Editing History of mathematics (section)

== Babylonian ==
{{Main|Babylonian mathematics}}
{{See also|Plimpton 322}}
[[Babylonia]]n mathematics refers to any mathematics of the peoples of [[Mesopotamia]] (modern [[Iraq]]) from the days of the early [[Sumer]]ians through the [[Hellenistic period]] almost to the dawn of [[Christianity]].<ref>{{Harv|Boyer|1991|loc="Mesopotamia" p. 24}}</ref> The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC ([[Seleucid]] period).<ref name="Boyer 1991 loc=Mesopotamia p. 26">{{Harv|Boyer|1991|loc="Mesopotamia" p. 26}}</ref> It is named Babylonian mathematics due to the central role of [[Babylon]] as a place of study. Later under the [[Caliphate|Arab Empire]], Mesopotamia, especially [[Baghdad]], once again became an important center of study for [[Islamic mathematics]].

[[File:Geometry problem-Sb 13088-IMG 0593-white.jpg|thumb|Geometry problem on a clay tablet belonging to a school for scribes; [[Susa]], first half of the 2nd millennium BC]]
In contrast to the sparsity of sources in [[Egyptian mathematics]], knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.<ref name="Boyer 1991 loc=Mesopotamia p. 25">{{Harv|Boyer|1991|loc="Mesopotamia" p. 25}}</ref> Written in [[Cuneiform script]], tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.<ref name="Boyer 1991 loc=Mesopotamia p. 41">{{Harv|Boyer|1991|loc="Mesopotamia" p. 41}}</ref>

The earliest evidence of written mathematics dates back to the ancient [[Sumer]]ians, who built the earliest civilization in Mesopotamia. They developed a complex system of [[metrology]] from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.<ref>{{Citation |last=Sharlach |first=Tonia |title=Calendars and Counting |url=http://dx.doi.org/10.4324/9780203096604.ch15 |work=The Sumerian World |year=2006 |pages=307–308 |access-date=2023-07-07 |publisher=Routledge |doi=10.4324/9780203096604.ch15 |isbn=978-0-203-09660-4}}</ref> From around 2500 BC onward, the Sumerians wrote [[multiplication table]]s on clay tablets and dealt with geometrical exercises and [[Division (mathematics)|division]] problems. The earliest traces of the Babylonian numerals also date back to this period.<ref>Melville, Duncan J. (2003). [http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html Third Millennium Chronology] {{Webarchive|url=https://web.archive.org/web/20180707213616/http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html |date=2018-07-07 }}, ''Third Millennium Mathematics''. [[St. Lawrence University]].</ref>

[[Image:Plimpton 322.jpg|thumb|left|The Babylonian mathematical tablet [[Plimpton 322]], dated to 1800 BC.]]
Babylonian mathematics were written using a [[sexagesimal]] (base-60) [[numeral system]].<ref name="Boyer 1991 loc=Mesopotamia p. 25"/> From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is thought the sexagesimal system was initially used by Sumerian scribes because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30,<ref name="Boyer 1991 loc=Mesopotamia p. 25"/> and for scribes (doling out the aforementioned grain allotments, recording weights of silver, etc.) being able to easily calculate by hand was essential, and so a sexagesimal system is pragmatically easier to calculate by hand with; however, there is the possibility that using a sexagesimal system was an ethno-linguistic phenomenon (that might not ever be known), and not a mathematical/practical decision.<ref name="Powell 1976 p. 418">{{Citation |last=Powell |first=M. |title=The Antecedents of Old Babylonian Place Notation and the Early History of Babylonian Mathematics |url=https://core.ac.uk/download/pdf/82557367.pdf |work=Historia Mathematica |volume=3 |pages=417–439 |year=1976 |access-date=July 6, 2023}}</ref> Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the [[decimal]] system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the [[Renaissance]], and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet [[YBC 7289]] gives an approximation of {{radic|2}} accurate to five decimal places.<ref name="Boyer 1991 loc=Mesopotamia p. 27">{{Harv|Boyer|1991|loc="Mesopotamia" p. 27}}</ref> The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.<ref name="Boyer 1991 loc=Mesopotamia p. 26"/> By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.<ref name="Boyer 1991 loc=Mesopotamia p. 26"/> This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.<ref name="Boyer 1991 loc=Mesopotamia p. 26"/>

Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of [[regular number]]s, and their [[Multiplicative inverse|reciprocal]] [[tuple|pairs]].<ref>{{cite book | author-link = Aaboe | last = Aaboe | first = Asger | title = Episodes from the Early History of Mathematics | year = 1998 | publisher = Random House | location = New York | pages = 30–31}}</ref> The tablets also include multiplication tables and methods for solving [[linear equation|linear]], [[quadratic equation]]s and [[cubic equation]]s, a remarkable achievement for the time.<ref>{{Harv|Boyer|1991|loc="Mesopotamia" p. 33}}</ref> Tablets from the Old Babylonian period also contain the earliest known statement of the [[Pythagorean theorem]].<ref>{{Harv|Boyer|1991|loc="Mesopotamia" p. 39}}</ref> However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for [[mathematical proof|proofs]] or logical principles.<ref name="Boyer 1991 loc=Mesopotamia p. 41"/>