Number Theory: An Introduction to Mathematics

3 Periodic Continued Fractions 191

To establish this, putqθ=p+δ,wherep∈Zand|δ|≤ 1 /2. Then

|e^2 πiqθ− 1 |= 2 |sinπqθ|= 2 |sinπδ|

and the result follows from the previous characterization, since(sinx)/xdecreases
from 1 to 2/πasxincreases from 0 toπ/2.

3 PeriodicContinuedFractions..................................

A complex numberζ is said to be aquadratic irrationalif it is a root of a monic
quadratic polynomialt^2 +rt+swith rational coefficientsr,s, but is not itself rational.
Sinceζ/∈Q, the rational numbersr,sare uniquely determined byζ.
Equivalently,ζis a quadratic irrational if it is a root of a quadratic polynomial

f(t)=At^2 +Bt+C

with integer coefficientsA,B,Csuch thatB^2 − 4 ACis not the square of an integer.
The integersA,B,Care uniquely determined up to a common factor and are uniquely
determined up to sign if we require that they have greatest common divisor 1. The
corresponding integerD=B^2 − 4 ACis then uniquely determined and is called the
discriminantofζ. A quadratic irrational is real if and only if its discriminant is positive.
It is readily verified that if a quadratic irrationalζis equivalent to a complex num-
berω,i.e.if

ζ=(αω+β)/(γω+δ),

whereα,β,γ,δ∈Zandαδ−βγ=±1, thenωis also a quadratic irrational. More-
over, ifζis a root of the quadratic polynomialf(t)=At^2 +Bt+C,whereA,B,Care
integers with greatest common divisor 1, thenωis a root of the quadratic polynomial

g(t)=A′t^2 +B′t+C′,

where

A′=α^2 A+αγB+γ^2 C,

B′= 2 αβA+(αδ+βγ)B+ 2 γδC,

C′=β^2 A+βδB+δ^2 C,

and hence

B′^2 − 4 A′C′=B^2 − 4 AC=D.

Since

A=δ^2 A′−γδB′+γ^2 C′,

B=− 2 βδA′+(αδ+βγ)B′− 2 αγC′,

C=β^2 A′−αβB′+α^2 C′,

A′,B′,C′also have greatest common divisor 1.