a     c      a+c
--- + --- => -----
 b     d      b+d

##</code><code>##

a/b = 0/1
c/d = 1/0 [This is infinity, larger than any number]

##</code><code>##

frac(3.14159265358979);
sub frac
{ my ($x,$a,$b,$c,$d,$e,$f) = (@_,0,1,1,0,1,1);
  while ($f<1000)
  { if ($e<$x*$f) { ($a,$b) = ($e,$f) }
    else          { ($c,$d) = ($e,$f) }
    ($e,$f) = ($a+$c, $b+$d);
    print "$a/$b < $x < $c/$d; => $e/$f\n";
} }

##</code><code>##

1/1 < 3.14159265358979 < 1/0; => 2/1
2/1 < 3.14159265358979 < 1/0; => 3/1
3/1 < 3.14159265358979 < 1/0; => 4/1
3/1 < 3.14159265358979 < 4/1; => 7/2
3/1 < 3.14159265358979 < 7/2; => 10/3
3/1 < 3.14159265358979 < 10/3; => 13/4
3/1 < 3.14159265358979 < 13/4; => 16/5
3/1 < 3.14159265358979 < 16/5; => 19/6
3/1 < 3.14159265358979 < 19/6; => 22/7
3/1 < 3.14159265358979 < 22/7; => 25/8
25/8 < 3.14159265358979 < 22/7; => 47/15
47/15 < 3.14159265358979 < 22/7; => 69/22
69/22 < 3.14159265358979 < 22/7; => 91/29
91/29 < 3.14159265358979 < 22/7; => 113/36
113/36 < 3.14159265358979 < 22/7; => 135/43
135/43 < 3.14159265358979 < 22/7; => 157/50
157/50 < 3.14159265358979 < 22/7; => 179/57
179/57 < 3.14159265358979 < 22/7; => 201/64
201/64 < 3.14159265358979 < 22/7; => 223/71
223/71 < 3.14159265358979 < 22/7; => 245/78
245/78 < 3.14159265358979 < 22/7; => 267/85
267/85 < 3.14159265358979 < 22/7; => 289/92
289/92 < 3.14159265358979 < 22/7; => 311/99
311/99 < 3.14159265358979 < 22/7; => 333/106
333/106 < 3.14159265358979 < 22/7; => 355/113
333/106 < 3.14159265358979 < 355/113; => 688/219
688/219 < 3.14159265358979 < 355/113; => 1043/332
1043/332 < 3.14159265358979 < 355/113; => 1398/445
1398/445 < 3.14159265358979 < 355/113; => 1753/558
1753/558 < 3.14159265358979 < 355/113; => 2108/671
2108/671 < 3.14159265358979 < 355/113; => 2463/784
2463/784 < 3.14159265358979 < 355/113; => 2818/897
2818/897 < 3.14159265358979 < 355/113; => 3173/1010