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Re: (- 2.8 1.6) ; 1.1999999999999997

From: Alexandre François Garreau
Subject: Re: (- 2.8 1.6) ; 1.1999999999999997
Date: Tue, 11 Feb 2020 02:50:42 +0100

Le mardi 11 février 2020, 02:31:56 CET Stefan Monnier a écrit :
> >     (- 2.8 1.6) ; 1.1999999999999997
> Emacs Lisp, like most common programming language doesn't support
> rational numbers (and even less real numbers, which can't be faithfully
> represented in a computer) but only floating point numbers.  And like
> most programming languages it uses the standard IEEE representation
> supported natively by most CPUs.  This representation can't represent
> 2.8 nor 1.6 (and nor 1.2) exactly.
> It so happens that "the representable number closest to 2.8" minus "the
> representable number closest to 1.6" is not equal to "the representable
> number closest to 1.2".
> > With the TI-84 Plus,
> > 
> >     2.8-1.6=1.2
> IIUC those calculators used a decimal floating point representation,
> which suffers from the same kinds of problems of course, but works
> better in this specific case.

To add precision: IEEE (hence most computers) store floating point number 
in base 2, so it’s like 2.8 = (2.5 = #b10.1) + (0.25 = #b0.01) - (0.05 = 
…)… etc. while 1.6 = (1.5 = #b1.1) + (0.1 = ~#b0.0001…) etc. and 1.2 = 
(1.25 = #b1.05) - 0.05, etc.

Base 2 is compatible with any base which is a power of 2 (2, 4, 16, etc.) 
without rounding errors.  We commonly day-to-day use base 10, which is 
2×5.  So some numbers convert smoothly without rounding error (“a half” 
translates in 0.5 in base 10, and 0.1 in base 2), other won’t ever: “a 
fivth” is tailor-made to be easily expressed in base 10: 0.2, but it 
*can’t* be expressed in base two (without rational numbers (eg. 

A good analogy to understand: in base 10, we can’t represent “a third” in 
decimal notation (it gives some rounding like “0.33333333…” etc.), neither 
it can in base 2 (because 3 is not a divisor of 2 nor of 10), but it *can* 
in base 12! so in base twelve 1/3 would simply be 1/4 (exactely like 4 
hours (or pennies or forgot how it worked) is already a third of a day).

So each base can turns out to be pretty arbitrary and incompatible with 
some others.  Some are more useful (base 12 have a lots of divisors), 
others more common (base 10 since arabic numbers became widely used)… and 
in case of computers, base 2 is ubiquitous.  You can’t make electronics 
efficiently with a base higher than 2 (afair CCCP made a base 3 computer 
once but it was more complex and more expensive).  So everything, even 
TI84+ is base 2.  The fact they store and treat base10 floating numbers 
helps for education, because you won’t see rounding errors and won’t have 
to learn about it.  But with real efficient computers, that would be a waste 
of both space and time.  Space because to store a base 10 digit, in base 
2, you anyway need 4 bits, that could store 16 different values (so you 
waste 6 of them who won’t ever be used), slower because algorithm that 
would naturally apply to base 2 won’t apply and you will have to do other 
more complicated stuff (taking into account there are 6 possibilities every 
4 bits that you are wasting and not taking into account).

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