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On the Approximation to Irrational Numbers By Rationals

Published online by Cambridge University Press:  03 November 2016

A.C. Heath
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Extract

The problem of deciding whether a given number is irrational or not is frequently one of some difficulty, as in the proof that π is irrational. Several numbers which are very probably irrational have never been proved so, e.g. y, eπ. However, we have one quite simple test for deciding whether a number is irrational or not when we have the number expressed as a continued fraction, viz. if the continued fraction terminates, the number is rational; if not, irrational. In what follows we assume our irrational number given as a continued fraction :

or shortly

Type
Research Article
Information
The Mathematical Gazette , Volume 13 , Issue 188 , May 1927 , pp. 362 - 366
Copyright
Copyright © Mathematical Association 1927

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References

Page 363 note * Heawood; I.e. infra, ser. 2, vol. 20, p. 233.