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{{noteTA|G1=Math|G2=IT}}
{{Otheruses}}
{{整数
| other list = {{Numbers digits|1|last=0|next=大数_(数学)|template=#invoke:NumberUtil{{!}}exp10link}}
| 花碼 = 一或〡
| 罗马数字 = Ⅰ
| 相反數 = [[−1]]
| 质因数分解 = [[單位元]]
| 因數 = 1
| 一進制 = {{進制|1|1|sub=1}}
| 三進制 = hide
| 四進制 = hide
| 五進制 = hide
|greek prefix=[[Wiktionary:mono-|mono-]]/[[Wiktionary:haplo-|haplo-]]
|latin prefix=[[Wiktionary:uni-|uni-]]
|lang1=[[英語]]
|lang1 symbol=one
|lang2=[[阿拉伯文数字|阿拉伯文]]、 [[中库尔德语]]、 [[波斯语]]、 [[信德语]]、 {{link-en|印度斯坦數字|Urdu numerals|印度斯坦语}}
|lang2 symbol={{resize|150%|١}}
|lang3=[[阿萨姆语]]和[[孟加拉语]]
|lang3 symbol={{resize|150%|১}}
|lang4=[[漢語]]
|lang4 symbol=一/弌/壹
|lang5=[[天城文]]
|lang5 symbol={{resize|150%|१}}
|lang6=[[吉茲字母|吉茲]]
|lang6 symbol={{resize|150%|፩}}
|lang7={{link-en|格鲁吉亚语数字|Georgian numerals|格鲁吉亚语}}
|lang7 symbol={{resize|130%|Ⴀ/ⴀ/ა}}([[Ani (letter)|Ani]])

|lang8=[[希伯來數字|希伯來語]]
|lang8 symbol=[[aleph|{{resize|150%|א}}]]
|lang9=[[日語數字|日語]]
|lang9 symbol={{lang|ja|一/壱}}
|lang10=[[卡纳达语]]
|lang10 symbol={{resize|150%|[[೧]]}}
|lang11=[[高棉數字]]
|lang11 symbol={{resize|150%|១}}
|lang13=[[马拉雅拉姆语]]
|lang13 symbol=൧
|lang14=[[曼尼普尔语]]
|lang14 symbol={{resize|150%|꯱}}
|lang15=[[泰文数字|泰文]]
|lang15 symbol={{resize|150%|๑}}
|lang16=[[泰米尔语]]
|lang16 symbol={{resize|150%|௧}}
|lang17=[[泰卢固语]]
|lang17 symbol={{resize|150%|೧}}
}}
{{高斯整數導航}}
'''1'''（一）是最小的[[正整数]]，也是继[[0]]第二小的[[自然数]]。阿拉伯数字“1”的形状由[[苏美尔|古苏美尔]]和[[巴比伦]]符号演变而来。

在数学中，1为乘法单位元，即任何数字与1相乘皆等于其本身。按照惯例，1不被视为素数。在数字电路中，1表示[[二进制]]中的“开启”状态，是计算技术的基础。在[[哲学]]领域，1在诸多传统中象征着终极现实或存在的本源。

== 在数学上 ==
数字1是[[0]]之后之首个[[自然数]]。每一自然数，含1，皆由后继法构造，即于前一自然数加1而成。数字1为[[整数]]、[[实数]]及[[复数 (数学)|复数]]之[[乘法]]单位元，意即任何数字<math>n</math>与1相乘保持其原值不变（<math>1\times n = n\times 1 = n</math>）。因此，[[正方形]]（<math>1^2=1</math>）、[[平方根]]（<math>\sqrt{1} = 1</math>）、1的任何其他[[幂]]始终等于1本身{{sfn|Colman|1912|loc=chapt.2|pp=9–10}}。1是它自己的[[阶乘]]（<math>1!=1</math>），而1也是0的阶乘（这是空积的特例）{{sfn|Graham|Knuth|Patashnik|1994|p=111}}。尽管1符合[[素数]]的定义，即只能被1和它本身（也是1）整除，但按照现代惯例，它既不是素数也不是[[合数]]{{sfn|Caldwell|Xiong|2012|pp=8–9}}。

