Using fractions instead of floats

A

andresj

I was doing some programming in Python, and the idea came to my mind:
using fractions instead of floats when doing 2/5.

The problem arises when you try to represent some number, like 0.4 in
a float. It will tell you that it's equal to 0.40000000000000002.
"This is easy to fix", you may say. "You just use the decimal.Decimal
class!". Well, firsly, there would be an excess of typing I would need
to do to calculate 0.4+0.6:

from decimal import Decimal
print Decimal("0.4")+Decimal("0.6")

Secondly, what happens if I need to sum 1/3 and 0.4? I could use
Decimal to represent 0.4 precisely, but what about 1/3? Sure, I could
use _another_ class which works in a base (binary, decimal, octal,
hexadecimal) in which 1/3 can be represented exactly... Not to mention
the problem of operating with those two different classes...

So the solution I think is using a fraction type/class, similar to the
one found in Common Lisp. If you have used CLisp before, you only need
to type:
(+ 1/3 6/10)
to get the exact result. (Yes, I also hate the (operator arg1 arg2)
syntax, but it's just an example). I would like to have something
similar in Python, in which dividing two numbers gives you a fraction,
instead of an integer (python 2.x) or a float (decided for python
3.x).

an implementation could be like this:
class frac(object): # PS: This (object) thing will be removed in
python 3.0, right?
def __init__(self, numerator, denominator):
pass
def __add__(self, other):
pass
#...

(I have an implementation of the frac class done (this meaning, it
works for me), and although it's pretty dirty, I'd be happy to post it
here if you want it.)

My idea, in summary would be that this python shell session is true:
I would like to get some feedback on this idea. Has this been posted
before? If so, was it rejected? and for what?
Also, I would like to know if you have improvements on the initial
design, and if it would be appropiate to send it as a PEP.
 
B

Ben Finney

andresj said:
The problem arises when you try to represent some number, like 0.4 in
a float.
Click to expand...

Which is really a specific case of the general problem that, for any
given number base, some non-integer numbers cannot be exactly
represented as fractions.
Secondly, what happens if I need to sum 1/3 and 0.4? I could use
Decimal to represent 0.4 precisely, but what about 1/3?
Click to expand...

What about the sum of π (pi) and √2 (sqrt(2))?
So the solution I think is using a fraction type/class
Click to expand...

As explained above, a fractional-number class only shifts the "exact
representation" problem, it doesn't solve it.
 
A

andresj

Which is really a specific case of the general problem that, for any
given number base, some non-integer numbers cannot be exactly
represented as fractions.
Click to expand...

Yes. That's what I meant to say, by giving an example.
What about the sum of (pi) and 2 (sqrt(2))?
Click to expand...

Mm... To be honest I hadn't thought of that in the moment I wrote the
last post...
But I think then, that I'll be more specific, so that my proposal only
deals with *rational numbers*, not irrationals.
As explained above, a fractional-number class only shifts the "exact
representation" problem, it doesn't solve it.
Click to expand...

I don't understand completely what you said, but I think you are
saying that it doesn't solve the problem with irrational numbers.
Which is exactly what this proposal doesn't solve. (Mainly because I
do not know of any way of doing it in "normal" math).

(While I write this, my mind reminds me of operating with roots " 2
(sqrt(2))", which I think could be solved in another similar way,
although it has some more specific use cases... But in any case,
that's not in the scope of this proposal, as they are irrational
numbers.)
 
M

mensanator

I was doing some programming in Python, and the idea came to my mind:
using fractions instead of floats when doing 2/5.

The problem arises when you try to represent some number, like 0.4 in
a float. It will tell you that it's equal to 0.40000000000000002.
"This is easy to fix", you may say. "You just use the decimal.Decimal
class!". Well, firsly, there would be an excess of typing I would need
to do to calculate 0.4+0.6:

from decimal import Decimal
print Decimal("0.4")+Decimal("0.6")

Secondly, what happens if I need to sum 1/3 and 0.4? I could use
Decimal to represent 0.4 precisely, but what about 1/3? Sure, I could
use _another_ class which works in a base (binary, decimal, octal,
hexadecimal) in which 1/3 can be represented exactly... Not to mention
the problem of operating with those two different classes...

So the solution I think is using a fraction type/class, similar to the
one found in Common Lisp. If you have used CLisp before, you only need
to type:
(+ 1/3 6/10)
to get the exact result. (Yes, I also hate the (operator arg1 arg2)
syntax, but it's just an example). I would like to have something
similar in Python, in which dividing two numbers gives you a fraction,
instead of an integer (python 2.x) or a float (decided for python
3.x).

an implementation could be like this:
class frac(object): # PS: This (object) thing will be removed in
python 3.0, right?
def __init__(self, numerator, denominator):
pass
def __add__(self, other):
pass
#...

(I have an implementation of the frac class done (this meaning, it
works for me), and although it's pretty dirty, I'd be happy to post it
here if you want it.)

