Editing Rounding (section)

==History==
The concept of rounding is very old, perhaps older than the concept of division itself. Some ancient [[clay tablet]]s found in [[Mesopotamia]] contain tables with rounded values of [[multiplicative inverse|reciprocals]] and square roots in base 60.<ref>[http://it.stlawu.edu/~dmelvill/mesomath/tablets/YBC7289.html Duncan J. Melville. "YBC 7289 clay tablet". 2006]</ref>
Rounded approximations to [[pi|{{pi}}]], the length of the year, and the length of the month are also ancient – see [[sexagesimal#Examples|base 60 examples]].

The ''round-half-to-even'' method has served as [[American Standards Association|American Standard]] Z25.1 and [[ASTM]] standard E-29 since 1940.<ref>{{citation |mode=cs1 |publisher=[[American Standards Association]] |title=Rules for Rounding Off Numerical Values |year=1940 |id=Z25.1-1940}} {{pb}} The standard arose from a committee of the ASA working to standardize inch–millimeter conversion. See: {{cite magazine |last=Agnew |first=P. G. |title=Man's Love Of Round Numbers |magazine=Industrial Standardization and Commercial Standards Monthly |date=Sep 1940 |volume=11 |number=9 |pages=230–233 |url=https://archive.org/details/sim_magazine-of-standards_1940-09_11_9/page/230/ }} {{pb}} The standard was also more concisely advertised in: {{cite magazine |url=https://archive.org/details/sim_power_1940-11_84_11/page/721/ |title=Rounding Off Decimals |magazine=Power |volume=84 |issue=11 |date=Nov 1940 |page=93 }} {{pb}} {{citation |mode=cs1 |publisher=ASTM |title=Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications |id=E-29 |year=2013 |orig-year=1940 |doi=10.1520/E0029-13 }} </ref> The origin of the terms ''unbiased rounding'' and ''statistician's rounding'' are fairly self-explanatory. In the 1906 fourth edition of ''Probability and Theory of Errors'' [[Robert Simpson Woodward]] called this "the computer's rule",<ref>{{cite book |title=Probability and theory of errors |last=Woodward |first=Robert S. |series=Mathematical Monographs |volume=7 |year=1906 |place=New York |publisher=J. Wiley & Son |page=42 |url=https://archive.org/details/probabilitytheor00wooduoft/page/42/ |quote=An important fact with regard to the error 1/2 for {{mvar|n}} even is that its sign is arbitrary, or is not fixed by the computation as is the case with all the other errors. However, the computer's rule, which makes the last rounded figure of an interpolated value even when half a unit is to be disposed of, will, in the long run, make this error as often plus as minus. }}</ref> indicating that it was then in common use by [[human computer]]s who calculated mathematical tables. For example, it was recommended in [[Simon Newcomb]]'s c. 1882 book ''Logarithmic and Other Mathematical Tables''.<ref>{{cite book |last=Newcomb |first=Simon |date=1882 |title=Logarithmic and Other Mathematical Tables with Examples of their Use and Hints on the Art of Computation |place=New York |publisher=Henry Holt |pages=14–15 |url=https://archive.org/details/logarithmicother00newcrich/page/n24 |quote=Here we have a case in which the half of an odd number is required. [...] A good rule to adopt in such a case is to ''write the nearest'' '''even''' ''number''. }}</ref> Lucius Tuttle's 1916 ''Theory of Measurements'' called it a "universally adopted rule" for recording physical measurements.<ref>{{cite book |last=Tuttle |first=Lucius |year=1916 |title=The Theory of Measurements |place=Philadelphia |publisher=Jefferson Laboratory of Physics |page=29 |url=https://archive.org/details/theoryofmeasurem00tuttrich/page/29 |quote=A fraction perceptibly less than a half should be discarded and more than a half should always be considered as one more unit, but when it is uncertain which figure is the nearer one the universally adopted rule is to ''record the nearest even number'' rather than the odd number that is equally near. The reason for this procedure is that in a series of several measurements of the same quantity it will be as apt to make a record too large as it will to make one too small, and so in the average of several such values will cause but a slight error, if any. }}</ref> [[Churchill Eisenhart]] indicated the practice was already "well established" in data analysis by the 1940s.<ref>{{cite book |title=Selected Techniques of Statistical Analysis for Scientific and Industrial Research, and Production and Management Engineering |publisher=McGraw-Hill |location=New York |pages=187–223 |chapter-url=https://archive.org/details/selectedtechniqu00colu |author=Churchill Eisenhart |editor1=Eisenhart |editor2=Hastay |editor3=Wallis |access-date=30 January 2014 |chapter=Effects of Rounding or Grouping Data |date=1947}}</ref>

The origin of the term ''bankers' rounding'' remains more obscure. If this rounding method was ever a standard in banking, the evidence has proved extremely difficult to find. To the contrary, section 2 of the European Commission report ''The Introduction of the Euro and the Rounding of Currency Amounts''<ref>{{cite web |title=The Introduction of the Euro and the Rounding of Currency Amounts |url=https://ec.europa.eu/economy_finance/publications/pages/publication1224_en.pdf |access-date=2011-08-19 |url-status=live |archive-url=https://web.archive.org/web/20101009203257/http://ec.europa.eu/economy_finance/publications/publication1224_en.pdf |archive-date=2010-10-09}}</ref> suggests that there had previously been no standard approach to rounding in banking; and it specifies that "half-way" amounts should be rounded up.

Until the 1980s, the rounding method used in floating-point computer arithmetic was usually fixed by the hardware, poorly documented, inconsistent, and different for each brand and model of computer. This situation changed after the IEEE 754 floating-point standard was adopted by most computer manufacturers. The standard allows the user to choose among several rounding modes, and in each case specifies precisely how the results should be rounded. These features made numerical computations more predictable and machine-independent, and made possible the efficient and consistent implementation of [[interval arithmetic]].

Currently, much research tends to round to multiples of 5 or 2. For example, [[Jörg Baten]] used [[age heaping]] in many studies, to evaluate the numeracy level of ancient populations. He came up with the [[Whipple's index#ABCC Index|ABCC Index]], which enables the comparison of the [[numeracy]] among regions possible without any historical sources where the population [[literacy]] was measured.<ref>{{cite journal |last1=Baten |first1=Jörg |title=Quantifying Quantitative Literacy: Age Heaping and the History of Human Capital |journal=Journal of Economic History |date=2009 |volume=69 |issue=3 |pages=783–808 |doi=10.1017/S0022050709001120 |hdl=10230/481 |s2cid=35494384 |url=http://repositori.upf.edu/bitstream/10230/481/1/996.pdf |hdl-access=free}}</ref>