Guillaume Evrard

Yesterday i was teaching Max to my students, and there are things i couldn't

1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000

2) in Max, a squared integer can be negative, and 65536*65536=0 ?!

i really need to understand what's going on... thanks !

Yesterday i was teaching Max to my students, and there are things i couldn't
explain to them :

1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000
2) in Max, a squared integer can be negative, and 65536*65536=0 ?!

yesterdayyy-all-my-trouble-seem-so-far-awayyyyy

> Yesterday i was teaching Max to my students, and there are things i 

> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000

decimal numbers are never exact in binary form, the same as 1/3 is never 

exact in decimal system. 4.2 is the offending number.

> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!

> i really need to understand what's going on... thanks !

> ------------------------------------------------------------------------

G. wrote:
> ... not really !
>
> Hi list !
>
> Yesterday i was teaching Max to my students, and there are things i 
> couldn't explain to them :
>
> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000
decimal numbers are never exact in binary form, the same as 1/3 is never 
exact in decimal system. 4.2 is the offending number.
> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!
>
> i really need to understand what's going on... thanks !
>
> g
>
> here is the patch :
>
>
>
> ------------------------------------------------------------------------
>
>


i can believe it... how max couldn't compute the exact result of 10.*4.2 ???

moreover, 10.*4.21 = 42.1000 ----- print : 42.100000

>> Yesterday i was teaching Max to my students, and there are things i

>> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000

> decimal numbers are never exact in binary form, the same as 1/3 is never

> exact in decimal system. 4.2 is the offending number.

>> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!

>> i really need to understand what's going on... thanks !

>> ------------------------------------------------------------------------

i can believe it... how max couldn't compute the exact result of 10.*4.2 ???
moreover, 10.*4.21 = 42.1000 ----- print : 42.100000
is 4.21 more exact than 4.2 ?

> G. wrote:
>
>> ... not really !
>>
>> Hi list !
>>
>> Yesterday i was teaching Max to my students, and there are things i
>> couldn't explain to them :
>>
>> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000
>>
> decimal numbers are never exact in binary form, the same as 1/3 is never
> exact in decimal system. 4.2 is the offending number.
>
>> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!
>>
>> i really need to understand what's going on... thanks !
>>
>> g
>>
>> here is the patch :
>>
>>
>>
>> ------------------------------------------------------------------------
>>
>>
>>
>
>

> i can believe it... how max couldn't compute the exact result of 10.*4.2

> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000

>>> Yesterday i was teaching Max to my students, and there are things i

>>> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000

>> decimal numbers are never exact in binary form, the same as 1/3 is never

>> exact in decimal system. 4.2 is the offending number.

>>> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!

>>> i really need to understand what's going on... thanks !

>>> ------------------------------------------------------------------------

> i can believe it... how max couldn't compute the exact result of 10.*4.2
> ???
> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000
> is 4.21 more exact than 4.2 ?
>
>
> 2008/6/7 atir ajnopse :
>
> G. wrote:
>>
>>> ... not really !
>>>
>>> Hi list !
>>>
>>> Yesterday i was teaching Max to my students, and there are things i
>>> couldn't explain to them :
>>>
>>> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000
>>>
>> decimal numbers are never exact in binary form, the same as 1/3 is never
>> exact in decimal system. 4.2 is the offending number.
>>
>>> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!
>>>
>>> i really need to understand what's going on... thanks !
>>>
>>> g
>>>
>>> here is the patch :
>>>
>>>
>>>
>>> ------------------------------------------------------------------------
>>>
>>>
>>>
>>
>>
>
>

Whether or not a rational number has a terminating expansion depends on 

the base. For example, in base-10 the number 1/2 has a terminating 

expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only 

rationals with denominators that are powers of 2 (such as 1/2 or 3/16) 

are terminating. Any rational with a denominator that has a prime factor 

other than 2 will have an infinite binary expansion. This means that 

numbers which appear to be short and exact when written in decimal 

format may need to be approximated when converted to binary 

floating-point. For example, the decimal number 0.1 is not representable 

in binary floating-point of any finite precision; the exact binary 

representation would have a "1100" sequence continuing endlessly:

/e/ = -4; /s/ = 1100110011001100110011001100110011...,

where, as previously, /s/ is the significand and /e/ is the exponent.

