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<h2 id="in-mathematics">In mathematics</h2> <p>The number 1 is the first natural number after <a href="/facts/0/4NBRtKHc">0</a>. Each <a href="/facts/Natural_number/u65og8JE">natural number</a>, including 1, is constructed by <a href="/facts/Successor_function/R1hhkSEG">succession</a>, that is, by adding 1 to the previous natural number. The number 1 is the <a href="/facts/Multiplicative_identity/9nss3xHY">multiplicative identity</a> of the <a href="/facts/Integer/PUwwrawx">integers</a>, <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, and <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, that is, any number n {\displaystyle n} multiplied by 1 remains unchanged ( 1 × n = n × 1 = n {\displaystyle 1\times n=n\times 1=n} ). As a result, the <a href="/facts/Square_(algebra)/Jzpq0R0z">square</a> ( 1 2 = 1 {\displaystyle 1^{2}=1} ), <a href="/facts/Square_root/AfrzfBdQ">square root</a> ( 1 = 1 {\displaystyle {\sqrt {1}}=1} ), and any other power of 1 is always equal to 1 itself.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> 1 is its own <a href="/facts/Factorial/kkfy1Bwe">factorial</a> ( 1 ! = 1 {\displaystyle 1!=1} ), and 0! is also 1. These are a special case of the <a href="/facts/Empty_product/mj7nhGeD">empty product</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> Although 1 meets the naïve definition of a <a href="/facts/Prime_number/L20FLkgn">prime number</a>, being evenly divisible only by 1 and itself (also 1), by modern convention it is regarded as neither a <a href="/facts/Prime_number/L20FLkgn">prime</a> nor a <a href="/facts/Composite_number/SPtW9ftC">composite number</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>Different mathematical constructions of the natural numbers represent 1 in various ways. In <a href="/facts/Giuseppe_Peano/VT2Wsmu7">Giuseppe Peano</a>'s original formulation of the <a href="/facts/Peano_axioms/DhhDnNug">Peano axioms</a>, a set of postulates to define the natural numbers in a precise and logical way, 1 was treated as the starting point of the sequence of natural numbers.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Peano later revised his axioms to begin the sequence with 0.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> In the <a href="/facts/Von_Neumann_cardinal_assignment/EHFlKG96">Von Neumann cardinal assignment</a> of natural numbers, where each number is defined as a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> that contains all numbers before it, 1 is represented as the <a href="/facts/Singleton_(mathematics)/LtNVfujT">singleton</a> { 0 } {\displaystyle \{0\}} , a set containing only the element 0.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The <a href="/facts/Unary_numeral_system/J37tMmyW">unary numeral system</a>, as used in <a href="/facts/Tally_mark/5rNQKmnM">tallying</a>, is an example of a "base-1" number system, since only one mark – the tally itself – is needed. While this is the simplest way to represent the natural numbers, base-1 is rarely used as a practical base for <a href="/facts/Counting/DsRhH9RT">counting</a> due to its difficult readability.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>In many mathematical and engineering problems, numeric values are typically <a href="/facts/Normalized_solution_(mathematics)/4wbNGuaF">normalized</a> to fall within the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> ([0,1]), where 1 represents the maximum possible value. For example, by definition 1 is the <a href="/facts/Probability/fgYvMhwb">probability</a> of an event that is absolutely or <a href="/facts/Almost_certain/fWaxzpBA">almost certain</a> to occur.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> Likewise, <a href="/facts/Vector_space/M1pxMLx2">vectors</a> are often normalized into <a href="/facts/Unit_vector/GyaXneDn">unit vectors</a> (i.e., vectors of magnitude one), because these often have more desirable properties. Functions are often normalized by the condition that they have <a href="/facts/Integral/V7lyV997">integral</a> one, maximum value one, or <a href="/facts/Square_integrable/IASblcJJ">square integral</a> one, depending on the application.