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 20160611 12:22:47 