自然数的不同数学构造以各种方式表示1。在[[朱塞佩·皮亞諾|朱塞佩·皮亚诺]]最初制定的[[皮亚诺公理]]（一组精确且逻辑地定义自然数的公设）中，1被视为自然数序列的起点{{sfn|Kennedy|1974|pp=389}}{{sfn|Peano|1889|p=1}}。皮亚诺后来修订了他的公理，将序列的起点改为0{{sfn|Kennedy|1974|pp=389}}{{sfn|Peano|1908|p=27}}。在[[冯·诺伊曼]]对自然数的基数赋值中，每个数字被定义为包含其之前所有数字的集合，1 被表示为单元素集<math>\{0\}</math>，即仅包含元素0的集合{{sfn|Halmos|1974|p=32}}。用于计数的一进制系统是“以1为基数”的系统的一个例子，因为只需要一个标记——计数本身。虽然这是表示自然数的最简单方法，但由于其可读性较差，以1为基数很少被用作实际计数的基数{{sfn|Hodges|2009|p=14}}{{sfn|Hext|1990}}。

在许多数学和工程问题中，数值通常会被归一化，使其落在单位区间([0,1]) 内，其中1表示最大可能值。例如，根据定义，1 表示某个事件绝对或几乎肯定会发生的概率{{sfn|Graham|Knuth|Patashnik|1994|p=381}}。同样，向量通常被归一化为单位向量（即幅值为1的向量），因为它们通常具有更理想的性质。函数通常会根据其定义域上积分为1、最大值为1或平方积分值为1的条件进行归一化，具体取决于其应用{{sfn|Blokhintsev|2012|p=35}}。

1是[[勒让德常数]]的值。该常数由[[阿德里安-马里·勒让德]]于1808年提出，用于描述素数计数[[函数]]的渐近行为{{sfn|Pintz|1980|pp=733-735}}。韦伊关于玉河数的猜想指出，对于所有单连通群（即路径连通且无“空洞”的群），其玉河数<math>\tau(G)</math>是一个关于定义在全局数域上的连通[[线性代数]]群的几何度量，其值为1{{sfn|Gaitsgory|Lurie|2019|pp=204–307}}{{sfn|Kottwitz|1988}}。

1是许多现实世界数值数据中最常见的首位数字。这是本福德定律的结果，该定律指出某一特定首位数字<math>d</math>出现的概率为<math display="inline"> \log_{10} \left(\frac{d+1}{d} \right) </math>。现实世界中的数字往往呈指数或对数增长，这使得它们的首位数字分布偏向较小的数字，其中数字1的出现频率约为30%{{sfn|Miller|2015|pp=3-4}}。

== 符号和表示 ==
=== 历史 ===
在语言学上，一是一个基数词，用于计数或表示一组事物中的数量{{sfn|Hurford|1994|pp=23–24}}。在已知最早的数字系统记录中，包括刻于泥板上的[[苏美尔]]十进六十进制系统，该系统的记载可追溯到公元前第三千年上半叶{{sfn|Conway|Guy|1996|p=17}}。在最早期的苏美尔数字中，表示数字1和60的符号均为横向的半圆形符号{{sfn|Chrisomalis|2010|p=241}}。而到了公元前约2350年，原有的弧形苏美尔数字已被[[楔形文字]]符号所取代，其中数字1和60均以相同的、主要呈垂直方向的符号来表示。
[[File:Babylonian 1.svg|50px|centre|]]
苏美尔的楔形文字数字系统是埃布拉语和亚述-巴比伦[[闪米特语族|闪米特语]]十进制楔形文字系统的直系先祖{{sfn|Chrisomalis|2010|p=244}}。现存的巴比伦文献大多来自古巴比伦（约公元前1500年）和[[塞琉古帝国|塞琉古]]（约公元前300年）两个时期{{sfn|Conway|Guy|1996|p=17}}。在巴比伦楔形文字的数字记法中，数字1和60仍使用与苏美尔系统相同的符号{{sfn|Chrisomalis|2010|p=249}}。