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p><p>1 is the value of <a href="/facts/Legendre%2527s_constant/29eqKCfU">Legendre's constant</a>, introduced in 1808 by <a href="/facts/Adrien-Marie_Legendre/2a7CAolM">Adrien-Marie Legendre</a> to express the <a href="/facts/Asymptotic_analysis/kt87qLpu">asymptotic behavior</a> of the <a href="/facts/Prime-counting_function/Dk1xJzsA">prime-counting function</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> The <a href="/facts/Weil%2527s_conjecture_on_Tamagawa_numbers/XP1Jeqf4">Weil's conjecture on Tamagawa numbers</a> states that the <a href="/facts/Tamagawa_number/NRYVaI7S">Tamagawa number</a> τ ( G ) {\displaystyle \tau (G)} , a geometrical measure of a connected linear <a href="/facts/Algebraic_group/DDtBCIvh">algebraic group</a> over a global <a href="/facts/Number_field/e0K0TXSh">number field</a>, is 1 for all simply connected groups (those that are <a href="/facts/Homotopical_connectivity/dbCgWgu3">path-connected</a> with no '<a href="/facts/Homotopical_connectivity/dbCgWgu3">holes</a>').<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>1 is the most common leading digit in many sets of real-world numerical data. This is a consequence of <a href="/facts/Benford%25E2%2580%2599s_law/CDfMJgdq">Benford’s law</a>, which states that the probability for a specific leading digit d {\displaystyle d} is log 10 ( d + 1 d ) {\textstyle \log _{10}\left({\frac {d+1}{d}}\right)} . The tendency for real-world numbers to grow exponentially or logarithmically biases the distribution towards smaller leading digits, with 1 occurring approximately 30% of the time.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h2 id="as-a-word">As a word</h2> <p class="note">See also: <a href="/facts/One_(pronoun)/TAuck6k5">One (pronoun)</a></p> <p><i>One</i> originates from the <a href="/facts/Old_English/9rsmQG2a">Old English</a> word <i>an</i>, derived from the <a href="/facts/Germanic_languages/BxAJX3Kf">Germanic</a> root <i>*ainaz</i>, from the <a href="/facts/Proto-Indo-European_root/L3p63vYW">Proto-Indo-European root</a> <i>*oi-no-</i> (meaning "one, unique").<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Linguistically, <i>one</i> is a <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a> used for counting and expressing the number of items in a collection of things.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> <i>One</i> is most commonly a <a href="/facts/English_determiners/SH13shAQ">determiner</a> used with <a href="/facts/Grammatical_number/zzs5AV6v">singular</a> countable <a href="/facts/English_nouns/FpxbYzYr">nouns</a>, as in <i>one day at a time</i>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> The determiner has two senses: numerical one (<i>I have one apple</i>) and singulative one (<i>one day I'll do it</i>).<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> <i>One</i> is also a gender-neutral <a href="/facts/One_(pronoun)/TAuck6k5">pronoun</a> used to refer to an unspecified <a href="/facts/Person/XD3Hhqup">person</a> or to people in general as in <i>one should take care of oneself</i>.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p><p>Words that derive their meaning from <i>one</i> include <i>alone</i>, which signifies <i>all one</i> in the sense of being by oneself, <i>none</i> meaning <i>not one</i>, <i>once</i> denoting <i>one time</i>, and <i>atone</i> meaning to become <i>at one</i> with the someone. Combining <i>alone</i> with <i>only</i> (implying <i>one-like</i>) leads to <i>lonely</i>, conveying a sense of solitude.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> Other common <a href="/facts/Numeral_prefix/b5sechhR">numeral prefixes</a> for the number 1 include <i>uni-</i> (e.g., <a href="/facts/Unicycle/MDf0yPAw">unicycle</a>, universe, unicorn), <i>sol-</i> (e.g., solo dance), derived from Latin, or <i>mono-</i> (e.g., <a href="/facts/Monorail/Mf8po5Vh">monorail</a>, monogamy, monopoly) derived from Greek.