Sequence of Best Rational Approximations to Square Root of 2/Historical Note
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Historical Note on Sequence of Best Rational Approximations to Square Root of 2
The sequence of best rational approximations to the square root of $2$ was known to Theon of Smyrna in the $2$nd century C.E.
More precisely, he knew that if $\dfrac p q$ is an approximation to $\sqrt 2$, then $\dfrac {p + 2 q} {p + q}$ is a better one, which result is demonstrated in Relation between Adjacent Best Rational Approximations to Root 2.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$