Reading the documentation

[Rustom Mody]

On Friday, August 25, 2017 at 9:58:15 AM UTC+5:30, Chris Angelico wrote:
> On Fri, Aug 25, 2017 at 1:23 PM, Rustom Mody wrote:
> > Early in my python classes I show this:
> >
> > $ python
> > Python 2.7.13 (default, Jan 19 2017, 14:48:08)
> > [GCC 6.3.0 20170118] on linux2
> > Type "help", "copyright", "credits" or "license" for more information.
> >>>> .1 + .1 == .2
> > True
> >>>> .1 + .1 + .1 == .3
> > False
> >>>>
> > --
> > https://mail.python.org/mailman/listinfo/python-list
> 
> Aside from the fact that I show it with Python 3 (or sometimes
> Node.js) instead, I've demonstrated something similar - but then go on
> to make the point that 0.1 is not "one tenth". The problem isn't with
> addition, which is completely correct here; the problem is with the
> assumption that writing "0.1" in your source code indicates the number
> "one tenth". If you want the details, Python is very helpful here:
> 
> >>> "%d/%d" % (0.1).as_integer_ratio()
> '3602879701896397/36028797018963968'
> >>> "%d/%d" % (0.2).as_integer_ratio()
> '3602879701896397/18014398509481984'
> >>> "%d/%d" % (0.3).as_integer_ratio()
> '5404319552844595/18014398509481984'
> 
> Or in hex:
> 
> >>> "%x/%x" % (0.1).as_integer_ratio()
> 'ccccccccccccd/80000000000000'
> >>> "%x/%x" % (0.2).as_integer_ratio()
> 'ccccccccccccd/40000000000000'
> >>> "%x/%x" % (0.3).as_integer_ratio()
> '13333333333333/40000000000000'
> 

Thanks for the hex tip… Useful!
> 
> Clearly the second one is exactly double the first. And equally
> clearly, the first two have been rounded up, while the second is
> rounded down. But you don't need that much detail to understand what's
> going on; most people can follow this analogy:
> 
> 0.6667 + 0.6667 + 0.6667 != 2.000
> 
> since, in grade school, most of us learned that the decimal expansion
> for two-thirds gets rounded up. It's the same thing, just in binary.
> 
> In fact, the ONLY way to create this confusion is to use (some
> derivative of) one fifth, which is a factor of base 10 but not of base
> 2. Any other fraction will either terminate in both bases (eg "0.125"
> in decimal or "0.001" in binary), or repeat in both (any denominator
> with any other prime number in it). No other rational numbers can
> produce this apparently-irrational behaviour, pun intended.

You almost make that sound like a rare exception <wink>