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| {{Numeral systems}}
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| The system of ancient Egyptian numerals''' '''was used in Ancient Egypt in around 3000 BC<ref>{{cite web|title=Egyptian numerals|url=http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html|accessdate=25 September 2013}}</ref> until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the higher power, written in [[Egyptian hieroglyph|hieroglyph]]s, but they had no concept of a [[place-valued system]] such as the [[Decimal|decimal system]] is.<ref>"The Story of Numbers" by John McLeish</ref> The [[hieratic]] form of numerals stressed an exact finite series notation, ciphered one to one onto the Egyptian alphabet.The Ancient Egyptian system used bases of ten.
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| ==Digits and numbers==
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| The following hieroglyphics were used to denote [[Exponentiation|powers]] of ten:
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| {| style="margin: 0.5em 1em 0.5em 1em; border: 1px solid #aaaaaa;" cellpadding=2px
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| |- style="background-color:#ccccff;"
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| !Value
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| |align="center"|1
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| |aligp="center"|10
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| |align="center"|100
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| |align="center"|1,000
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| |align="center"|10,000
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| |align="center"|100,000
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| |align="center"|1 [[million]], or<br>[[many]]
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| |-
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| !Hieroglyph
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| |align="center"|<hiero>Z1</hiero>
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| |align="center"|<hiero>V20</hiero>
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| |align="center"|<hiero>V1</hiero>
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| |align="center"|<hiero>M12</hiero>
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| |align="center"|<hiero>D50</hiero>
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| |align="center"|<hiero>I8</hiero><br>or<br><hiero>I7</hiero>
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| |align="center"|<hiero>C11</hiero>
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| |-
| |
| !Description
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| | Single stroke
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| | Heel bone
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| | Coil of [[rope]]
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| | [[Nymphaea lotus|Water lily]]<br/>(also called Lotus)
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| | Bent Finger
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| | [[Tadpole]]<br/>or [[Frog]]
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| | Man with both hands raised,<br/> perhaps [[Huh (god)|Heh]].<ref>Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. Hoboken, NJ: John Wiley, 2011, p. 10</ref>
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| |}
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| Multiples of these values were expressed by repeating the symbol as many times as needed. For instance, a stone carving from [[Karnak]] shows the number 4622 as
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| {| align="center"
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| |<hiero>M12-M12-M12-M12-!-V1*V1*V1:V1*V1*V1-!-V20-V20-Z1-Z1</hiero>
| |
| |}
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| Egyptian hieroglyphs could be written in [[Bi-directional text|both directions]] (and even vertically). This example is written left-to-right and top-down; on the original stone carving, it is right-to-left, and the signs are thus reversed
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| | |
| ==Zero and negative numbers==
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| {| style="float:right; clear:right; text-align:center; border: 1px solid" align=right cellspacing=0 cellpadding=8
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| |-
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| !nfr<br>
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| |heart with trachea<br>beautiful, pleasant, good
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| |<hiero>F35</hiero>
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| |}
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| By 1740 BCE, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.<ref>{{cite book|page=86 |title=The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition) |author=George Gheverghese Joseph |isbn=978-0-691-13526-7 |publisher=Princeton |year=2011}}</ref>
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| ==Fractions==
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| {{Main|Egyptian fraction}}
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| | |
| [[Rational number]]s could also be expressed, but only as sums of [[unit fraction]]s, ''i.e.'', sums of [[Reciprocal (mathematics)|reciprocal]]s of positive integers, except for 2/3 and 3/4. The hieroglyph indicating a fraction looked like a mouth, which meant "part":
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| {| align="center"
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| |<hiero>D21</hiero>
| |
| |}
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| Fractions were written with this fractional [[Slash (punctuation)|solidus]], ''i.e.'', the [[numerator]] 1, and the positive [[denominator]] below. Thus, 1/3 was written as:
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| {| align="center"
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| |<hiero>D21:Z1*Z1*Z1</hiero>
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| | <math>= \frac{1}{3}</math>
| |
| |}
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| There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently):
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| {| align="center"
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| |<hiero>Aa13</hiero>
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| |<math>= \frac{1}{2}</math>
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| |
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| |<hiero>D22</hiero>
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| |<math>= \frac{2}{3}</math>
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| |
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| |<hiero>D23</hiero>
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| |<math>= \frac{3}{4}</math>
| |
| |}
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| If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":
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| {| align="center"
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| |<hiero>D21:V1*V1*V1-V20*V20:V20*Z1</hiero>
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| |<math>= \frac{1}{331}</math>
| |
| |}
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| | |
| ==Addition and subtraction==
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| For [[plus and minus signs]], the hieroglyphs <hiero>D54-and-D55</hiero> were used: if the feet pointed into the direction of writing, it signified [[addition]], otherwise [[subtraction]].<ref name="historyOfMathematicalNotations">{{Cite book|last= Cajori |first= Florian |authorlink= Florian Cajori |title= A History of Mathematical Notations |pages= ''pp.'' 229–230 |year= 1993 |origyear= 1929 |publisher= [[Dover Publications]] |isbn= 0-486-67766-4}}
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| </ref>
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| | |
| ==Written numbers==
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| As with most modern day languages, the ancient [[Egyptian language]] could also write out numerals as words phonetically, just like one can write ''thirty'' instead of "30" in [[English language|English]]. ''Thirty'', for instance, was written as
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| {| align="center"
| |
| |<hiero>Aa15:D36-D58</hiero>
| |
| |}
| |
| while the ''number'' 30 was
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| {| align="center"
| |
| |<hiero>V20-V20-V20</hiero>
| |
| |}
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| | |
| This was, however, uncommon for most numbers other than one and two and the signs were used most of the time.
