186 IV Continued Fractions and Their Uses
Thus we now assumeμ=0. Sinceq≤qn,λandμcannot both be positive and hence,
sinceq>0,λμ <0. Then
qξ−p=λ(qn− 1 ξ−pn− 1 )+μ(qnξ−pn)and both terms on the right have the same sign. Hence
|qξ−p|=|λ(qn− 1 ξ−pn− 1 )|+|μ(qnξ−pn)|
≥|qn− 1 ξ−pn− 1 |.This proves the first statement of the proposition. The second statement follows,
since
|ξ−p/q|=q−^1 |qξ−p|>q−^1 |qnξ−pn|
=(qn/q)|ξ−pn/qn|
≥|ξ−pn/qn|. To illustrate the application of Proposition 3, consider the continued fraction ex-
pansion ofπ= 3. 14159265358 ....We easily find that it begins [3, 7 , 15 , 1 , 292 ,...].
It follows that the first five convergents ofπare
3 / 1 , 22 / 7 , 333 / 106 , 355 / 113 , 103993 / 33102.Using the inequality|ξ−pn/qn|< 1 /qnqn+ 1 and choosingn=3sothatan+ 1 is
large, we obtain
0 < 355 / 113 −π< 0. 000000267 ···.The approximation 355/113 toπ was first given by the Chinese mathematician
Zu Chongzhi in the 5th century A.D. Proposition 3 shows that it is a better approx-
imation toπthan any other rational number with denominator≤113.
In general, a rational numberp′/q′,wherep′,q′are integers andq′>0, may be
said to be abest approximationto a real numberξif
|ξ−p/q|>|ξ−p′/q′|for all different rational numbersp/qwhose denominatorqsatisfies 0<q≤q′. Thus
Proposition 3 says that any convergentpn/qn(n≥1) ofξis a best approximation
ofξ. However, these are not the only best approximations. It may be shown that, if
pn− 2 /qn− 2 andpn− 1 /qn− 1 are consecutive convergents ofξ, then any rational number
of the form
(cpn− 1 +pn− 2 )/(cqn− 1 +qn− 2 ),wherecis an integer such thatan/ 2 <c≤anis a best approximation ofξ.Further-
more, every best approximation ofξhas this form if, whenanis even, one allows also
c=an/2.
It follows that 355/113 is a better approximation toπthan any other rational num-
ber with denominator less than 16604, since 292/ 2 =146 and 146× 113 + 106 =
16604.