Number Theory: An Introduction to Mathematics

2 Diophantine Approximation 187

The complete continued fraction expansion ofπis not known. However, it was
discovered by Cotes (1714) and then proved by Euler (1737) that the complete
continued fraction expansion ofe = 2. 71828182459 ...is given bye− 1 =
[1, 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 ,...].
The preceding results may also be applied to the construction of calendars. The
solar year has a length of about 365.24219 mean solar days. The continued fraction
expansion ofλ=( 0. 24219 )−^1 begins [4, 7 , 1 , 3 , 24 ,...]. Hence the first five conver-
gents ofλare

4 / 1 , 29 / 7 , 33 / 8 , 128 / 31 , 3105 / 752.

It follows that

0 < 128 / 31 −λ< 0. 0000428

and 128/31 is a better approximation toλthan any other rational number with denom-
inator less than 380. The Julian calendar, by adding a day every 4 years, estimated the
year at 365.25 days. The Gregorian calendar, by adding 97 days every 400 years, esti-
mates the year at 365.2425 days. Our analysis shows that, if we added instead 31 days
every 128 years, we would obtain the much more precise estimate of 365.2421875
days.
Best approximations also find an application in the selection of gear ratios, and con-
tinued fractions were already used for this purpose by Huygens (1682) in constructing
his planetarium (a mechanical model for the solar system).
The next proposition describes another way in which the continued fraction expan-
sion provides good rational approximations.

Proposition 4If p,q are coprime integers with q> 0 such that, for some real num-
berξ,

|ξ−p/q|< 1 / 2 q^2 ,

then p/q is a convergent ofξ.

Proof Letpn/qnbe the convergents ofξand assume thatp/qis not a convergent. We
show first thatq0. This is obvious ifξis irrational. Ifξ=pN/qN
is rational, then

1 /qN≤|qpN−pqN|/qN=|qξ−p|< 1 / 2 q.

Henceq0.
It follows thatqn− 1 ≤q0. By Proposition 3,
|qn− 1 ξ−pn− 1 |≤|qξ−p|< 1 / 2 q.

Hence

1 /qqn− 1 ≤|qpn− 1 −pqn− 1 |/qqn− 1

=|pn− 1 /qn− 1 −p/q|

≤|pn− 1 /qn− 1 −ξ|+|ξ−p/q|

< 1 / 2 qqn− 1 + 1 / 2 q^2.

But this impliesq<qn− 1 , which is a contradiction.