Inventing Your Own Number System

Inventing Your Own Number System Through the ages people have invented many different ways to name, write, and compute with numbers. Our current number system is based on place values corresponding to powers of ten. In principle, place values could correspond to any sequence of numbers. For example, the places could have values corresponding to the sequence of square numbers, triangular numbers, multiples of six, Fibonacci numbers, prime numbers, or factorials. The Roman numeral system does not use place values, but the position of numerals does matter when determining the number represented. Tally marks are a simple system, but representing large numbers requires many strokes. In our number system, symbols for digits and the positions they are located combine to represent the value of the number. It is possible to create a system where symbols stand for operations rather than values. For example, the system might always start at a default number and use symbols to stand for operations such as doubling, adding one, taking the reciprocal, dividing by ten, squaring, negating, or any other specific operations. Create your own number system. What symbols will you use for your numbers? How will your system work? Demonstrate how your system could be used to perform some of the following functions. • Count from 0 up to 100 • Compare the sizes of numbers • Add and subtract whole numbers • Multiply and divide whole numbers • Represent fractional values • Represent irrational numbers (such as π) What are some of the advantages of your system compared with other systems? What are some of the disadvantages? If you met aliens that had developed their own number system, how might their mathematics be similar to ours and how might it be different? Make a list of some math facts and procedures that you have learned. Which items on the list would probably be the same no matter what number system was used? Which facts and procedures would depend on the number system?.

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Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
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Zero Displacement Ternary Number System: the Most Economical Way of Representing Numbers
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This page intentionally left blank Numerical Notation Th is book is a cross-cultural reference volume of all attested numerical notation systems (graphic, nonphonetic systems for representing numbers), encompassing more than 100 such systems used over the past 5,500 years. Using a typology that defi es progressive, unilinear evolutionary models of change, Stephen Chrisomalis identifi es fi ve basic types of numerical notation systems, using a cultural phylo- genetic framework to show relationships between systems and to create a general theory of change in numerical systems. Numerical notation systems are prima- rily representational systems, not computational technologies. Cognitive factors that help explain how numerical systems change relate to general principles, such as conciseness and avoidance of ambiguity, which also apply to writing systems. Th e transformation and replacement of numerical notation systems relate to spe- cifi c social, economic, and technological changes, such as the development of the printing press and the expansion of the global world-system. Stephen Chrisomalis is an assistant professor of anthropology at Wayne State Uni- versity in Detroit, Michigan. He completed his Ph.D. at McGill University in Montreal, Quebec, where he studied under the late Bruce Trigger. Chrisomalis’s work has appeared in journals including Antiquity, Cambridge Archaeological Jour- nal, and Cross-Cultural Research. He is the editor of the Stop: Toutes Directions project and the author of the academic weblog Glossographia. Numerical Notation A Comparative History Stephen Chrisomalis Wayne State University CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521878180 © Stephen Chrisomalis 2010 This publication is in copyright.
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