Unexpected behaviour of math.floor, round and int functions (rounding)

[Avi Gross]

Sorry Chris,

I was talking mathematically where a number like pi or like 1/7 conceptually
have an infinite number of digits needed that are added to a growing sum
using ever smaller powers of 10, in the decimal case.

In programming, and in the binary storage, the number of such is clearly
limited.

Is there any official limit on the maximum size of a python integer other
than available memory?

And replying sparsely, yes, pretty much nothing can be represent completely
in base e other than integral multiples of e, perhaps. No other numbers,
especially integers,  can be linear combinations of e or e raised to an
integral power.

Having said that, if you throw in another transcendental called pi and
expand to include the complex number i, then you can weirdly combine them
another way to make -1. I am sure you have seen equations like:

e**(pi*i) +1 = 0

By extension, you can make any integer by adding multiple such entities
together.

On another point, an indefinitely continued repeated fraction is sort of
similar to indefinitely summed series. Both can exist and demonstrate a
regularity when the actual digits of the number seemingly show no patterns.



-----Original Message-----
From: Python-list <python-list-bounces+avigross=verizon.net at python.org> On
Behalf Of Chris Angelico
Sent: Saturday, November 20, 2021 8:03 PM
To: python-list at python.org
Subject: Re: Unexpected behaviour of math.floor, round and int functions
(rounding)

On Sun, Nov 21, 2021 at 11:39 AM Avi Gross via Python-list
<python-list at python.org> wrote:
>
> Can I suggest a way to look at it, Grant?
>
> In base 10, we represent all numbers as the (possibly infinite) sum of 
> ten raised to some integral power.

Not infinite. If you allow an infinite sequence of digits, you create
numerous paradoxes, not to mention the need for infinite storage.

> 123 is 3 times 1 (ten to the zero power) plus
> 2 times 10 (ten to the one power) plus
> 1 times 100 (ten to the two power)
>
> 123.456 just extends this with
> 4 times 1/10 (ten to the minus one power) plus
> 5 times 1/100 (10**-2) plus
> 6 time 1/1000 (10**-3)
>
> In binary, all the powers are not powers of 10 but powers of two.
>
> So IF you wrote something like 111 it means 1 times 1 plus 1 times 2 
> plus 1 times 4 or 7. A zero anywhere just skips a 2 to that power. If 
> you added a decimal point to make 111.111 the latter part would be 1/2 
> plus 1/4 plus 1/8 or 7/8 which combined might be 7 and 7/8. So any 
> fractions of the form something over 2**N can be made easily and 
> almost everything else cannot be made in finite stretches. How would you
make 2/3 or 3 /10?

Right, this is exactly how place value works.

> But the opposite is something true. In decimal, to make the above it 
> becomes
> 7.875 and to make other fractions of the kind, you need more and more 
> As it happens, all such base-2 compatible streams can be made because 
> each is in some sense a divide by two.
>
> 7/16 = 1/2 * .875 = .4375
> 7/32 = 1/2 * .4375 = .21875
>
> and so on. But this ability is a special case artifact caused by a 
> terminal digit 5 always being able to be divided in tow to make a 25 a 
> unit longer and then again and again. Note 2 and 5 are factors of 10.  
> In the more general case, this fails. In base 7, 3/7 is written easily 
> as 0.3 but the same fraction in decimal is a repeating copy of 
> .428571... which never terminates. A number like 3/7 + 4/49 + 5/343 
> generally cannot be written in base 7 but the opposite is also true 
> that only a approximation of numbers in base 2 or base 10 can ever be 
> written. I am, of course, talking about the part to the right of the 
> decimal. Integers to the left can be written in any base. It is fractional
parts that can end up being nonrepeating.

If you have a number with a finite binary representation, you can guarantee
that it can be represented finitely in decimal too.
Infinitely repeating expansions come from denominators that are coprime with
the numeric base.

> What about pi and e and the square root of 2? I suspect all of them 
> have an infinite sequence with no real repetition (over long enough 
> stretches) in any base! I mean an integer base, of course. The 
> constant e in base e is just 1.

More than "suspect". This has been proven. That's what transcendental means.

I don't think "base e" means the same thing that "base ten" does.
(Normally you'd talk about a base e *logarithm*, which is a completely
different concept.) But if you try to work with a transcendental base like
that, it would be impossible to represent any integer finitely.

(Side point: There are other representations that have different
implications about what repeats and what doesn't. For instance, the decimal
expansion for a square root doesn't repeat, but the continued fraction for
the same square root will. For instance, 7**0.5 is 2;1,1,1,4,1,1,1,4... with
an infinitely repeating four-element unit.)

> I think there have been attempts to use a decimal representation in 
> some accounting packages or database applications that allow any 
> decimal numbers to be faithfully represented and used in calculations. 
> Generally this is not a very efficient process but it can handle 0.3 
> albeit still have no way to deal with transcendental numbers.

Fixed point has been around for a long time (the simplest example being
"work in cents and use integers"), but actual decimal floating-point is
quite unusual. Some databases support it, and REXX used that as its only
numeric form, but it's not hugely popular.

> Let me leave you with Egyptian mathematics. Their use of fractions, 
> WAY BACK WHEN, only had the concept of a reciprocal of an integer. As 
> in for any integer N, there was a fraction of 1/N. They had a concept 
> of 1/3 but not of
> 2/3 or 4/9.
>
> So they added reciprocals to make any more complex fractions. To make 
> 2/3 they added 1/2 plus 1/6 for example.
>
> Since they were not stuck with any one base, all kinds of such 
> combined fractions could be done but of course the square root of 2 or 
> pi were a bit beyond them and for similar reasons.
>
> https://en.wikipedia.org/wiki/Egyptian_fraction

It's interesting as a curiosity, but it makes arithmetic extremely
difficult.

> My point is there are many ways humans can choose to play with numbers 
> and not all of them can easily do the same thing. Roman Numerals were 
> (and
> remain) a horror to do much mathematics with and especially when they 
> play games based on whether a symbol like X is to the left or right of 
> another like C as XC is 90 and CX is 110.

Of course, there are myriad ways to do things. And each one has
implications. Which makes it even more surprising that, when someone sits
down at a computer and asks it to do arithmetic, they can't handle the
different implications.

ChrisA
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