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| | |
| ==Hieratic numerals==
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| As administrative and accounting texts were written on [[papyrus]] or [[Ostracon|ostraca]], rather than being carved into hard stone (as were hieroglyphic texts), the vast majority of texts employing the Egyptian numeral system utilize the [[hieratic]] script. Instances of numerals written in hieratic can be found as far back as the [[Early Dynastic Period of Egypt|Early Dynastic Period]]. The Old Kingdom [[Abusir Papyri]] are a particularly important corpus of texts that utilize hieratic numerals.
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| | |
| Boyer proved 50 years ago{{When|date=December 2013}} that [[hieratic]] script used a different numeral system, using individual signs for the numbers 1 to 9, multiples of 10 from 10 to 90, the hundreds from 100 to 900, and the thousands from 1000 to 9000. A large number like 9999 could thus be written with only four signs—combining the signs for 9000, 900, 90, and 9—as opposed to 36 hieroglyphs. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history. Greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian.
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| In the oldest hieratic texts the individual numerals were clearly written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, repeated as Roman numerals practiced. However, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing; this process continued into [[Demotic Egyptian|Demotic]] as well.
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| | |
| Two famous mathematical papyri using hieratic script are the [[Moscow Mathematical Papyrus]] and the [[Rhind Mathematical Papyrus]].
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| | |
| ==Egyptian words for numbers==
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| The following table shows the reconstructed Middle Egyptian forms of the numerals<ref>John B. Callender, ''Middle Egyptian'', 1975</ref> (which are indicated by a preceding asterisk), followed by the [[Egyptian transliteration|transliteration]] of the [[Egyptian hieroglyphs|hieroglyphs]] used to write them, and finally the [[Coptic language|Coptic]] numerals which descended from them and which give Egyptologists clues as to the [[vowel|vocalism]] of the original Egyptian numbers. The majuscule letter "A" in some reconstructed forms means that the quality of that vowel remains uncertain:
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| | |
| {| class="wikitable"
| |
| |-
| |
| ! [[Ancient Egyptian language|Egyptian]] [[Transliteration]] !! [[English language|English]] [[Translation]] !! [[Coptic language|Coptic]] (Sahidic dialect)
| |
| |-
| |
| | '''{{unicode|*wiʻyaw}}''' ‹ {{unicode|wꜥ.w}} ([[grammatical gender|masc.]])<br>'''{{unicode|*wiʻīyat}} ''' ‹ {{unicode|wꜥ.t}} ([[grammatical gender|fem.]]) || one || oua (masc.)<br>ouei (fem.)
| |
| |-
| |
| | '''{{unicode|*sínway}}''' ‹ {{unicode|sn.wy}} (masc.)<br>'''{{unicode|*síntay}}''' ‹ {{unicode|sn.ty}} (fem.) || two || snau (masc.)<br>snte (fem.)
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| |-
| |
| | '''{{unicode|*ḫámtaw}}''' ‹ {{unicode|ḫmt.w}} (masc.)<br>'''{{unicode|*ḫámtat}}''' ‹ {{unicode|ḫmt.t}} (fem.) || three || šomnt (masc.)<br>šomte (fem.)
| |
| |-
| |
| | '''{{unicode|*yAfdáw}}''' ‹ {{unicode|ỉfd.w}} (masc.)<br>'''{{unicode|*yAfdát}}''' ‹ {{unicode|ỉfd.t}} (fem.) || four || ftoou (masc.)<br>ftoe (fem.)
| |
| |-
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| | '''{{unicode|*dīyaw}}''' ‹ {{unicode|dỉ.w}} (masc.)<br>'''{{unicode|*dīyat}}''' ‹ {{unicode|dỉ.t}} (fem.) || five || tiou (masc.)<br>tie (fem.)
| |
| |-
| |
| | '''{{unicode|*yAssáw}}''' ‹ {{unicode|sỉs.w}} or {{unicode|ỉs.w}} (?) (masc.)<br>'''{{unicode|*yAssát}}''' ‹ {{unicode|sỉs.t}} or {{unicode|ỉs.t}} (?) (fem.) || six || soou (masc.)<br>soe (fem.)
| |
| |-
| |
| | '''{{unicode|*sáfḫaw}}''' ‹ {{unicode|sfḫ.w}} (masc.)<br>'''{{unicode|*sáfḫat}}''' ‹ {{unicode|sfḫt}} (fem.) || seven || šašf(masc.)<br>šašfe (fem.)
