[Python-ideas] Python Numbers as Human Concept Decimal System

[Steven D'Aprano]

On Thu, Mar 06, 2014 at 04:17:55PM -0800, Mark H. Harris wrote:

> I just had one more thought along this line;  consider this:
> 
> >>> from pdeclib import *
> >>> s1=sqrt(2.01)
> >>> s1
> Decimal('1.41774468787578244511883132198668766452744')
> >>> s2=sqrt(d(2.01))
> >>> s2
> Decimal('1.41774468787578252029556185427085779261123')
> >>> 
> >>> s1**2
> Decimal('2.00999999999999978683717927196994423866272')
> >>> s2**2
> Decimal('2.01000000000000000000000000000000000000000')

If you skip the conversion to Decimal, you actually get the right answer 
using floats:

py> (2.01**0.5)**2
2.01

So the problem here isn't the binary float, but that Decimal 
by default has *too much precision* and consequently it ends 
up keeping digits that the user doesn't care about:

py> from decimal import Decimal as D
py> (D.from_float(2.01)**D("0.5"))**2
Decimal('2.009999999999999786837179272')

Floats have about 14 significant base-10 figures of precision (more in 
base-2), so if we tell Decimal to use the same, we should get the same 
result:

py> import decimal
py> ct = decimal.getcontext()
py> ct.prec = 14
py> (D.from_float(2.01)**D("0.5"))**2
Decimal('2.0100000000000')


Decimal is not a panacea. Both Decimal and binary floats have the same 
limitations, they just occur in different places for different numbers. 
All floating point numbers have these same issues. Fixed point numbers 
have different issues, rationals have their own issues, and symbolic 
computations have a different set of issues.

Computer maths is a leaky abstraction. No matter what you do, how you 
implement it, the abstraction leaks. Not even Mathematica can entirely 
hide the fact that it is computing rather than performing a Platonic 
ideal of mathematics.


-- 
Steven