If I enter the expression

35+42√2

into my graphing calculator and then ask it for the fractional (rational) equivalent it cannot do so. Since the number is irrational it displays the first few decimal digits:

94.39696962

However, if I enter the expression

55+42√2

into the same graphing calculator and then ask it for the fractional equivalent it displays:

37751
-----
 330 

But this second number I entered is clearly not rational either.

What's going on?

The problem Isosceles Leg Length
http://perplexus.info/show.php?pid=10183
uses the very similar number 85+42√2
which my calculator thinks is 47651/330

What is it about 42√2?
42√2≈59.39696962≈59.396969696...=19601/330
which implies the excellent approximation
√2 ≈ 19601/13860
1.414213562 ≈ 1.414213564

The fraction is actually one of the convergents to √2
see
http://oeis.org/A001333
http://oeis.org/A000129

Posted by Jer on 2016-06-11 12:22:47