| |
| |-
| |
| | '''{{unicode|*ḫAmānaw}}''' ‹ {{unicode|ḫmnw}} (masc.)<br>'''{{unicode|*ḫAmānat}}''' ‹ {{unicode|ḫmnt}} (fem.) || eight || šmoun (masc.)<br>šmoune (fem.)
| |
| |-
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| | '''{{unicode|*pAsīḏaw}}''' ‹ {{unicode|psḏw}} (masc.)<br>'''{{unicode|*pAsīḏat}}''' ‹ {{unicode|psḏt}} (fem.) || nine || psis (masc.)<br>psite (fem.)
| |
| |-
| |
| | '''{{unicode|*mūḏaw}}''' ‹ {{unicode|mḏw}} (masc.)<br>'''{{unicode|*mūḏat}}''' ‹ {{unicode|mḏt}} (fem.) || ten || mēt (masc.)<br>mēte (fem.)
| |
| |-
| |
| | '''{{unicode|*ḏubāʻatay}}''' ‹ {{unicode|ḏbꜥ.ty}} || twenty || jōt (masc.)<br>jōti (fem.)
| |
| |-
| |
| | '''{{unicode|*máʻbAʼ}}''' ‹ {{unicode|mꜥbꜣ}} (masc.)<br>'''{{unicode|*máʻbAʼat}}''' ‹ {{unicode|mꜥbꜣ.t}} (fem.) || thirty || maab (masc.)<br>maabe (fem.)
| |
| |-
| |
| | '''{{unicode|*ḥAmí}}''' (?) ‹ {{unicode|ḥm.w}} (masc.) || forty || xme
| |
| |-
| |
| | '''{{unicode|*díywu}}''' ‹ {{unicode|dy.w}} || fifty || taeiou
| |
| |-
| |
| | '''{{unicode|*yAssáwyu}}''' ‹ {{unicode|sỉsy.w}} or {{unicode|ỉswy.w}} (?) || sixty || se
| |
| |-
| |
| | '''{{unicode|*safḫáwyu}}''' ‹ {{unicode|sfḫy.w}} (masc.)|| seventy || šfe
| |
| |-
| |
| | '''{{unicode|*ḫamanáwyu}}''' ‹ {{unicode|ḫmny.w}} (masc.)|| eighty || xmene
| |
| |-
| |
| | '''{{unicode|*pAsiḏawyu}}''' ‹ {{unicode|psḏy.w}} (masc.)|| ninety || pstaiou
| |
| |-
| |
| | '''{{unicode|*šáwat}}''' ‹ {{unicode|š.t}} || one hundred || še
| |
| |-
| |
| | '''{{unicode|*šūtay}}''' ‹ {{unicode|š.ty}} || two hundred || šēt
| |
| |-
| |
| | '''{{unicode|*ḫaʼ}}''' ‹ {{unicode|ḫꜣ}} || one thousand || šo
| |
| |-
| |
| | '''{{unicode|*ḏubaʻ}}''' ‹ {{unicode|ḏbꜣ}}{{Dubious|date=September 2010}}<br>{{fix|text=these do not match}} || ten thousand || tba
| |
| |-
| |
| | ‹ {{unicode|hfn}} || one hundred thousand
| |
| |-
| |
| | '''{{unicode|*ḥaḥ}}''' ‹ {{unicode|ḥḥ}} || one million || xax "many"
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Ancient Egypt]]
| |
| *[[Egyptian language]]
| |
| *[[Egyptian mathematics]]
| |
| | |
| ==References==
| |
| *Allen, James Paul. 2000. ''Middle Egyptian: An Introduction to the Language and Culture of Hieroglyphs''. Cambridge: Cambridge University Press. Numerals discussed in §§9.1–9.6.
| |
| *Gardiner, Alan Henderson. 1957. ''Egyptian Grammar; Being an Introduction to the Study of Hieroglyphs''. 3rd ed. Oxford: Griffith Institute. For numerals, see §§259–266.
| |
| *Goedicke, Hans. 1988. ''Old Hieratic Paleography''. Baltimore: Halgo, Inc.
| |
| *Möller, Georg. 1927. ''Hieratische Paläographie: Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit.'' 3 vols. 2nd ed. Leipzig: J. C. Hinrichs'schen Buchhandlungen. (Reprinted Osnabrück: Otto Zeller Verlag, 1965)
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| | |
| ==Notes==
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| <references/>
| |
| | |
| ==External links==
| |
| {{commons category|Hieroglyphs of Egypt: Numerals}}
| |
| *[http://www.greatscott.com/hiero/numbers.html Introduction]
| |
| *[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html Egyptian numerals]
| |
| *[http://www.egyptvoyager.com/hieroglyph_lesson7.htm Numbers and dates]
| |
| <!--*[http://myweb.tiscali.co.uk/davel/egyptian%20numbers.htm Hieratic numerals] link is dead.-->
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| *http://egyptianmath.blogspot.com
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| | |
| {{Ancient Egypt topics}}
| |
| | |
| {{DEFAULTSORT:Egyptian Numerals}}
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| [[Category:Egyptian hieroglyphs]]
| |
| [[Category:Numeral systems]]
| |
| [[Category:Numerals]]
| |
| [[Category:Egyptian mathematics]]
| |
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