Approximation of Irrationals

a

2

m

10

show fractional parts for small m

point size

If

a

is irrational and

m

is any positive integer, there is a fraction

p

q

with

q≤m

and for which

a-

p

q

<

1

qm

≤

1

2

q

.

Proof. Let

m

be a positive integer. Then by the pigeonhole principle, among the

m+1

points

{0a},{1a},{2a},…,{ma}

(where

{x}

denotes the fractional part of

x

), there are at least two numbers

u≠v

such that

|{ua}-{va}|<1/m

. Then

|(u-v)a-p|<1/m

for some integer

p

. The statement is proved if we put

q=|u-v|

.