Editing Exponentiation (section)

==History==
In ''[[The Sand Reckoner]]'', [[Archimedes]] proved the law of exponents, {{math|1=10<sup>''a''</sup> · 10<sup>''b''</sup> = 10<sup>''a''+''b''</sup>}}, necessary to manipulate powers of {{math|10}}.<ref>
Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press. {{doi|10.1017/CBO9780511695124.017}}.</ref> He then used powers of {{math|10}} to estimate the number of grains of sand that can be contained in the universe.

In the 9th century, the Persian mathematician [[Al-Khwarizmi]] used the terms مَال (''māl'', "possessions", "property") for a [[Square (algebra)|square]]—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"<ref name="worldwidewords"/>—and كَعْبَة (''[[Kaaba|Kaʿbah]]'', "cube") for a [[cube (algebra)|cube]], which later [[Mathematics in the medieval Islamic world|Islamic]] mathematicians represented in [[mathematical notation]] as the letters ''[[mīm]]'' (m) and ''[[kāf]]'' (k), respectively, by the 15th century, as seen in the work of [[Abu'l-Hasan ibn Ali al-Qalasadi]].<ref>{{MacTutor|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>
[[Nicolas Chuquet]] used a form of exponential notation in the 15th century, for example {{math|12<sup>2</sup>}} to represent {{math|12''x''<sup>2</sup>}}.<ref>{{cite book |last=Cajori |first=Florian |title=A History of Mathematical Notations |date=1928 |publisher=The Open Court Company |volume=1 |page=102 |url=https://archive.org/details/historyofmathema031756mbp}}</ref> This was later used by [[Henricus Grammateus]] and [[Michael Stifel]] in the 16th century. In the late 16th century, [[Jost Bürgi]] would use Roman numerals for exponents in a way similar to that of Chuquet, for example {{Overset|iii|4}} for {{math|4''x''<sup>3</sup>}}.<ref name="cajori">{{cite book |last=Cajori |first=Florian |author-link=Florian Cajori |date=1928 |title=A History of Mathematical Notations |location=London |publisher=[[Open Court Publishing Company]] |page=[https://archive.org/details/historyofmathema031756mbp/page/n363 344] |volume=1 |url=https://archive.org/details/historyofmathema031756mbp}}</ref>

In 1636, [[James Hume (mathematician)|James Hume]] used in essence modern notation, when in ''L'algèbre de Viète'' he wrote {{math|''A''<sup>iii</sup>}} for {{math|''A''<sup>3</sup>}}.<ref>{{cite book |last=Cajori |first=Florian |title=A History of Mathematical Notations |date=1928 |publisher=The Open Court Company |volume=1 |pages=204 |url=https://archive.org/details/historyofmathema031756mbp}}</ref> Early in the 17th century, the first form of our modern exponential notation was introduced by [[René Descartes]] in his text titled ''[[La Géométrie]]''; there, the notation is introduced in Book I.<ref>{{cite book |last=Descartes |first=René |author-link=René Descartes |date=1637 |title=Discourse de la méthode [...] |location=Leiden |publisher=Jan Maire |page=299 |chapter=''[[La Géométrie]]'' |url=http://gallica.bnf.fr/ark:/12148/btv1b86069594/f383.image |quote=Et ''aa'', ou {{math|''a''<sup>2</sup>}}, pour multiplier {{math|''a''}} par soy mesme; Et {{math|''a''<sup>3</sup>}}, pour le multiplier encore une fois par {{math|''a''}}, & ainsi a l'infini}} (And {{math|''aa''}}, or {{math|''a''<sup>2</sup>}}, in order to multiply {{math|''a''}} by itself; and {{math|''a''<sup>3</sup>}}, in order to multiply it once more by {{math|''a''}}, and thus to infinity).</ref>

{{Blockquote|text=I designate ... {{math|''aa''}}, or {{math|''a''<sup>2</sup>}} in multiplying {{math|''a''}} by itself; and {{math|''a''<sup>3</sup>}} in multiplying it once more again by {{math|''a''}}, and thus to infinity.|author=René Descartes|title=La Géométrie}}

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write [[polynomial]]s, for example, as {{math|''ax'' + ''bxx'' + ''cx''<sup>3</sup> + ''d''}}.

[[Samuel Jeake]] introduced the term ''indices'' in 1696.<ref name="MacTutor">{{MacTutor|class=Miscellaneous|id=Mathematical_notation|title=Etymology of some common mathematical terms}}</ref> The term ''involution'' was used synonymously with the term ''indices'', but had declined in usage<ref>The most recent usage in this sense cited by the OED is from 1806 ({{cite OED |involution}}).</ref> and should not be confused with [[involution (mathematics)|its more common meaning]].

In 1748, [[Leonhard Euler]] introduced variable exponents, and, implicitly, non-integer exponents by writing:{{blockquote|Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not [[algebraic function]]s, since in those the exponents must be constant.<ref name="Euler_1748"/>}}

=== 20th century ===
As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example [[Konrad Zuse]] introduced [[floating-point arithmetic]] in his 1938 computer Z1. One [[register (computer)|register]] contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier [[Leonardo Torres Quevedo]] contributed ''Essays on Automation'' (1914) which had suggested the floating-point representation of numbers. The more flexible [[decimal floating-point]] representation was introduced in 1946 with a [[Bell Laboratories]] computer. Eventually educators and engineers adopted [[scientific notation]] of numbers, consistent with common reference to [[order of magnitude]] in a [[ratio scale]].<ref>Janet Shiver & Terri Wiilard "[https://www.visionlearning.com/en/library/Math-in-Science/62/Scientific-Notation/250 Scientific notation: working with orders of magnitude] from [[Visionlearning]]</ref>

For instance, in 1961 the [[School Mathematics Study Group]] developed the notation in connection with units used in the [[metric system]].<ref>School Mathematics Study Group (1961) ''Mathematics for Junior High School'', volume 2, part 1, [[Yale University Press]]</ref><ref>Cecelia Callanan (1967) "Scientific Notation", ''[[The Mathematics Teacher]]'' 60: 252–6 [https://www.jstor.org/stable/27957540 JSTOR]</ref>

Exponents also came to be used to describe [[units of measurement]] and [[quantity dimension]]s. For instance, since [[force]] is mass times acceleration, it is measured in kg m/sec<sup>2</sup>. Using M for mass, L for length, and T for time, the expression M L T<sup>–2</sup> is used in [[dimensional analysis]] to describe force.<ref>[[Edwin Bidwell Wilson]] (1920) [https://archive.org/details/aeronauticsclass00wilsrich/page/182/mode/2up Theory of Dimensions], chapter 11 in ''Aeronautics: A Class Text'', via Internet Archive</ref><ref>{{cite book |last1= Bridgman|first1= Percy Williams|title= Dimensional Analysis|publisher= Yale University Press|location= New Haven|date= 1922|oclc= 840631|authorlink=Percy Bridgman|url= https://archive.org/details/dimensionalanaly00bridrich }}</ref>