/e/ = -4; /s/ = 110011001100110011001101,

which is actually 0.100000001490116119384765625 in decimal.

> i can believe it... how max couldn't compute the exact result of 

> Yesterday i was teaching Max to my students, and there are

> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of

> decimal numbers are never exact in binary form, the same as 1/3 is

> never exact in decimal system. 4.2 is the offending number.

Whether or not a rational number has a terminating expansion depends on 
the base. For example, in base-10 the number 1/2 has a terminating 
expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only 
rationals with denominators that are powers of 2 (such as 1/2 or 3/16) 
are terminating. Any rational with a denominator that has a prime factor 
other than 2 will have an infinite binary expansion. This means that 
numbers which appear to be short and exact when written in decimal 
format may need to be approximated when converted to binary 
floating-point. For example, the decimal number 0.1 is not representable 
in binary floating-point of any finite precision; the exact binary 
representation would have a "1100" sequence continuing endlessly:

    /e/ = -4; /s/ = 1100110011001100110011001100110011...,

    /e/ = -4; /s/ = 110011001100110011001101,

G. wrote:
> i can believe it... how max couldn't compute the exact result of 
> 10.*4.2 ???
> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000
> is 4.21 more exact than 4.2 ?
>
>
> 2008/6/7 atir ajnopse >:
>
>     G. wrote:
>
>         ... not really !
>
>         Hi list !
>
>         Yesterday i was teaching Max to my students, and there are
>         things i couldn't explain to them :
>
>         1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of
>         44.000000
>
>     decimal numbers are never exact in binary form, the same as 1/3 is
>     never exact in decimal system. 4.2 is the offending number.
>
>         2) in Max, a squared integer can be negative, and 65536*65536=0 ?!
>
>         i really need to understand what's going on... thanks !
>
>         g
>
>         here is the patch :
>
>
>
>         ------------------------------------------------------------------------
>
>         maxmsp@cycling74.com 
>
>
>     maxmsp@cycling74.com 
>
>
> ------------------------------------------------------------------------
>
>


ok, i'm really unhappy with that... so i just can't make thing like that :

and what about the second part of the patch i sent in the first message? how

> Whether or not a rational number has a terminating expansion depends on the

> base. For example, in base-10 the number 1/2 has a terminating expansion

> (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals

> with denominators that are powers of 2 (such as 1/2 or 3/16) are

> terminating. Any rational with a denominator that has a prime factor other

> than 2 will have an infinite binary expansion. This means that numbers which

> appear to be short and exact when written in decimal format may need to be

> approximated when converted to binary floating-point. For example, the

> decimal number 0.1 is not representable in binary floating-point of any

> finite precision; the exact binary representation would have a "1100"

> /e/ = -4; /s/ = 1100110011001100110011001100110011...,

> where, as previously, /s/ is the significand and /e/ is the exponent.

> /e/ = -4; /s/ = 110011001100110011001101,

> which is actually 0.100000001490116119384765625 in decimal.

>> i can believe it... how max couldn't compute the exact result of 10.*4.2

>> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000

>> Yesterday i was teaching Max to my students, and there are

>> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of

>> decimal numbers are never exact in binary form, the same as 1/3 is

ok, i'm really unhappy with that... so i just can't make thing like  that :

and what about the second part of the patch i sent in the first message? how
do you explain that?