在现代西方世界中，最常用于表示数字1的符号是[[阿拉伯数字]]，其形状通常为一条竖线，上方常带有衬线，有时底部还带有一条短横线。这一符号的起源可以追溯到古印度的[[婆罗米文]]，大约公元前250年，[[阿育王]]在其《[[阿育王詔書|阿育王诏书]]》中就曾以一条简单的竖线表示数字1<ref>{{cite journal|doi=10.3126/jie.v14i1.20077 |title=Evidences of Hierarchy of Brahmi Numeral System |date=2018 |last1=Acharya |first1=Eka Ratna |journal=Journal of the Institute of Engineering |volume=14 |issue=1 |pages=136–142 |doi-access=free }}</ref>。这种文字体系中的数字形状在中世纪经由[[马格里布]]和[[安达卢斯]]传入[[欧洲]]{{sfn|Schubring|2008|pp=147}}。阿拉伯数字以及用于表示数字一的其他字形（例如罗马数字（Ⅰ）、中文数字（一））都是表意文字。它们不依赖语音分解，而是直接表示“一”这个概念{{sfn|Crystal|2008|pp=289}}。1在[[原始印欧语]]词根*oi-no-（意为“一，独一无二”）<ref name="etymonline">{{cite web |title=Online Etymology Dictionary |url=http://www.etymonline.com/index.php?term=one |website=etymonline.com |publisher=Douglas Harper |access-date=2013-12-30 |archive-date=2013-12-30 |archive-url=https://web.archive.org/web/20131230234708/http://www.etymonline.com/index.php?term=one |url-status=live }}</ref>。
=== 现代字体 ===
{{multiple image
| total_width = 400
| image1 = Woodstock typewriter, 1940s, daylight - keyboard.jpg
| caption1 = 这台20世纪40年代的伍德斯托克打字机缺少数字1的单独按键。
| image2 = Mediaevalziffern.svg
| caption2 = [[Hoefler Text]]是一种于1991年设计的字体，它将数字1表示为类似于小号大写字母I。
}}
在现代字体中，数字1的字符形状通常排版为带上升部的等高数字，以使数字的高度和宽度与大写字母相同。然而，在带有文本数字的字体（也称为旧式数字或非等高数字）中，字形通常为[[x字高]]，并设计为遵循小写字母的节奏，例如[[File:TextFigs148.svg|50px|alt=Horizontal guidelines with a one fitting within lines, a four extending below guideline, and an eight poking above guideline]]{{sfn|Cullen|2007|p=93}}。在旧式字体（例如，Hoefler Text）中，数字1的字体类似于I的小写版本，顶部和底部有平行的衬线，而大写字母I保留了全高形式。这是罗马数字的遗产，其中Ⅰ代表1<ref>{{Cite web|url=https://www.typography.com/|title=Fonts by Hoefler&Co.|website=www.typography.com|access-date=2023-11-21|archive-date=2024-11-23|archive-url=https://web.archive.org/web/20241123092348/https://www.typography.com/|url-status=live}}</ref>。许多老式打字机没有数字1的专用键，需要使用小写字母L或大写字母Ⅰ来代替<ref name="medium-typewriters">{{Cite web|url=https://medium.com/@PostHasteCo/why-old-typewriters-lack-a-1-key-83d777f1e9d0|title=Why Old Typewriters Lack A "1" Key|first=|last=|date=2017-04-02|work=Post Haste Telegraph Company}}</ref>{{sfn|Polt|2015|pp=203}}{{sfn|Chicago|1993|pp=52}}{{sfn|Guastello|2023|pp=453}}。
[[File:Clock 24 J.jpg|thumb|alt=Decorative clay/stone circular off-white sundial with bright gold stylized sunburst in center of the 24-hour clock face, one through twelve clockwise on right, and one through twelve again clockwise on left, with J shapes where ones' digits would be expected when numbering the clock hours. Shadow suggests 3 PM toward the lower left.|[[威尼斯]]的24小时塔鐘，其使用J代替1。]]
小写字母“j”可以被认为是小写[[罗马数字]]“i”的斜体变体，常用于小写罗马数字的最后一个字母“i”。历史上也曾用j或J代替阿拉伯数字“1”<ref>{{Cite web|url=https://books.google.com/books?id=QO5UAAAAcAAJ&dq=%22JO+JJ+J2+J3%22&pg=PA70|title=Der allzeitfertige Rechenmeister|first=Christian|last=Köhler|date=1693-11-23|via=Google Books|page=70}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=MIW8-UrpEwIC&dq=%22JO+JJ+J2+J3%22&pg=PA341|title=Naeuw-keurig reys-boek: bysonderlijk dienstig voor kooplieden, en reysende persoonen, sijnde een trysoor voor den koophandel, in sigh begrijpende alle maate, en gewighte, Boekhouden, Wissel, Asseurantie ... : vorders hoe men ... kan reysen ... door Neederlandt, Duytschlandt, Vrankryk, Spanjen, Portugael en Italiën ...|date=1679-11-23|publisher=by Jan ten Hoorn|via=Google Books|page=341}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=UJ-VoRZUhaYC&dq=JO+JJ&pg=PA3|title=Articvli Defensionales Peremptoriales & Elisivi, Bvrgermaister vnd Raths zu Nürmberg, Contra Brandenburg, In causa die Fraiszlich Obrigkait [et]c: Produ. 7. Feb. Anno [et]c. 33|date=1586-11-23|publisher=Heußler|via=Google Books|page=3|access-date=2023-12-02|archive-date=2024-11-13|archive-url=https://web.archive.org/web/20241113172327/https://books.google.com/books?id=UJ-VoRZUhaYC&dq=JO+JJ&pg=PA3|url-status=live}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=gc9TAAAAcAAJ&dq=j0+jj+jz+j3&pg=PA285|title=Gustavi Seleni Cryptomenytices Et Cryptographiae Libri IX.: In quibus & planißima Steganographiae a Johanne Trithemio ... magice & aenigmatice olim conscriptae, Enodatio traditur; Inspersis ubique Authoris ac Aliorum, non contemnendis inventis|first=Braunschweig-Lüneburg|last=August (Herzog)|date=1624-11-23|publisher=Johann & Heinrich Stern|via=Google Books|page=285}}</ref>。在[[德语]]中，顶部的衬线可以延长成与竖线等长的上划。这种变化可能会与其他国家/地区表示“7”的字形混淆，因此为了在视觉上区分两者，数字“7”可以用一条横线穿过竖线来书写{{sfn|Huber|Headrick|1999|pp=181}}。
== 其他领域 ==
在数字电路中，数据通过二进制代码表示，即采用以2为底的数制，由1和0构成的序列来表示各种数值。数字化数据在物理设备中（如[[电子计算机]]）通常以电脉冲的形式体现，通过[[晶体管]]或[[邏輯閘|逻辑门]]等开关器件进行处理，其中“1”代表“开”的状态。因此，在许多[[编程语言]]中，[[布爾值]]中的“真”（true）对应的数值就是1{{sfn|Woodford|2006|p=9}}{{sfn|Godbole|2002|p=34}}。在[[λ演算]]和可计算性理论中，自然数通过丘奇数以函数的形式来表示。其中，数字1的丘奇数表示为一个函数<math>f</math>，它对某个参数<math>x</math>应用一次，亦即{{nobr|(1<math>fx=fx</math>)}}{{sfn|Hindley|Seldin|2008|p=48}}。

