Approximation of Irrationals
Approximation of Irrationals
If is irrational and is any positive integer, there is a fraction with and for which .
a
m
p
q
q≤m
a-<≤
p
q
1
qm
1
2
q
Proof. Let be a positive integer. Then by the pigeonhole principle, among the points (where denotes the fractional part of ), there are at least two numbers such that . Then for some integer . The statement is proved if we put .
m
m+1
{0a},{1a},{2a},…,{ma}
{x}
x
u≠v
|{ua}-{va}|<1/m
|(u-v)a-p|<1/m
p
q=|u-v|