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Formulae for the convergents to some irrationals

Published online by Cambridge University Press:  01 August 2016

Martin Griffiths
Martin Griffiths*
Affiliation:
Colchester County High School for Girls, Norman Way, Colchester CO3 3US
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Extract

One of the questions that I set my students in a recent A level revision lesson was to express (√2 - 1)6 in the form a + b√2, where a and b are integers. While they were calculating the answer, it occurred to me that, as a consequence of the fact that (√2 - 1)n → 0 as n → ∞ (since 0 < √2 - 1 < 1 ), a and b could be used to give us a ‘good’ rational approximation to √2. Within a minute or so everyone agreed that (√2 - 1)6 = 99 - 70√2, allowing us to write √2 ≈ 99/70.

Type
Articles
Information
The Mathematical Gazette , Volume 88 , Issue 511 , March 2004 , pp. 28 - 38
Copyright
Copyright © The Mathematical Association 2004

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References

1. Burton, David M. Elementary number theory, McGraw-Hill (1998).Google Scholar
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