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| | Andrew Berryhill is what his wife enjoys to contact him and he completely digs that name. Alaska is the only place I've been residing in but now I'm contemplating other options. Distributing production has been his occupation for some time. What I adore doing is soccer but I don't have the time recently.<br><br>Here is my webpage ... [http://www.khuplaza.com/dent/14869889 online psychic] |
| {{Other people2|James Anderson (disambiguation)}}
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| '''James Anderson''' is an [[academic staff]] member in the School of Systems Engineering at the [[University of Reading]], England. He is currently teaching [[compilers]], [[algorithms]], and [[computer algebra]], and in the past he has taught [[Computer programming|programming]] and [[computer graphics]].<ref name="reading">{{Cite web
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| | url=http://www.reading.ac.uk/sse/about/staff/j-anderson.aspx
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| | title=Computer Science at Reading - Dr. James Anderson
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| | accessdate=28 February 2011
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| | work=[[University of Reading]]
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| }}</ref>
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| Anderson quickly gained publicity in December 2006 in the United Kingdom when the regional [[BBC South Today]] reported his claim of "having solved a 1200 year old problem", namely that of [[division by zero]]. However, commentators quickly responded that his ideas are just a variation of the standard [[IEEE 754]] concept of [[NaN]] (Not a Number), which has been commonly employed on [[computer]]s in [[floating point]] [[arithmetic]] for many years.<ref name="Good Math, Bad Math">{{Cite web|url=http://scienceblogs.com/goodmath/2006/12/nullity_the_nonsense_number_1.php|title=Nullity: The Nonsense Number|work=Good Math, Bad Math|date=7 December 2006|author=Mark C. Chu-Carroll|accessdate=7 December 2006}}</ref> Dr Anderson defended against the criticism of his claims on BBC Berkshire on 12 December 2006, saying, "If anyone doubts me I can hit them over the head with a computer that does it."<ref name=BBC2>{{Cite web|url=http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml|title='Nullity is a number, and that makes a difference'|accessdate=12 December 2006|date=12 December 2006|work=[[BBC News]]}}</ref>
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| ==Research and background==
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| Anderson is a member of the [[British Computer Society]], the British Machine Vision Association, [[Eurographics]], and the British Society for the Philosophy of Science.<ref>{{Cite web
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| | url = http://www.api.reading.ac.uk/about.htm
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| | title = About the Ambient & Pervasive Intelligence Research Group
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| | accessdate = 16 January 2007
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| | work = [[University of Reading]]
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| }}</ref> He is also a teacher in the Computer Science department (School of Systems Engineering) at the [[University of Reading]].<ref name="reading" /> He was
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| a psychology graduate who worked in the Electrical and Electronic Engineering departments at the [[University of Sussex]] and Plymouth Polytechnic (now the [[University of Plymouth]]). His doctorate is from the University of Reading for (in Anderson's words) "developing a canonical description of the perspective transformations in whole numbered dimensions".
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| He has written two papers on [[division by zero]]<ref name=BBC1>{{Cite news|url=http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml|title=1200-year-old problem "easy"|date=7 December 2006|author=Ben Moore|coauthors=Ollie Williams|work=[[BBC News]]|quote=Schoolchildren from Caversham have become the first to learn a brand new theory that dividing by zero is possible using a new number—"nullity". But the suggestion has left many mathematicians cold.}}.</ref><ref name="Slashdot">{{Cite web|url=http://science.slashdot.org/article.pl?sid=06/12/07/0416223|title=Professor Comes Up With a Way to Divide by Zero|accessdate=7 December 2006|work=[[Slashdot]]}}</ref> and has invented what he calls the "Perspex machine".
