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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.

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Type
Articles
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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
2. Neill, Hugh and Quadling, Douglas Pure Mathematics 5 & 6, Cambridge University Press (2001).Google Scholar
3. McBride, Adam Remarks on Pell’s equation and square root algorithms, Math. Gaz. 83 (March 1999), pp. 47–52.Google Scholar
4. Desbrow, Darrell Approximation to square roots – further to Note 81.41, Math. Gaz. 84 (July 2000), pp. 282–287.Google Scholar