"Why do calculators get this wrong?" (YouTube)

[EdS2]

Nice video! And a pretty interesting question... when does the calculator 'succeed' in 'simplifying' and when does it not. Two examples from your description

11^6 ÷ 13 is presented as (156158413/3600)π
11^6 ÷ 17 is not presented as (119415257/3600)π

So, we wonder, what about the second approximation is different, to fail to 'simplify'? I pick these two examples because the denominator is the same. (I wonder if 3600 is an attractive denominator: the calculator goes off to seek a pi-based simplification because there are 360 degrees in a circle, so this might be an angle.)

In the first case, the LHS is
136273.923076923076...
and the RHS is
136273.923076955795...
but no doubt the calculator will have calculated some specific number of digits.

In the second case, the LHS is
104209.470588235294...
and the RHS is
104209.470588260314...

Looks like the calculator displays 10 digits (when it is so inclined) so is probably calculating to at least 12, although it might have the last digit wrong, and might do some rounding off to fewer digits than it calculates (but more than it displays.)

Having said which, mostly likely it will first divide by pi and then seek a rational approximation, so these are not the right numbers to compare.

And having said that, I see a commenter has found that the denominator needs to be a factor of 25200 for the search to work.

We should, then, calculate
11^6 / (13π) and compare to (156158413/3600)
11^6 / (17π) and compare to (119415257/3600)
probably to 12 or 14 digits.

So, first case is to compare
43377.3369444340
43377.3369444444
and the second case
33170.9047222142
33170.9047222222

which is still kind of interesting because the discrepancy falls the same way both times.

And yet: really we need to do the whole chained calculation in the way the Casio does, to the right number of digits, with the right error, and the right rounding. Will need one of those Casios to find out what that is!