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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.

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
2. Liu, A. C. F. Lattice points and Pick’s theorem, Mathematics Magazine 52 (1979), pp. 232–235.Google Scholar
3. DeTemple, D. and Robertson, J. M. The equivalence of Euler’s and Pick’s theorems, Mathematics Teacher 67 (1974), pp. 222–226.Google Scholar
4. Hardy, G. H. and Wright, E. M. An introduction to the theory of numbers, 2nd edition, Clarendon Press (1945).Google Scholar
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
7. Beck, A. Bleicher, M. N. and Crowe, D. W. Excursions into mathematics, fourth reprint of the 1969 edition, Worth, New York (1976).Google Scholar