[Tutor] Floating Confusion

[Alan Gauld]

"wormwood_3" <wormwood_3 at yahoo.com> wrote 

>>>> 1.1
> 1.1000000000000001


>  "1.1" is a float type, and apparently it cannot be represented by 
> binary floating point numbers accurately. 
> I must admit that I do not understand why this is the case. 

This isn't just a problem in binary. Consider using normal 
decimal the cases of 1/7 or 1/3. There cannot be represented
by any fixed number of decimal digits. 1/3 is 0.3333333.... to 
infinity and 1/7 is 0.142857142857..... repeating.

In binary the denominator of the fraction must be a power 
of 2 (2,4,8,16 etc)  for it to be accurately represented, otherwise 
you will get an approximation. Thus 1/10 could be approximated 
by 102/1024 = 0.09961 with a fairly small error (4/1024 = 0.00039). 
We can reduce the error still further by using bigger numbers, 
thus 6554/65536 = 0.100006 The best we can do in Python is 
use 2**64 as the denominator and the best approximation of 
2**64/10 as the numerator, which gives the result above:

>>> n= (2**64)/10
>>> n
1844674407370955161L
>>> float(n)/(2**64)
0.10000000000000001

But we can never eliminate the error entirely because no power 
of two is also a power of 10.

HTH,

-- 
Alan Gauld
Author of the Learn to Program web site
http://www.freenetpages.co.uk/hp/alan.gauld