> from wikipedia:
>
> Whether or not a rational number has a terminating expansion depends on the
> base. For example, in base-10 the number 1/2 has a terminating expansion
> (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals
> with denominators that are powers of 2 (such as 1/2 or 3/16) are
> terminating. Any rational with a denominator that has a prime factor other
> than 2 will have an infinite binary expansion. This means that numbers which
> appear to be short and exact when written in decimal format may need to be
> approximated when converted to binary floating-point. For example, the
> decimal number 0.1 is not representable in binary floating-point of any
> finite precision; the exact binary representation would have a "1100"
> sequence continuing endlessly:
>
>   /e/ = -4; /s/ = 1100110011001100110011001100110011...,
>
> where, as previously, /s/ is the significand and /e/ is the exponent.
>
> When rounded to 24 bits this becomes
>
>   /e/ = -4; /s/ = 110011001100110011001101,
>
> which is actually 0.100000001490116119384765625 in decimal.
>
>
>
> G. wrote:
>
>> i can believe it... how max couldn't compute the exact result of 10.*4.2
>> ???
>> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000
>> is 4.21 more exact than 4.2 ?
>>
>>
>> 2008/6/7 atir ajnopse >:
>>
>>
>>    G. wrote:
>>
>>        ... not really !
>>
>>        Hi list !
>>
>>        Yesterday i was teaching Max to my students, and there are
>>        things i couldn't explain to them :
>>
>>        1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of
>>        44.000000
>>
>>    decimal numbers are never exact in binary form, the same as 1/3 is
>>    never exact in decimal system. 4.2 is the offending number.
>>
>>        2) in Max, a squared integer can be negative, and 65536*65536=0 ?!
>>
>>        i really need to understand what's going on... thanks !
>>
>>        g
>>
>>        here is the patch :
>>
>>
>>
>>
>>  ------------------------------------------------------------------------
>>
>>        maxmsp@cycling74.com 
>>
>>
>>    maxmsp@cycling74.com 
>>
>>
>> ------------------------------------------------------------------------
>>
>>
>>
>
>

Quote: G. wrote on Sat, 07 June 2008 10:43

----------------------------------------------------

> Yesterday i was teaching Max to my students, and there are things i couldn't

this behavior is because of "rounding errors". look it up. it is normal behavior in any system using floating point calculations.

any integer getting too big will wrap around. the range of numbers that can be represented in 32 bit is -2,147,483,648 to +2,147,483,647.

Search the archives on this (rounding error, limit of 32 bit system et cetera) . It has been asked many times, and Peter Castine and others have explained it ad nauseam...

Quote: G. wrote on Sat, 07 June 2008 10:43
----------------------------------------------------
> ... not really !
>
> Hi list !
>
> Yesterday i was teaching Max to my students, and there are things i couldn't
> explain to them :
>
> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000

therefore:
2,147,483,647 + 1 = -2,147,483,648
2,147,483,647 + 2 = -2,147,483,647
2,147,483,647 + 3 = -2,147,483,646
et cetera.

http://babbage.cs.qc.edu/IEEE-754/Decimal.html

You might want to use expr for better precision, JavaScript, Java or C.

> i can believe it... how max couldn't compute the exact result of  
> 10.*4.2 ???
> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000
> is 4.21 more exact than 4.2 ?

Ogg: Re: [maxmsp] Yesterdayyy, all my trouble seem so far 

expansion (0.5) while the number 1/3 does 

floating-point. For example, the decimal number 

actually 0.100000001490116119384765625 in decimal.

believe it... how max couldn't compute the exact result of 

> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 

------------------------------------------------------------------------

Hi thanks for reply,
I try chhecking and changing the cable
I tryed 
once yesterday, but probably
was too anxious to check it clearly
lt you 
know in a few days
thanks a lot
Umberto

----Messaggio originale----
Da: atir.ajnopse@gmx.de
Data: 7-giu-2008 11.46
A:
com>
Ogg: Re: [maxmsp] Yesterdayyy, all my trouble seem so far 
awayyyyy...

Whether or not a rational number has a 
terminating expansion depends on 
the base. For example, in base-10 the 
number 1/2 has a terminating 
expansion (0.5) while the number 1/3 does 
not (0.333...). In base-2 only 
rationals with denominators that are 
powers of 2 (such as 1/2 or 3/16) 
are terminating. Any rational with a 
denominator that has a prime factor 
other than 2 will have an infinite 
binary expansion. This means that 
numbers which appear to be short and 
exact when written in decimal 
format may need to be approximated when 
converted to binary 
floating-point. For example, the decimal number 
0.1 is not representable 
in binary floating-point of any finite 
precision; the exact binary 
representation would have a "1100" 
sequence continuing endlessly:

    /e/ = -4; /s/ = 
1100110011001100110011001100110011...,

where, as previously, /s/ is 
the significand and /e/ is the exponent.

which is 
actually 0.100000001490116119384765625 in decimal.