My idea, in summary would be that this python shell session is true:


I would like to get some feedback on this idea. Has this been posted
before? If so, was it rejected? and for what?
Also, I would like to know if you have improvements on the initial
design, and if it would be appropiate to send it as a PEP.
Click to expand...

The gmpy module has unlimited precision rationals.
Works pretty good, too.
 
A

Andres Riofrio

PS: Sorry, George Sakkis, for the double emailing... I forgot to add
python-list in the To: field the first time. :)

Haha. Ok. Thank you for pointing me to those links :). I hadn't
thought of searching for the word 'rational' instead of 'decimal'...
From what I've read, seems that the principal reason for rejecting the
Click to expand...
PEP is that there was not much need (enthusiasm)... Well, then I have
a question: Is there a way to make 5/2 return something other than an
integer? I can do:
class int(int):
def __add__(self, other):
pass
but that will only work if I do int(5)/int(2)...

(setting __builtin__.int=int doesn't work, either)

What I'd like is to be able to implement what I put in the proposal,
as I don't think it's a really big language change...
 
T

Terry Reedy

| a question: Is there a way to make 5/2 return something other than an
| integer?
2.5

tjr
 
G

Gabriel Genellina

En Mon, 01 Oct 2007 00:10:05 -0300, Andres Riofrio
From what I've read, seems that the principal reason for rejecting the
PEP is that there was not much need (enthusiasm)... Well, then I have
a question: Is there a way to make 5/2 return something other than an
integer? I can do:
class int(int):
def __add__(self, other):
pass
but that will only work if I do int(5)/int(2)...
Click to expand...

I'm afraid not. But if you are crazy enough you can preprocess your source
using something similar to the tokenize example
<http://docs.python.org/lib/module-tokenize.html>
 
S

Stargaming

On Sun, 30 Sep 2007 20:10:05 -0700, Andres Riofrio wrote:

[snip]
PEP is that there was not much need (enthusiasm)... Well, then I have a
question: Is there a way to make 5/2 return something other than an
integer? I can do:
class int(int):
def __add__(self, other):
pass
but that will only work if I do int(5)/int(2)...

(setting __builtin__.int=int doesn't work, either)

What I'd like is to be able to implement what I put in the proposal, as
I don't think it's a really big language change...
Click to expand...
[snip]

You could make up an example implementation by using fractionizing_int
(1) / fractionizing_int(3) (or whatever name you come up with).

I don't know of any particularly nice way to Just Let It Work anywhere in
python (perhaps some ugly byte code hacks, automatically wrapping ints).
So, you would have to implement it in C if you want to propose this for
CPython. And that's no 5 minute task, I guess.

Still, having this `int` behaviour working globally would include those
previously mentioned irrational numbers (but Pi is rational in any
visualization anyways, so FWIW that wouldn't be too much of a blockade).
But where would you put normal integer division then?
Introducing a new type seems best to me.

You could, as well, try to implement this in PyPy. If this would hit the
broad masses is another question, though. ;)

Regards,
stargaming
 
A

Arnaud Delobelle

[snip Rational numbers in Python]
I would like to get some feedback on this idea. Has this been posted
before? If so, was it rejected? and for what?
Also, I would like to know if you have improvements on the initial
design, and if it would be appropiate to send it as a PEP.
Click to expand...

As pointed out by others, implementations of rationals in Python
abound. Whereas there is a canonical representation of floats and ints
(and even longints) in the machine, it is not the case for rationals.
Moreover most programming tasks do not need rationals, so why burden
the language with them? If one needs them, there are perfectly
adequate modules to import (even though I, like many others I suspect,
have my own implementation in pure Python). Finally, arithmetic would
become very confusing if there were three distinct numeric types; it
already causes enough confusion with two!
 
N

Neil Cerutti

Finally, arithmetic would become very confusing if there were
three distinct numeric types; it already causes enough
confusion with two!
Click to expand...

Scheme says: It's not that bad.
 
A

Arnaud Delobelle

Scheme says: It's not that bad.
Click to expand...

Scheme has prefix numeric operators, so that 1/2 is unambiguously (for
the interpreter and the user) a litteral for 'the fraction 1/2'. You
can't avoid the confusion in python, as binary operators are infix. Of
course, we could create a new kind of litteral. Let's see, / and //
are already operators, so why not use /// ? ;)
 
A

andresj

Well, yeah... I get what you are saying, that would be confusing...

Terry Reedy and Laurent Pointal:
I know from __future__ import division changes the behaivour to return
floats instead of ints, but what I meant is to be able to implement a
function (or class/method) which would return what _I_ want/decide.

Gabriel Genellina, thanks for the suggestion of wrapping my code. I
think it could be one way... But I think it would be too much trouble,
I'll just go with writing fractions/rationals explicitly.

