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?
42*sqrt(2) ~= 59.39696961967
Add 35 and it still has two digits before the decimal point and so is 94.39696961 or 94.39696962 when truncated or rounded, respectively, to 10 significant digits. The 1 or the 2 breaks the apparent repetition of 969696.
When 55 is added, rather than 35, there are now 3 digits to the left of the decimal, so the last digit to the right of the decimal is lost: 114.3969696. Whoever programmed the calculator's rational recognition routine apparently thought three repetitions of a pair of digits was enough to justify the assumption that the repetition continued forever and calculated the appropriate numerator and denominator for that continuation, reduced to lowest terms.
It's also an indication that the calculator works in binary coded decimal: groups of 4 bits don't go any higher than 9 and the groups (halfbytes or nybbles) are considered as decimal digits.

Posted by Charlie
on 20150529 10:09:45 