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> </p> <h2 id="symbols-and-representation">Symbols and representation</h2> <h3>History</h3> <p class="note">See also: <a href="/facts/History_of_the_Hindu%25E2%2580%2593Arabic_numeral_system/PSFvjx4m">History of the Hindu–Arabic numeral system</a></p> <p>Among the earliest known records of a numeral system, is the <a href="/facts/Sumer/bxRz093q">Sumerian</a> decimal-<a href="/facts/Sexagesimal/bspNRWdR">sexagesimal</a> system on <a href="/facts/Clay_tablet/9mbScsAn">clay tablets</a> dating from the first half of the <a href="/facts/3rd_millennium_BC/csYpGrX5">third millennium BCE</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols, <a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> by c. 2350 BCE, the older Sumerian curviform numerals were replaced with <a href="/facts/Cuneiform/MgIQoGu8">cuneiform</a> symbols, with 1 and 60 both represented by the same mostly vertical symbol. </p> <p>The Sumerian cuneiform system is a direct ancestor to the <a href="/facts/Eblaite/GuBooU5A">Eblaite</a> and <a href="/facts/Assyro-Babylonian/esQDlrFw">Assyro-Babylonian</a> <a href="/facts/Semitic_languages/WCYrz4Qm">Semitic</a> cuneiform <a href="/facts/Decimal/7omfVNIM">decimal</a> systems.<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> Surviving Babylonian documents date mostly from Old Babylonian (c. 1500 BCE) and the Seleucid (c. 300 BCE) eras.<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system.<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> </p><p>The most commonly used glyph in the modern Western world to represent the number 1 is the <a href="/facts/Arabic_numerals/LryNhh26">Arabic numeral</a>, a vertical line, often with a <a href="/facts/Serif/Ef9KtHfp">serif</a> at the top and sometimes a short horizontal line at the bottom. It can be traced back to the <a href="/facts/Brahmi_numerals/N93e0dST">Brahmic</a> script of ancient India, as represented by <a href="/facts/Ashoka/YsxDJyF2">Ashoka</a> as a simple vertical line in his <a href="/facts/Edicts_of_Ashoka/sVdwz3Tm">Edicts of Ashoka</a> in c. 250 BCE.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> This script's numeral shapes were transmitted to Europe via the <a href="/facts/Maghreb/oIcU24Mz">Maghreb</a> and <a href="/facts/Al-Andalus/JTNmWTzP">Al-Andalus</a> during the Middle Ages <a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> The Arabic numeral, and other glyphs used to represent the number one (e.g., Roman numeral (I ), Chinese numeral (一)) are <a href="/facts/Logogram/wWYAITGc">logograms</a>. These symbols directly represent the concept of 'one' without breaking it down into phonetic components.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> </p> <h3>Modern typefaces</h3> <p>In modern <a href="/facts/Typeface/It41hIjl">typefaces</a>, the shape of the character for the digit 1 is typically typeset as a <i>lining figure</i> with an <a href="/facts/Ascender_(typography)/F4z0lTpa">ascender</a>, such that the digit is the same height and width as a <a href="/facts/Capital_letter/KQcwaUUD">capital letter</a>. However, in typefaces with <a href="/facts/Text_figures/c3A2PVnB">text figures</a> (also known as <i>Old style numerals</i> or <i>non-lining figures</i>), the glyph usually is of <a href="/facts/X-height/lXpkGuqV">x-height</a> and designed to follow the rhythm of the lowercase, as, for example, in .<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> In <i>old-style</i> typefaces (e.g., <a href="/facts/Hoefler_Text/kcBAp4r1">Hoefler Text</a>), the typeface for numeral 1 resembles a <a href="/facts/Small_caps/SVH5prpT">small caps</a> version of I, featuring parallel serifs at the top and bottom, while the capital I retains a full-height form. This is a relic from the <a href="/facts/Roman_numerals/7BA8nn78">Roman numerals</a> system where <a href="/facts/I/2w7mqI7e">I</a> represents 1.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> Many older <a href="/facts/Typewriter/CHE43FWP">typewriters</a> do not have a dedicated key for the numeral 1, requiring the use of the lowercase letter <i><a href="/facts/L/0Ge4MIc9">L</a></i> or uppercase <i><a href="/facts/I/2w7mqI7e">I</a></i> as substitutes.<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a><a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a><a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> </p> <p>The lower case "<a href="/facts/J/I3pUoxSq">j</a>" can be considered a <a href="/facts/Swash_(typography)/4u0ClDkS">swash</a> variant of a lower-case Roman numeral "<a href="/facts/I/2w7mqI7e">i</a>", often employed for the final i of a "lower-case" Roman numeral. It is also possible to find historic examples of the use of <i>j</i> or <i>J</i> as a substitute for the Arabic numeral 1.