And thanks to mensanator and Gabriel for the suggestion of gmpy, and
to Nick for the example (It really helps to have an example, because
it usually takes me hours to get to what I want in the
documentations :). I think I will use that, when i get it working in
Ubuntu. Do you guys know of any pre-made package or guides for getting
it working in Ubuntu?

Well, I guess my idea was not as good as I thought it was :). But
anyways... I look forward to Python 3.0 (Specially the
__future__.with_statement and the standardization of names-- cStringIO
is just too ugly for my eyes!)
 
R

richyjsm

mpq(14,15)
Click to expand...

Golly! That's quite impressive. And more than a little bit magic as
well, since 0.6 is definitely not the same as 3/5. How on earth does
this work?

Richard
 
M

mensanator

Well, yeah... I get what you are saying, that would be confusing...

Terry Reedy and Laurent Pointal:
I know from __future__ import division changes the behaivour to return
floats instead of ints, but what I meant is to be able to implement a
function (or class/method) which would return what _I_ want/decide.

Gabriel Genellina, thanks for the suggestion of wrapping my code. I
think it could be one way... But I think it would be too much trouble,
I'll just go with writing fractions/rationals explicitly.

And thanks to mensanator and Gabriel for the suggestion of gmpy, and
to Nick for the example (It really helps to have an example,
Click to expand...

Would you like to see a more thorough example?

First, go check out the Wikipedia article:
<http://en.wikipedia.org/wiki/Collatz_conjecture>

And scroll down the the section "Iterating on rational numbers
with odd denominators". I added the section beginning with
"The complete cycle being:..." through "And this is because...".

Here's the Python program I used to develop that section.

import gmpy

def calc_pv_xyz(pv):
"""calculate Hailstone Function Parameters
using Parity Vector instead of Sequence Vector
(defined using (3n+1)/2)

calc_pv_xyz(pv)
pv: parity vector
returns HailstoneFunctionParameters (x,y,z)
"""
ONE = gmpy.mpz(1)
TWO = gmpy.mpz(2)
TWE = gmpy.mpz(3)
twee = gmpy.mpz(sum(pv))
twoo = gmpy.mpz(len(pv))
x = TWO**twoo
y = TWE**twee
z = gmpy.mpz(0)
c = gmpy.mpz(sum(pv)-1)
for i,j in enumerate(pv):
if j:
z += TWE**c * TWO**i
c -= ONE
return (x,y,z)

def iterating_on_rational(n): # Collatz for rational numbers
# is numerator odd?
if n.numer() % 2 == 1:
n = (n*3 + 1)/2
else:
n = n/2
print n,
return n

pv = [1,0,1,1,0,0,1]

for i in xrange(len(pv)+1):
print pv
# get Hailstone Function parameters X,Y,Z
xyz = calc_pv_xyz(pv)
# calculate the Crossover Point Z/(X-Y)
cp = gmpy.mpq(xyz[2],xyz[0]-xyz[1]) # as a rational number
# start of loop cycle
print cp,
# start the cycle...
n = iterating_on_rational(cp)
# ...which MUST return to the starting point
# since ALL rationals are valid Crossover Points
# not just integers
while n != cp:
n = iterating_on_rational(n)
print
print
# step through the cyclic permutations
p = pv.pop(0)
pv.append(p)

## parity vector cyclic permutations
##
## [1, 0, 1, 1, 0, 0, 1]
## 151/47 250/47 125/47 211/47 340/47 170/47 85/47 151/47
##
## [0, 1, 1, 0, 0, 1, 1]
## 250/47 125/47 211/47 340/47 170/47 85/47 151/47 250/47
##
## [1, 1, 0, 0, 1, 1, 0]
## 125/47 211/47 340/47 170/47 85/47 151/47 250/47 125/47
##
## [1, 0, 0, 1, 1, 0, 1]
## 211/47 340/47 170/47 85/47 151/47 250/47 125/47 211/47
##
## [0, 0, 1, 1, 0, 1, 1]
## 340/47 170/47 85/47 151/47 250/47 125/47 211/47 340/47
##
## [0, 1, 1, 0, 1, 1, 0]
## 170/47 85/47 151/47 250/47 125/47 211/47 340/47 170/47
##
## [1, 1, 0, 1, 1, 0, 0]
## 85/47 151/47 250/47 125/47 211/47 340/47 170/47 85/47
##
## [1, 0, 1, 1, 0, 0, 1]
## 151/47 250/47 125/47 211/47 340/47 170/47 85/47 151/47
 
M

mensanator

Golly! That's quite impressive. And more than a little bit magic as
well, since 0.6 is definitely not the same as 3/5.
Click to expand...

It's not? Since when?
3/5

How on earth does this work?
Click to expand...

The rationals are always reduced to lowest terms.
 
R

richyjsm

It's not? Since when?
Click to expand...

The 0.6 above is a floating point number, mathematically very close to
0.6 but definitely not equal to it, since 0.6 can't be represented
exactly as a float.
 

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