改變積分中上下限的顯示方式；如果沒有它，上下限就會顯示在積分符號旁邊，而不是上下方： ```latex \[ \int_0^1 x^2 + y^2 \ dx \] ``` [在 Overleaf 中開啟此範例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%5Cint_0%5E1+x%5E2+%2B+y%5E2+%5C+dx+%5C%5D%0A%5Cend%7Bdocument%7D) 這段 LaTeX 程式碼會產生： $$\int\_0^1 x^2 + y^2 \ dx$$ ## 更詳細的範例 這些符號 `_` 和 `^` 也可以在同一個表達式中組合，例如： ```latex \[ a_1^2 + a_2^2 = a_3^2 \] ``` [在 Overleaf 中開啟此範例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+2\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+a_1%5E2+%2B+a_2%5E2+%3D+a_3%5E2+%5C%5D%0A%5Cend%7Bdocument%7D) 這段 LaTeX 程式碼會產生： $$a\_1^2 + a\_2^2 = a\_3^2$$ 如果表達式包含較長的上標或下標，則需要用大括號包起來，因為 LaTeX 通常只會將數學命令 `^` 和 `_` 套用到後面的單一字元： ```latex \[ x^{2 \alpha} - 1 = y_{ij} + y_{ij} \] ``` [在 Overleaf 中開啟此範例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+3\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+x%5E%7B2+%5Calpha%7D+-+1+%3D+y_%7Bij%7D+%2B+y_%7Bij%7D++%5C%5D%0A%5Cend%7Bdocument%7D) 這段 LaTeX 程式碼會產生： $$x^{2 \alpha} - 1 = y\_{ij} + y\_{ij}$$ 下標和上標可以以各種方式巢狀與組合。不過，在巢狀下標／上標時，請記得每個命令都必須對應到單一元素；這可以是單一字母或數字，如上例所示，或是用大括號或中括號括起來的較複雜數學表達式。例如： ```latex \[ (a^n)^{r+s} = a^{nr+ns} \] ``` [在 Overleaf 中開啟此範例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+4\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%28a%5En%29%5E%7Br%2Bs%7D+%3D+a%5E%7Bnr%2Bns%7D++%5C%5D%0A%5Cend%7Bdocument%7D) 這段 LaTeX 程式碼會產生： $$(a^n)^{r+s} = a^{nr+ns}$$ ## 使用下標和上標的運算子 某些數學運算子可能需要下標和上標。最常見的情況是積分 `\int` （請參見 [引言](#introduction)）以及求和（`\sum`）運算子，其上下界會精確地以下標和上標排版。 ```latex \[ \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} \] ``` [在 Overleaf 中開啟此範例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+5\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%0A%3D+%5Cprod_p+%5Cfrac%7B1%7D%7B1+-+p%5E%7B-s%7D%7D+%5C%5D%0A%5Cend%7Bdocument%7D) 這段 LaTeX 程式碼會產生： $$\sum\_{n=1}^{\infty} \frac{1}{n^s} = \prod\_p \frac{1}{1 - p^{-s}}$$ 其他需要下標／上標的常用運算子請參見 [參考指南](#reference-guide). ## 在 Overleaf 中開啟所有程式碼片段 使用下方提供的連結，將上面的所有範例作為單一 Overleaf 專案開啟： ```latex 以下是下標和上標的簡單用法範例： \[ \int\limits_0^1 x^2 + y^2 \ dx \] \vspace{1cm} 在同一個表達式中使用上標和下標 \[ a_1^2 + a_2^2 = a_3^2 \] \vspace{1cm} 較長的下標和上標： \[ x^{2 \alpha} - 1 = y_{ij} + y_{ij} \] \vspace{1cm} 巢狀下標和上標 \[ (a^n)^{r+s} = a^{nr+ns} \] \vspace{1cm} 含有下標和上標的數學方程式範例 \[ \sum_{i=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} \] \vspace{1cm} 平方根的用法 \[ \sqrt[4]{4ac} = \sqrt{4ac}\sqrt{4ac} \] ``` [將上面的所有範例作為單一 Overleaf 專案開啟。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+6\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Ctitle%7BSubscripts+and+Superscripts%7D%0A%5Cauthor%7BExamples+from+Overleaf%7D%0A%5Cbegin%7Bdocument%7D%0A%5Cmaketitle%0AHere+are+some+examples+of+simple+usage+of+subscripts+and+superscripts%3A%0A%0A%5C%5B+%5Cint%5Climits_0%5E1+x%5E2+%2B+y%5E2+%5C+dx+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0AUsing+superscript+and+subscripts+in+the+same+expression%0A%0A%5C%5B+a_1%5E2+%2B+a_2%5E2+%3D+a_3%5E2+