Simultaneous Approximation of Two Real Numbers by Rationals

a

1

π



2

3

-1

cos

1

3

+2

-1

cos

1

3

a

2

π

0.135914

2

3

1+

5

choose m > 1/ϵ

3

point size

0.02

a

1

≈



3

,

a

2

≈

2

3

This Demonstration illustrates the simultaneous approximation of two real numbers by rational numbers, because there is always a small square with at least two points.