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a> In German, the serif at the top may be extended into a long upstroke as long as the vertical line. This variation can lead to confusion with the glyph used for <a href="/facts/7/4GV2hz09">seven</a> in other countries and so to provide a visual distinction between the two the digit 7 may be written with a horizontal stroke through the vertical line.<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> </p> <h2 id="in-other-fields">In other fields</h2> <p>In digital technology, data is represented by <a href="/facts/Binary_code/v5YF0sQl">binary code</a>, i.e., a <a href="/facts/Radix/J5RGWRJF">base</a>-2 numeral system with numbers represented by a sequence of 1s and <a href="/facts/0_(number)/4NBRtKHc">0s</a>. Digitised data is represented in physical devices, such as <a href="/facts/Computer/SQe1aK05">computers</a>, as pulses of electricity through switching devices such as <a href="/facts/Transistor/Bmr8zwL0">transistors</a> or <a href="/facts/Logic_gate/RHLulvKI">logic gates</a> where "1" represents the value for "on". As such, the numerical value of <a href="/facts/Boolean_data_type/0GWYLDoC">true</a> is equal to 1 in many <a href="/facts/Programming_language/cWTYbgWa">programming languages</a>.<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a><a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a> In <a href="/facts/Lambda_calculus/DaD3RuQ4">lambda calculus</a> and <a href="/facts/Computability_theory/cqDfXwdf">computability theory</a>, natural numbers are represented by <a href="/facts/Church_encoding/IlLPzswH">Church encoding</a> as functions, where the Church numeral for 1 is represented by the function f {\displaystyle f} applied to an argument x {\displaystyle x} once (1 f x = f x {\displaystyle fx=fx} ).<a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> </p><p>In <a href="/facts/Physics/a7aH2y08">physics</a>, selected <a href="/facts/Physical_constant/uI2QvAHw">physical constants</a> are set to 1 in <a href="/facts/Natural_unit/nf8megYV">natural unit</a> systems in order to simplify the form of equations; for example, in <a href="/facts/Planck_units/xDU6KlDM">Planck units</a> the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> equals 1.<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a> <a href="/facts/Dimensionless_quantities/TINrpQIq">Dimensionless quantities</a> are also known as 'quantities of dimension one'.<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a> In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the normalization condition for <a href="/facts/Wavefunction/KgM5ykjY">wavefunctions</a> requires the integral of a wavefunction's squared modulus to be equal to 1.<a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> In chemistry, <a href="/facts/Hydrogen/BoYHjl0J">hydrogen</a>, the first element of the <a href="/facts/Periodic_table/umdGIfUb">periodic table</a> and the most <a href="/facts/Abundance_of_the_elements/CE9WehBl">abundant element</a> in the known <a href="/facts/Universe/XbfpnDmM">universe</a>, has an <a href="/facts/Atomic_number/bgAr581S">atomic number</a> of 1. Group 1 of the periodic table consists of hydrogen and the <a href="/facts/Alkali_metal/IkWbCLk2">alkali metals</a>.<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a> </p><p>In philosophy, the number 1 is commonly regarded as a symbol of unity, often representing God or the universe in <a href="/facts/Monotheism/scRUM2It">monotheistic</a> traditions.<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a> The Pythagoreans considered the numbers to be plural and therefore did not classify 1 itself as a number, but as the origin of all numbers. In their number philosophy, where odd numbers were considered male and even numbers female, 1 was considered neutral capable of transforming even numbers to odd and vice versa by addition.<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a> The <a href="/facts/Neopythagoreanism/PUpCgfcv">Neopythagorean</a> philosopher <a href="/facts/Nicomachus/EdlB4L3G">Nicomachus of Gerasa</a>'s number treatise, as recovered by <a href="/facts/Boethius/9k2rqLps">Boethius</a> in the Latin translation <i><a href="/facts/Introduction_to_Arithmetic/HLodboz4">Introduction to Arithmetic</a></i>, affirmed that one is not a number, but the source of number.