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| Anderson claims that "mathematical arithmetic is sociologically invalid" and that [[IEEE floating point standard|IEEE floating-point arithmetic]], with NaN, is also faulty.<ref name ="investor presentation">{{Cite web|url=http://www.angletechnology.com/events/investorday/Dr%20James%20Anderson%20Transreal%20presentation.pdf|title=Transreal Computing Research and Portfolio — Company Showcase|author=Dr James A.D.W. Anderson|format=PDF|accessdate=11 December 2006}}</ref>
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| ==Transreal arithmetic==
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| {{BLP sources|date=May 2010}}
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| {| class="wikitable" style="margin:10px; float:right; border: inset; border-collapse:collapse;" border="2" width="15%"
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| |+ Zero divided by zero
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| |-
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| | In mathematical analysis, the following limits can be found:
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| :<math> \lim_{x \to 0}\frac 0 x = 0 </math>
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| : <math> \lim_{x \to 0^{+}}\frac 1 x = +\infty</math>
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| : <math> \lim_{x \to 0^{-}}\frac 1 x = -\infty</math>
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| : <math> \lim_{x \to 0}\frac{\sin x}{x} = 1 </math>
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| : <math> \lim_{x \to 0}\frac{1 - \cos x}{x} = 0 </math>
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| <math>0^0</math> is also an [[indeterminate form]]. See [[Exponentiation#Zero to the zero power|exponentiation]].
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| |-
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| | In IEEE floating-point arithmetic:
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| : <math> \frac{0}{0} = NaN </math>
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| : by definition
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| In several computer programming languages, including [[C (programming language)|C]]'s <code>pow</code> function, <math>0^0</math> is defined to be <math>1</math>, as that is the most convenient value for numerical analysis programs, since it makes <math>f(x) = x^0</math> (and many other functions) [[continuous function|continuous]] at zero, with the notable exception of <math>f(x) = 0^x</math>.<ref>{{Cite journal|title=Rationale for International Standard — Programming Languages — C|version=Revision 5.10|date=April 2003|author=John Benito|url=http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf|page=182}}</ref>
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| |-
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| | In transreal arithmetic:
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| : <math> \frac{0}{0} = \Phi </math>
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| : by definition
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| : <math> 0^0 = \Phi \,</math>
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| : by Anderson's proof, reported on by the BBC, that:
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| : <math> 0^0 = \frac{0}{0} </math>
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| |}
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| Anderson's transreal numbers were first mentioned in a 1997 publication,<ref name=Pub1>{{Cite web| url=http://adsabs.harvard.edu/abs/1997RSPTB.352.1129A|title=Representing Geometrical Knowledge}}</ref> and made well-known on the [[Internet]] in 2006, but not accepted as useful by the mathematics community. These numbers are used in his concept of '''transreal arithmetic''' and the Perspex machine. According to Anderson, transreal numbers include all of the [[real number]]s, plus three others: [[infinity]] (<math>\infty</math>), negative infinity (<math>-\infty</math>) and "nullity" (<math>\Phi</math>), a numerical representation of a non-number that lies outside of the [[affinely extended real number line]]. ([[Null space|Nullity]], confusingly, has an existing mathematical meaning.)
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| Anderson intends the [[axiom]]s of transreal arithmetic to complement the axioms of standard arithmetic; they are supposed to produce the same result as standard arithmetic for all calculations where standard arithmetic defines a result. In addition, they are intended to define a consistent numeric result for the calculations which are undefined in standard arithmetic, such as [[division by zero]].<ref name="perplex8">{{cite conference|url=http://www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf|format=PDF|title=Perspex Machine VIII: Axioms of Transreal Arithmetic|author=J A D W Anderson|year=2006|booktitle=Vision Geometry XV: Proceedings of SPIE|volume=6499|editor=[[Longin Jan Latecki]], David M. Mount, and Angela Y. Wu.}}</ref>
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| ===Transreal arithmetic and other arithmetics===
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| "Transreal arithmetic" closely resembles IEEE floating point arithmetic, a [[floating point]] arithmetic commonly used on [[computer]]s. IEEE floating point arithmetic, like transreal arithmetic, uses affine infinity (two separate infinities, one positive and one negative) rather than [[projectively extended real numbers|projective infinity]] (a single unsigned infinity, turning the number line into a loop). Division of any non-zero finite number by zero results in either positive or negative infinity.
