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A visual approach to some elementary number theory

Published online by Cambridge University Press:  01 August 2016

Maxim Bruckheimer  and
Abraham Arcavi
Maxim Bruckheimer
Affiliation:
Department of Science Teaching, Weizmann Institute of Science, Rehovot 76100, Israel
Abraham Arcavi
Affiliation:
Department of Science Teaching, Weizmann Institute of Science, Rehovot 76100, Israel
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Extract

Euclidean geometry and fractions not only make unlikely bedfellows, but would seem to be a sure recipe for boredom. However, in this paper, with the help of a little history, we hope to prove the opposite.

Given a (rectangular) lattice, then a lattice polygon is a polygon whose vertices are lattice points. Pick's theorem [1] - the area of a simple lattice polygon is given by ½b + i – 1 where b is the number of lattice points on the boundary of the polygon, and i is the number in its interior – is well known and has been proved many times and in many ways.

Information

Type
Articles
Information
The Mathematical Gazette , Volume 79 , Issue 486 , November 1995 , pp. 471 - 478
Copyright
Copyright © The Mathematical Association 1995

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References

1. Pick, G. Geometrisches zur Zahlenlehre, Zeitschrift das Vereines ‘Lotos’ 19 (1899), pp. 311–319.Google Scholar
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5. Coxeter, H. S. M. Introduction to geometry, 2nd edition, Wiley, New York (1969).Google Scholar
6. Bruckheimer, M. and Arcavi, A. Farey series and Pick’s theorem, The Mathematical Intelligencer, in press.Google Scholar
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