<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a> In the philosophy of <a href="/facts/Plotinus/BQDNMGDA">Plotinus</a> (and that of other <a href="/facts/Neoplatonist/Tyq4uTG0">neoplatonists</a>), 'The One' is the ultimate reality and source of all existence.<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a> <a href="/facts/Philo/Ieyu5Igv">Philo of Alexandria</a> (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers.<a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/%25E2%2588%25921/WtI4msbk">−1</a></li> <li><a href="/facts/0.999.../fdtw8eBL">0.999...</a> – Alternative decimal expansion of 1</li></ul> <h2 id="sources">Sources</h2> <ul><li>Blokhintsev, D. I. (2012). <a href="https://books.google.com/books?id=9_nwCAAAQBAJ&pg=PAPA35"><i>Quantum Mechanics</i></a>. Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-9401097116.</li> <li>Caldwell, Chris K.; Xiong, Yeng (2012). <a href="https://www.emis.de///journals/JIS/VOL15/Caldwell1/cald5.html">"What is the smallest prime?"</a>. <i><a href="/facts/Journal_of_Integer_Sequences/fNv6NIRD">Journal of Integer Sequences</a></i>. 15 (9, Article 12.9.7). Waterloo, CA: <a href="/facts/University_of_Waterloo/NANSHN5Y">University of Waterloo</a> <a href="/facts/David_R._Cheriton_School_of_Computer_Science/6Pny1UMm">David R. Cheriton School of Computer Science</a>: 1–14. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1209.2007">1209.2007</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3005530">3005530</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1285.11001">1285.11001</a>. <a href="https://web.archive.org/web/20231216130155/https://www.emis.de///journals/JIS/VOL15/Caldwell1/cald5.html">Archived</a> from the original on 2023-12-16. Retrieved 2023-12-16.</li> <li>Chicago, University of (1993). <i>The Chicago Manual of Style</i> (14th ed.). University of Chicago Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-10389-7.</li> <li>Chrisomalis, Stephen (2010). <a href="/facts/Numerical_Notation%3a_A_Comparative_History/ymVmFwxw"><i>Numerical Notation: A Comparative History</i></a>. New York: Cambridge University Press. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511676062">10.1017/CBO9780511676062</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-87818-0.</li> <li>Colman, Samuel (1912). Coan, C. Arthur (ed.). <a href="https://archive.org/details/naturesharmonic00coangoog/page/n26/mode/2up"><i></i>Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form<i></i></a>. New York and London: G.P. Putnam's Sons.</li> <li>Crystal, D. (2008). <i>A Dictionary of Linguistics and Phonetics</i> (6th ed.). Malden, MA: Wiley-Blackwell. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0631226642.</li> <li>Conway, John H.; Guy, Richard K. (1996). <a href="https://link.springer.com/book/10.1007/978-1-4612-4072-3"><i></i>The Book of Numbers<i></i></a>. New York: Copernicus Publications. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-4072-3">10.1007/978-1-4612-4072-3</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0614971667. <a href="https://web.archive.org/web/20241118194359/https://link.springer.com/book/10.1007/978-1-4612-4072-3">Archived</a> from the original on 2024-11-18. Retrieved 2023-12-17.</li> <li>Cullen, Kristin (2007). <a href="https://books.google.com/books?id=d2M_I7EXu0UC&pg=PA93"><i>Layout Workbook: A Real-World Guide to Building Pages in Graphic Design</i></a>. Gloucester, MA: Rockport Publishers. pp. 1–240. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-592-533-527.</li> <li>Emsley, John (2001). <i>Nature's Building Blocks: An A-Z Guide to the Elements</i> (illustrated, reprint ed.). 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"Processes of Algebraization". <i>Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture</i>. By Radford, Luis; Schubring, Gert; Seeger, Falk. Kaiser, Gabriele (ed.). Semiotic Perspectives in the Teaching and Learning of Math Series. Vol. 1. Netherlands: Sense Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-9087905972.