在[[物理学]]中，为了简化[[方程]]的形式，一些自然单位制中会将特定的物理常数设定为1。例如，在[[普朗克單位制|普朗克单位制]]中，[[光速]]被定为1{{sfn|Glick|Darby|Marmodoro|2020|pp=99}}。无量纲量也称为量纲为1的量{{sfn|Mills|1995|pp=538-539}}。在[[量子力学]]中，波函数的归一化条件要求[[波函数]]的模平方的积分等于1{{sfn|McWeeny|1972|pp=14}}。在[[化學]]中，[[氢]]是[[元素周期表]]的第一个元素，也是已知宇宙中最丰富的元素，它的原子序数为1。元素周期表的第一族由氢和[[碱金属]]组成{{sfn|Emsley|2001}}。

在哲学中，数字1通常被视为统一的象征，在[[一神論]]传统中经常代表[[上帝]]或宇宙{{sfn|Stewart|2024}}。[[毕达哥拉斯学派]]认为数字是复数，因此不将1本身归类为数字，而是将其归类为所有数字的起源。在他们的数字哲学中，[[奇数]]被认为是阳性，[[偶数]]被认为是阴性，而1被认为是中性的，能够通过加法将偶数转化为奇数，反之亦然<ref>{{cite journal|url=https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/from-abacus-to-algorism-theory-and-practice-in-medieval-arithmetic/7DFF144C90C127E715CA40083254E601#access-block|title=From Abacus to Algorism: Theory and Practice in Medieval Arithmetic|journal=The British Journal for the History of Science|volume=10|issue=2|date=1977-07-01|page=Abstract|doi=10.1017/S0007087400015375|publisher=Cambridge University Press|author=British Society for the History of Science|s2cid=145065082|access-date=2021-05-16|archive-date=2021-05-16|archive-url=https://web.archive.org/web/20210516110812/https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/from-abacus-to-algorism-theory-and-practice-in-medieval-arithmetic/7DFF144C90C127E715CA40083254E601#access-block|url-status=live|url-access=subscription}}</ref>。在[[普罗提诺]]及其他[[新柏拉图主义]]者的哲学中，“一者”（The One）被视为终极的实在，是一切存在的根源{{sfn|Halfwassen|2014|pp=182–183}}。[[亚历山大的斐洛]]则将数字一视为上帝之数，是所有数字的基础<ref>"De Allegoriis Legum", ii.12 [i.66]</ref>。