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| However, in IEEE arithmetic, division of zero by zero is still considered [[indeterminate form|indeterminate]]. The reason for this is simple: A statement about the quotient of two numbers is understood in mathematics as another statement about multiplication. Specifically, if
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| <math>a \div b = c</math>
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| this is understood as simply another way of saying that
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| <math>a = b \times c</math>
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| Thus, if for some number <math>c</math>
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| <math>0 \div 0 = c</math>
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| then this is just another way of saying that
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| <math>0 = 0 \times c</math>
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| But in fact this is true for '''all''' real numbers <math>c</math>. And that is precisely the reason that mathematicians do not assign a single value to <math>0 \div 0</math> but rather label it "indeterminate". Assigning a value to <math>0 \div 0</math>, even a newly fabricated "number", misses the point entirely.
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| In IEEE arithmetic, the value of <math>0 \div 0</math> is therefore represented by the symbol [[NaN|Not a Number (NaN)]] (Not a Number). NaN is '''not''' meant to be a number, but rather an error message conveying the fact that the arithmetical operation the computer just attempted cannot be assigned a single number as an answer – even if <math>+\infty</math> and <math>-\infty</math> are considered numbers. Because <math>NaN</math> is an error message and not a number, it is not considered equal to anything, even itself. That is, the comparison <math>NaN = NaN</math> evaluates to false.
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| Here are some [[identity (mathematics)|identities]] in transreal arithmetic with the IEEE equivalents: | |
| {| class="wikitable" border="3"
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| ! Transreal arithmetic !! IEEE standard floating point arithmetic
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| |-
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| | <math>0 \div 0 = \Phi</math> || <math>0 \div 0 = NaN</math>
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| |-
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| | <math>\infty \times 0 = \Phi</math> || <math>\infty \times 0 = NaN</math>
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| |-
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| | <math>\infty - \infty = \Phi</math> || <math>\infty - \infty = NaN</math>
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| |-
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| | <math>\Phi + a = \Phi \ </math> || <math>NaN + a = NaN</math>
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| |-
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| | <math>\Phi \times a = \Phi</math> || <math>NaN \times a = NaN</math>
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| |-
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| | <math>-\Phi = \Phi \ </math> || <math>-NaN = NaN</math> (i.e. applying unary negation to NaN yields NaN)
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| |-
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| | <math>+1 \div 0 = +\infty</math> || <math>1 \div +0 = +\infty</math>
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| |-
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| | <math>-1 \div 0 = -\infty</math> || <math>1 \div -0 = -\infty</math>
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| |-
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| | <math>\Phi = \Phi \Rightarrow True \ </math> || <math>NaN = NaN \Rightarrow False </math>
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| |}
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| The main difference between transreal arithmetic and IEEE floating-point arithmetic is thus that nullity compares equal to nullity, whereas NaN does not compare equal to NaN.
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| Anderson's analysis of the properties of transreal algebra is given in his paper on "perspex machines".<ref name="perplex9">{{cite conference|url=http://www.bookofparagon.com/Mathematics/PerspexMachineIX.pdf|format=PDF|title=Perspex Machine IX: Transreal Analysis|author=J A D W Anderson|year=2006|booktitle=Vision Geometry XV: Proceedings of SPIE|volume=6499|editor=[[Longin Jan Latecki]], David M. Mount, and Angela Y. Wu.}}</ref>
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| Due to the more expansive definition of numbers in transreal arithmetic, several identities and theorems which apply to all numbers in standard arithmetic are not universal in transreal arithmetic. For instance, in transreal arithmetic, <math>a-a=0</math> is not true for all <math>a</math>, since <math>\Phi-\Phi=\Phi</math>. That problem is addressed in ref.<ref name="perplex9"/> pg. 7. Similarly, it is not always the case in transreal arithmetic that a number can be cancelled with its [[Multiplicative inverse|reciprocal]] to yield <math>1</math>. Cancelling zero with its reciprocal in fact yields nullity.