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0ALonger+subscripts+and+superscripts%3A%0A%0A%5C%5B+x%5E%7B2+%5Calpha%7D+-+1+%3D+y_%7Bij%7D+%2B+y_%7Bij%7D++%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0ANested+subscripts+and+superscripts%0A%0A%5C%5B+%28a%5En%29%5E%7Br%2Bs%7D+%3D+a%5E%7Bnr%2Bns%7D+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0AExample+of+a+mathematical+equation+with+subscripts+and+superscripts%0A%0A%5C%5B+%5Csum_%7Bi%3D1%7D%5E%7B%5Cinfty%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%3D+%5Cprod_p+%5Cfrac%7B1%7D%7B1+-+p%5E%7B-s%7D%7D+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0ASquared+root+usage%0A%0A%5C%5B+%5Csqrt%5B4%5D%7B4ac%7D+%3D+%5Csqrt%7B4ac%7D%5Csqrt%7B4ac%7D+%5C%5D%0A%5Cend%7Bdocument%7D) ## 參考指南 其他範例與運算子 | LaTeX 標記 | 顯示為 | | --------------------- | ------------------------------------------------------------------------------------------------------------------------------------------- | | `a_{n_i}` | a n i {\displaystyle a\_{n\_{i}}} ![{\displaystyle a\_{n\_{i}}}](/files/bd8ac7e1362e87c3edd0e756567f977249d1d82c) | | `\int_{i=1}^n` | ∫ i = 1 n {\displaystyle \int \_{i=1}^{n}} ![{\displaystyle \int \_{i=1}^{n}}](/files/e64fc62c45d9c6867558a51cbc4afa507fe81582) | | `\sum_{i=1}^{\infty}` | ∑ i = 1 ∞ {\displaystyle \sum \_{i=1}^{\infty }} ![{\displaystyle \sum \_{i=1}^{\infty }}](/files/c84c7ff94eacf20bd931b5bd2d16f59fa3e635f3) | | `\prod_{i=1}^n` | ∏ i = 1 n {\displaystyle \prod \_{i=1}^{n}} ![{\displaystyle \prod \_{i=1}^{n}}](/files/d6aff4a3defefebff06692284684c0c740c147e6) | | `\cup_{i=1}^n` | ∪ i = 1 n {\displaystyle \cup \_{i=1}^{n}} ![{\displaystyle \cup \_{i=1}^{n}}](/files/68110fa95101867b0f676e0f4f6e35b58519ca0a) | | `\cap_{i=1}^n` | ∩ i = 1 n {\displaystyle \cap \_{i=1}^{n}} ![{\displaystyle \cap \_{i=1}^{n}}](/files/b85c1a38d74a59a10424a459c00699d5a8761f48) | | `\oint_{i=1}^n` | ∮ i = 1 n {\displaystyle \oint \_{i=1}^{n}} ![{\displaystyle \oint \_{i=1}^{n}}](/files/f5f7434cd9b0c605d551ff15cdff6cdd2801a9c2) | | `\coprod_{i=1}^n` | ∐ i = 1 n {\displaystyle \coprod \_{i=1}^{n}} ![{\displaystyle \coprod \_{i=1}^{n}}](/files/a392f0c12ebf2b7ffeed2f31127665e4e04b4065) | 另外還有 `bigcup` 和 `bigcap` 與 `cup` 和 `cap` 類似的命令，但那些是用於較大的表達式。 ## 延伸閱讀 更多資訊請參見 * [希臘字母與數學符號列表](/latex/zh-tw/shu-xue/11-list-of-greek-letters-and-math-symbols.md) * [運算子](/latex/zh-tw/shu-xue/07-operators.md) * [積分、求和與極限](/latex/zh-tw/shu-xue/09-integrals-sums-and-limits.md) * [數學字型](/latex/zh-tw/shu-xue/12-mathematical-fonts.md) * [中括號與小括號](/latex/zh-tw/shu-xue/03-brackets-and-parentheses.md) * [LaTeX 符號大全](http://www.rpi.edu/dept/arc/training/latex/LaTeX_symbols.pdf) （amssymb **amssymb** 套件中的可用數學符號完整清單） * [LaTeX2ε 不那麼簡短的入門介紹](http://www.ctan.org/tex-archive/info/lshort/) * [使用 amsmath 對齊方程式](/latex/zh-tw/shu-xue/06-aligning-equations-with-amsmath.md) --- # Agent Instructions This documentation is published with GitBook. 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