>Hi to all,
>
>Here is the controversy:

12.345   -> 12.3
>         -> 12.4

In my opinion, the first one (12.3) is the correct 'mathematical rounding' (to 1 decimal here). However, some colleagues say the second one is correct.
>
>Please, tell me that I'm correct! ;)

Natural definition for natural round in a algebric group:

This is the standard definition
 on base 10 ( a algebric group )
 any element is a polynomial as:
x=N(k)*10^K+...+N(0)*10^0+N(-1)*10^-1+N(-2)*10^-2....
the ROUND(x,h) is
ROUND(x,h)=N(k)*10^K+...+N(0)*10^0+N(-1)*10^-1+N(-2)*10^-2....N'(-h)*10^(-h)
where 
    N'(-h)=N(-h)   if 0<=N(-h)<5  ( base/2) ( 0,1,2,3,4 ) 5 weights
    N'(-h)=N(-h)+1 if 5<=N(-h)    ( base/2) ( 5,6,7,8,9 ) 5 weights

but this is a simplified definition for b=10.
If the base is b=2*i+1 ( odd )
the definition is not therefore immediate.

I prefer this:
It is more syntetic and general ( for base b odd or even is ) definition is:
    N'(-h)=N(-h)+CARRY(x+x,-h)
where CARRY(x+x,-h) is the carry at power -h of the x+x addition.

example in base 3:
ROUND(12.112,1)=12.2
ROUND(12.111,1)=12.1

You have to observe a point:

* this is not a valid rule 
ROUND(x,h)=ROUND(ROUND(x,h+1),h)

then:

ROUND(12.345,1)=12.3<>12.4=ROUND(12.35,1)=ROUND(12.345,2),1)

This roundig rule don't minimize the computation error,
with 5 round down the average error is the same,
but with this you have the same probability
of apply or not apply the CARRY() value.

Fabio