192 IV Continued Fractions and Their Uses
Ifζis a quadratic irrational, we define theconjugateζ′ofζto be the other root of
the quadratic polynomialf(t)which hasζas a root. If
ζ=(αω+β)/(γω+δ),whereα,β,γ,δ∈Zandαδ−βγ=±1, then evidently
ζ′=(αω′+β)/(γω′+δ).Suppose now thatζ = ξ is real and that the integersA,B,Care uniquely
determined by requiring not only(A,B,C)=1butalsoA>0. The real quadratic
irrationalξis said to bereducedifξ>1and− 1 <ξ′<0. Ifξis reduced then, since
ξ>ξ′,wemusthave
ξ=(−B+√
D)/ 2 A,ξ′=(−B−√
D)/ 2 A.
Thus the inequalitiesξ>1and− 1 <ξ′<0imply
0 <√
D+B< 2 A<
√
D−B.
Conversely, if the coefficientsA,B,Coff(t)satisfy these inequalities, whereD=
B^2 − 4 AC>0, then one of the roots off(t)is reduced. ForB< 0 <Aand so the
rootsξ,ξ′off(t)have opposite signs. Ifξis the positive root, thenξandξ′are given
by the preceding formulas and henceξ> 1 ,− 1 <ξ′<0. It should be noted also that
ifξis reduced, thenB^2 <Dand henceC<0.
We return now to continued fractions. Ifξis a real quadratic irrational, then its
complete quotientsξnare all quadratic irrationals and, conversely, if some complete
quotientξnis a quadratic irrational, thenξis also a quadratic irrational.
The continued fraction expansion [a 0 ,a 1 ,a 2 ,...] of a real numberξis said to be
eventually periodicif there exist integersm≥0andh>0 such that
an=an+h for alln≥m.The continued fraction expansion is then conveniently denoted by
[a 0 ,a 1 ,...,am− 1 ,am,...,am+h− 1 ].The continued fraction expansion is said to beperiodicif it is eventually periodic with
m=0.
Equivalently, the continued fraction expansion ofξis eventually periodic ifξm=
ξm+hfor somem≥0andh>0, and periodic if this holds withm=0. Theperiodof
the continued fraction expansion, in either case, is the least positive integerhwith this
property.
We are going to show that there is a close connection between real quadratic irra-
tionals and eventually periodic continued fractions.
Proposition 7A real numberξis a reduced quadratic irrational if and only if its
continued fraction expansion is periodic.
Moreover, ifξ=[a 0 ,...,ah− 1 ],then− 1 /ξ′=[ah− 1 ,...,a 0 ].