G. wrote:
> i can 
believe it... how max couldn't compute the exact result of 
> 10.*4.2 
???
> moreover, 10.*4.21 = 42.1000 ----- print : 42.100000
> is 4.21 
more exact than 4.2 ?
>
>
> 2008/6/7 atir ajnopse
>:
>
>     G. wrote:
>
>         ... not 
really !
>
>         Hi list !
>
>         Yesterday i was teaching Max 
to my students, and there are
>         things i couldn't explain to 
them :
>
>         1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 
instead of
>         44.000000
>
>     decimal numbers are never exact 
in binary form, the same as 1/3 is
>     never exact in decimal system. 
4.2 is the offending number.
>
>         2) in Max, a squared integer 
can be negative, and 65536*65536=0 ?!
>
>         i really need to 
understand what's going on... thanks !
>
>         g
>
>         here 
is the patch :
>
>
>
>
------------------------------------------------------------------------
>
>
>         maxmsp@cycling74.com
maxmsp@cycling74.com>
>         http://www.cycling74.
com/mailman/listinfo/maxmsp
>
>
>
>     maxmsp mailing 
list
>     maxmsp@cycling74.com 
>
>
>
>
------------------------------------------------------------------------
>
> maxmsp mailing 
list
> http://www.cycling74.
com/mailman/listinfo/maxmsp
>

"the range of numbers that can be represented in 32 bit is -2,147,483,648 to +2,147,483,647."

My CPU core and operating system works at 64bit. And is not a "brand new gamers ultimate XPTO" cpu... it has about 5 years.

So, why Cycling, (and other software companies) still making software in 32? it's like using spectrum basic software on 486 machine... or 486 software on current 32bit machines!

By the way, and if I'm not totally wrong, why Max can't use recent GPU to do "better" maths, once they have better floating points!?!?! note: IF I'M NOT WRONG.... don't lol on me!

Quote: ..::RFC::.. wrote on Sun, 08 June 2008 02:55

> My CPU core and operating system works at 64bit. And is not a "brand new gamers ultimate XPTO" cpu... it has about 5 years.

Communication of data in Max is based around a data structure called an Atom. An Atom has exactly 32 bits reserved for data. No more. No less. It also has space reserved for specifying the type of data (integer, float, pointer to symbol, pointer to object, tokens such as comma/dollar sign/etc.)

This fundamental data structure has been central to Max since its inception twenty years ago. It is not likely to change in a hurry. The fact that a CPU may have registers of a different size is actually pretty irrelevant. We've had 64 and 80 bit floating point integers in many processors far older than your beloved 80486.

I could imagine that the Atom data structure might be expanded some day, but frankly don't see a lot of benefit at this stage. Sure, 64 bit ints would buy you pow(10,10) for integers, but pow(10,22) will exhibit the same problem. You still need to explain that digital integers are limited to a finite range and that floats have finite precision.

Absolutely no surprises here. And you missed a great opportunity to explain the limitations of digital arithmetic to your students. Shame.

You should maybe take a look at Peter Elsea's tutorials.

Quote: ..::RFC::.. wrote on Sun, 08 June 2008 02:55
----------------------------------------------------
> My CPU core and operating system works at 64bit. And is not a "brand new gamers ultimate XPTO" cpu... it has about 5 years.
----------------------------------------------------

I could imagine that the Atom data structure might be expanded some day, but frankly don't see a lot of benefit at this stage. Sure, 64 bit ints would buy you pow(10,10) for integers, but pow(10,22) will exhibit the same problem. You still need to explain that digital integers are limited to a finite range and that floats have finite precision. 

OP:
> Yesterday i was teaching Max to my students, and there are things i couldn't
> explain to them :

> 1) 10. * 4.2 + 2. = 44. ---- print : 43.9999998 instead of 44.000000
> 2) in Max, a squared integer can be negative, and 65536*65536=0 ?!

You should maybe take a look at Peter Elsea's tutorials.


Thanks for the explanation. I didn't know that.

Thanks for the explanation. I didn't know that.
Guess we are always learning!

Yesterdayyy, all my trouble seem so far awayyyyy...