periodic simple continued fractions

Question 1

periodic simple continued fraction:

Answer 1

iscf w/ the form [x0:x1,…,xk,y1,…,yn,y1…], denoted [x0:x1,…,xk,y1,…,yn] with a single line over the ys

Question 2

quadratic irrational:

Answer 2

an irrational real number that’s a solution of a quadratic equation with integer coefficients - can always be written r+sroot(d) with r,s rational, s!=0, and d a positive integer that isn’t a perfect square

Question 3

pscfs and quadratic irrationals:

Answer 3

a real number is/has one iff it has/is the other

Question 4

purely periodic:

Answer 4

a pscf is a pure pscf if it has the form [x0:x1,…,xk] with a line over the whole thing, the whole thing repeats including x0

Question 5

conjugate of a quadratic irrational:

Answer 5

if a=r+sroot(d), the conjugate a*=r-sroot(d)

Question 6

reduced quadratic irrational:

Answer 6

a>1, -1<a*<0

Question 7

ppscfs and quadratic irrationals:

Answer 7

a quadratic irrational has a ppscf iff it’s reduced

Question 8

root(d) in continued fractions:

Answer 8

[⌊d⌋: x1,…,x(r-1),2⌊root(d)⌋] (everything past ⌊d⌋ has a line over it, so x1 to the end) for some r>=1 and x1,…,x(r-1) in N