Quiz 2 Lemmas/Theorems Flashcards
a/b is reduced if…
1) GCD(a,b) = 1
2) b > 0
Prime decomp. of fractions
If a/b is reduced we write its prime decomposition as
a/b = p2^e2 x p3^e3…
ep>0 when p|a and
ep<0 when p|b
Proposition about rational roots
Let a/b be reduced fraction. Then for n∈Z+, nsqrt(a/b) is rational if and only if a and b are nth powers
Rational Roots Theorem
if a/b is reduced, and we have polynomial, then a|c0 and b|cd
Proposition about diophantine approx.
Let X∈R and let b∈Z+. The there exists a rational a/b∈Q, such that
|x - a/b| <= 1/2b
Generalization about diophantine approx.
Let x be rational. Then there exists Cx∈R such that for all a/b =/ x, a/b∈Q, b>0
|x - a/b| >= Cx(1/b)
Definition: Kissing Fractions
a/b kisses c/d, if ad-bc = +/- 1
Two fractions are kissing when their ford circle are tangent
Definition: For circle of a/b
Ford circle for a/b is circle of diameter 1/b^2 sitting directly above a/b on number line
Mediant of a/b and c/d
= a+b/c+d and lies between a/b and c/d
Proposition about mediant and kissing
if a/b kisses c/d, then the mediant kisses both a/b and c/d
Oct 17, first point
If a/b kisses a+1/b, then b = 1
Oct 17, second point
If a/b kisses c/d, then a+b/b kisses c+d/d
Oct 17, third point
If a/b rational with b>0, then there are exactly two fractions that kiss a/b and have smaller denominators. One is bigger than a/b and one is smaller.
Oct 17, fourth point
If s∈R is rational, and a/b < s < c/d, where a/b kisses c/d, there there exists e/f∈Q such that
|s - e/f| <= 1/2f^2 and f>= b +d
Dirtchlet’s Approximation Theorem
If x is irrational, then there exists infinitely many fraction a/b such that |x-a/b| < 1/2b^2
(infinitely many fractions whose Ford circles lies above x