== 参考 ==
{{reflist|30em}}
== 书目 ==
{{refbegin|30em}}
*{{Cite book|last=Blokhintsev|first=D. I.|title=Quantum Mechanics|year=2012|publisher=Springer Science & Business Media|isbn=978-9401097116|ref=harv|url={{Google books|9_nwCAAAQBAJ|page=PA35|plainurl=yes}}}}
*{{Cite journal |last1=Caldwell |first1=Chris K. |last2=Xiong |first2=Yeng |title=What is the smallest prime? |url=https://www.emis.de///journals/JIS/VOL15/Caldwell1/cald5.html |journal=[[Journal of Integer Sequences]] |publisher=[[University of Waterloo]] [[David R. Cheriton School of Computer Science]] |volume=15 |issue=9, Article 12.9.7 |location=Waterloo, CA |year=2012 |pages=1–14 |mr=3005530 |zbl=1285.11001 |arxiv=1209.2007 |archive-date=2023-12-16 |access-date=2023-12-16 |archive-url=https://web.archive.org/web/20231216130155/https://www.emis.de///journals/JIS/VOL15/Caldwell1/cald5.html |url-status=live |ref=harv}}
*{{cite book |last=Chicago |first=University of |title=The Chicago Manual of Style|year=1993|publisher=University of Chicago Press|edition=14th|isbn=0-226-10389-7|ref=harv}}
*{{cite book |last=Chrisomalis|first=Stephen|title=Numerical Notation: A Comparative History |title-link=Numerical Notation: A Comparative History |publisher=Cambridge University Press|year=2010|location=New York|isbn=978-0-521-87818-0|doi=10.1017/CBO9780511676062|ref=harv}}
*{{cite book| last1=Colman| first1=Samuel| editor-last=Coan| editor-first=C. Arthur| title=''Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form''| publisher=G.P. Putnam's Sons| location=New York and London| year=1912| url=https://archive.org/details/naturesharmonic00coangoog/page/n26/mode/2up|ref=harv}}
*{{cite book| last=Crystal| first=D.| year=2008 |title=A Dictionary of Linguistics and Phonetics| edition=6th| location=Malden, MA|publisher=Wiley-Blackwell|isbn=978-0631226642|ref=harv}}
*{{cite book| last1=Conway| first1=John H.| last2=Guy| first2=Richard K.| title=''The Book of Numbers''| publisher=Copernicus Publications| location=New York| year=1996| isbn=0614971667| doi=10.1007/978-1-4612-4072-3| url=https://link.springer.com/book/10.1007/978-1-4612-4072-3| archive-date=2024-11-18| access-date=2023-12-17| archive-url=https://web.archive.org/web/20241118194359/https://link.springer.com/book/10.1007/978-1-4612-4072-3| url-status=live|ref=harv}}
*{{Cite book |last=Cullen |first=Kristin |title=Layout Workbook: A Real-World Guide to Building Pages in Graphic Design |url=https://books.google.com/books?id=d2M_I7EXu0UC&pg=PA93 |publisher=Rockport Publishers |location=Gloucester, MA |year=2007 |pages=1–240 |isbn=978-1-592-533-527 |ref=harv}}
*{{Cite book|last=Emsley|first=John|title=Nature's Building Blocks: An A-Z Guide to the Elements|edition=illustrated, reprint|publisher=Oxford University Press|location=Oxford, UK|year=2001|isbn=0198503415|ref=harv}}
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{{refend}}
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[[Category:整数]]
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imported>JackoChenWei 来自 2026年2月23日 (一) 16:46