</li> <li>Stewart, Ian (2024). <a href="https://www.britannica.com/topic/number-symbolism">"Number Symbolism"</a>. <i>Brittanica</i>. <a href="https://web.archive.org/web/20080726140908/http://www.britannica.com/eb/article-248155/number-symbolism">Archived</a> from the original on 2008-07-26. Retrieved 2024-08-21.</li> <li><a href="/facts/Chris_Woodford_(author)/1XJfANg7">Woodford, Chris</a> (2006). <a href="https://books.google.com/books?id=My7Zr0aP2L8C&pg=PA9"><i>Digital Technology</i></a>. Evans Brothers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-237-52725-9. Retrieved 2016-03-24.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Colman 1912, pp. 9–10, chapt.2. - Colman, Samuel (1912). Coan, C. Arthur (ed.). Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form. New York and London: G.P. Putnam's Sons. <a href="https://archive.org/details/naturesharmonic00coangoog/page/n26/mode/2up" target="_blank">https://archive.org/details/naturesharmonic00coangoog/page/n26/mode/2up</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Graham, Knuth & Patashnik 1994, p. 111. - Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics (2 ed.). Reading, MA: Addison-Wesley. ISBN 0-201-14236-8. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Caldwell & Xiong 2012, pp. 8–9. - Caldwell, Chris K.; Xiong, Yeng (2012). "What is the smallest prime?". Journal of Integer Sequences. 15 (9, Article 12.9.7). Waterloo, CA: University of Waterloo David R. Cheriton School of Computer Science: 1–14. arXiv:1209.2007. MR 3005530. Zbl 1285.11001. Archived from the original on 2023-12-16. Retrieved 2023-12-16. <a href="https://www.emis.de///journals/JIS/VOL15/Caldwell1/cald5.html" target="_blank">https://www.emis.de///journals/JIS/VOL15/Caldwell1/cald5.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kennedy 1974, pp. 389. - Kennedy, Hubert C. (1974). "Peano's concept of number". Historia Mathematica. 1 (4): 387–408. doi:10.1016/0315-0860(74)90031-7. <a href="https://doi.org/10.1016/0315-0860(74)90031-7" target="_blank">https://doi.org/10.1016/0315-0860(74)90031-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Peano 1889, p. 1. - Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Turin: Fratres Bocca. pp. xvi, 1–20. JFM 21.0051.02. <a href="https://archive.org/details/arithmeticespri00peangoog" target="_blank">https://archive.org/details/arithmeticespri00peangoog</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Kennedy 1974, pp. 389. - Kennedy, Hubert C. (1974). "Peano's concept of number". Historia Mathematica. 1 (4): 387–408. doi:10.1016/0315-0860(74)90031-7. <a href="https://doi.org/10.1016/0315-0860(74)90031-7" target="_blank">https://doi.org/10.1016/0315-0860(74)90031-7</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Peano 1908, p. 27. - Peano, Giuseppe (1908). Formulario Mathematico [Mathematical Formulary] (V ed.). Turin: Fratres Bocca. pp. xxxvi, 1–463. JFM 39.0084.01. <a href="https://archive.org/details/formulairedemat04peangoog/page/n8/mode/2up" target="_blank">https://archive.org/details/formulairedemat04peangoog/page/n8/mode/2up</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Halmos 1974, p. 32. - Halmos, Paul R. (1974). Naive Set Theory. Undergraduate Texts in Mathematics. Springer. pp. vii, 1–104. doi:10.1007/978-1-4757-1645-0. ISBN 0-387-90092-6. MR 0453532. <a href="https://link.springer.com/book/10.1007/978-1-4757-1645-0" target="_blank">https://link.springer.com/book/10.1007/978-1-4757-1645-0</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Hodges 2009, p. 14. - Hodges, Andrew (2009). One to Nine: The Inner Life of Numbers. New York, NY: W. W. Norton & Company. pp. 1–330. ISBN 9780385672665. S2CID 118490841. <a href="https://books.google.com/books?id=5WErLc4rwm8C&pg=PA14" target="_blank">https://books.google.com/books?id=5WErLc4rwm8C&pg=PA14</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Hext 1990. - Hext, Jan (1990). Programming Structures: Machines and programs. Vol. 1. Prentice Hall. p. 33. ISBN 9780724809400. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Graham, Knuth & Patashnik 1994, p. 381. - Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics (2 ed.). Reading, MA: Addison-Wesley. ISBN 0-201-14236-8. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Blokhintsev 2012, p. 35. - Blokhintsev, D. I. (2012). Quantum Mechanics. Springer Science & Business Media. 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