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| Examining the [[axiom]]s provided by Anderson,<ref name="perplex8"/> it is easy to see that any term which contains an occurrence of the constant <math>\Phi</math> is provably equivalent to <math>\Phi</math>. Formally, let <math>t</math> be any term with a sub-term <math>\Phi</math>, then
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| <math>t=\Phi</math> is a [[theorem]] of the theory proposed by Anderson.
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| ==Media coverage==
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| Anderson's transreal arithmetic, and concept of "nullity" in particular, were introduced to the public by the [[BBC]] with its report in December 2006<ref name=BBC1 /> where Anderson was featured on a BBC television segment teaching schoolchildren about his concept of "nullity". The report implied that Anderson had ''discovered'' the solution to division by zero, rather than simply attempting to formalize it. The report also suggested that Anderson was the first to solve this problem, when in fact the result of zero divided by zero has been expressed formally in a number of different ways (for example, [[NaN]]).
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| The BBC was criticized for irresponsible journalism, but the producers of the segment defended the BBC, stating that the report was a light-hearted look at a mathematical problem aimed at a mainstream, regional audience for [[BBC South Today]] rather than at a global audience of mathematicians. The BBC later posted a follow-up giving Anderson's response to many claims that the theory is flawed.<ref name="BBC2"/>
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| ==Applications==
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| Anderson has been trying to market his ideas for transreal arithmetic and "Perspex machines" to investors. He claims that his work can produce computers which run "orders of magnitude faster than today's computers".<ref name ="investor presentation" /><ref name="company">{{Cite web|url=http://www.transrealcomputing.com/|title=Transreal Computing Ltd.|accessdate=12 December 2006}}</ref> He has also claimed that it can help solve such problems as [[quantum gravity]],<ref name ="investor presentation" /> the [[mind-body connection]],<ref name="paragon">http://www.bookofparagon.com/</ref> [[consciousness]]<ref name="paragon"/> and [[free will]].<ref name="paragon"/>
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| ==See also==
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| *[[Wheel theory]]
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| *[[Bottom type]] and [[bottom element]]
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| *[[Division by zero]]
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| ==References==
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| {{Wikinews|British computer scientist's new "nullity" idea provokes reaction from mathematicians}}
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| {{Reflist}}
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| ==Further reading==
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| *{{Cite web|url=http://www.jgc.org/blog/2006/12/midas-number-or-why-divide-by-zero.html|title= The Midas Number (or why divide by zero?) |date=11 December 2006|author=John Graham-Cumming}}
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| *{{Cite web|url=http://www.1729.com/blog/ZeroDividedByZero.html|title=Zero Divided By Zero: Application to Spherical Coordinates|author=Philip Dorrell|date=16 December 2006}}
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| ==External links==
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| *[http://www.reading.ac.uk/sse/about/staff/j-anderson.aspx Reading University Profile page]
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| *[http://www.bookofparagon.com/ Book of Paragon — personal homepage]
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME =Anderson, James
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION =British computer scientist
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| | DATE OF BIRTH =
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| | DATE OF DEATH =
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| }}
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| {{DEFAULTSORT:Anderson, James}}
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| [[Category:Year of birth missing (living people)]]
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| [[Category:Living people]]
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| [[Category:Alumni of the University of Reading]]
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| [[Category:Academics of the University of Reading]]
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| [[Category:English computer programmers]]
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| [[Category:English computer scientists]]
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| [[Category:Members of the British Computer Society]]
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| [[Category:Computer arithmetic]]
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| [[Category:Place of birth missing (living people)]]
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