Editing Ancient Egypt (section)

===Mathematics===
{{Main|Ancient Egyptian mathematics}}
[[File:Senenmut-Grab.JPG|right|thumb|Facsimile of the [[Astronomical ceiling of Senenmut's Tomb|Astronomical chart in Senemut's tomb]], 18th dynasty{{sfnp|Astronomical Ceiling|2020}}]]

The earliest attested examples of mathematical calculations date to the predynastic [[Naqada]] period, and show a fully developed [[Egyptian numerals|numeral system]].{{efn|Understanding of Egyptian mathematics is incomplete due to paucity of available material and lack of exhaustive study of the texts that have been uncovered ({{harvp|Imhausen|2007|p=13}}).}} The importance of mathematics to an educated Egyptian is suggested by a New Kingdom fictional letter in which the writer proposes a scholarly competition between himself and another scribe regarding everyday calculation tasks such as accounting of land, labor, and grain.{{sfnp|Imhausen|2007|p=11}} Texts such as the [[Rhind Mathematical Papyrus]] and the [[Moscow Mathematical Papyrus]] show that the ancient Egyptians could perform the four basic mathematical operations—addition, subtraction, [[Ancient Egyptian multiplication|multiplication]], and division—use fractions, calculate the areas of rectangles, triangles, and circles and compute the volumes of boxes, columns and pyramids. They understood basic concepts of [[Egyptian algebra|algebra]] and [[Egyptian geometry|geometry]], and could solve [[systems of equations]].{{sfnp|Clarke|Engelbach|1990|p=222}}

{{hiero | {{frac|2|3}} | <hiero>D22</hiero>| align=right| era=default}}

[[Mathematical notation]] was decimal, and based on hieroglyphic signs for each power of ten up to one million. Each of these could be written as many times as necessary to add up to the desired number; so to write the number eighty or eight hundred, the symbol for ten or one hundred was written eight times respectively.{{sfnp|Clarke|Engelbach|1990|p=217}} Because their methods of calculation could not handle most fractions with a numerator greater than one, they had to write [[Egyptian fraction|fractions]] as the sum of several fractions. For example, they resolved the fraction ''two-fifths'' into the sum of ''one-third'' + ''one-fifteenth''. Standard tables of values facilitated this.{{sfnp|Clarke|Engelbach|1990|p=218}} Some [[Vulgar fraction|common fractions]], however, were written with a special glyph—the equivalent of the modern two-thirds is shown on the right.{{sfnp|Gardiner|1957|p=197}}

Ancient Egyptian mathematicians knew the [[Pythagorean theorem]] as an empirical formula. They were aware, for example, that a triangle had a right angle opposite the [[hypotenuse]] when its sides were in a 3–4–5 ratio.{{sfnp|Strouhal|1989|p=241}} They were able to estimate the area of a [[circle]] by subtracting one-ninth from its diameter and squaring the result:

:Area ≈ [({{frac|8|9}})''D'']<sup>2</sup> = ({{frac|256|81}})''r''<sup>2</sup> ≈ 3.16''r''<sup>2</sup>,

a reasonable approximation of the formula {{nowrap|'''[[Pi|π]]'''''r''<sup>2</sup>}}.{{sfnmp|1a1=Strouhal|1y=1989|1p=241|2a1=Imhausen|2y=2007|2p=31}}