Working with the set of real numbers (was: Finding size of Variable)

Albert van der Horst albert at spenarnc.xs4all.nl
Wed Mar 5 03:15:14 CET 2014


In article <mailman.7687.1393902132.18130.python-list at python.org>,
Chris Angelico  <rosuav at gmail.com> wrote:
>On Tue, Mar 4, 2014 at 1:45 PM, Albert van der Horst
><albert at spenarnc.xs4all.nl> wrote:
>>>No, the Python built-in float type works with a subset of real numbers:
>>
>> To be more precise: a subset of the rational numbers, those with a denominator
>> that is a power of two.
>
>And no more than N bits (53 in a 64-bit float) in the numerator, and
>the denominator between the limits of the exponent. (Unless it's
>subnormal. That adds another set of small numbers.) It's a pretty
>tight set of restrictions, and yet good enough for so many purposes.
>
>But it's a far cry from "all real numbers". Even allowing for
>continued fractions adds only some more; I don't think you can
>represent surds that way.

Adding cf's adds all computable numbers in infinite precision.
However that is not even a drop in the ocean, as the computable
numbers have measure zero.
A cf object yielding its coefficients amounts to a program that generates
an infinite amount of data (in infinite time), so it is not
very surprising it can represent any computable number.

Pretty humbling really.

>
>ChrisA

Groetjes Albert
-- 
Albert van der Horst, UTRECHT,THE NETHERLANDS
Economic growth -- being exponential -- ultimately falters.
albert at spe&ar&c.xs4all.nl &=n http://home.hccnet.nl/a.w.m.van.der.horst




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