## Bug #2121

### mathn/rational destroys Fixnum#/, Fixnum#quo and Bignum#/, Bignum#quo

Status: | Assigned | ||
---|---|---|---|

Priority: | Normal | ||

Assignee: | Keiju Ishitsuka | ||

Category: | lib | ||

Target version: | - | ||

ruby -v: | Any 1.8.6 or higher | Backport: |

**Description**

=begin

I've known this for a while, but only now realized this is actually a terrible bug.

The mathn library replaces Fixnum#/ and Bignum#/ causing them to return a different value. When the result of a division is not an integral value, the default versions will return 0. I can think of many algorithms that would use this expectation, and most other languages will not upconvert integral numeric types to floating-point or fractional types without explicit consent by the programmer.

When requiring 'mathn', Fixnum#/ and Bignum#/ are replaced with versions that return a fractional value ('quo') causing a core math operator to return not just a different type, but *a different value*.

No core library should be allowed to modify the return value of core numeric operators, or else those operators are worthless; you can't rely on them to return a specific value *ever* since someone else could require 'mathn' or 'rational'.

Note also that 'rational' destroys Fixnum#quo and Bignum#quo. This is also a bug that should be fixed, though it is less dangerous because they're not commonly-used operators.

The following code should not fail; Fixnum#/ should never return a value of a different magnitude based on which libraries are loaded:

{{{

require 'test/unit'

class TestFixnumMath < Test::Unit::TestCase

# 0 to ensure it runs first, for illustration purposes

def test*0*without*mathn
assert*equal 0, 1/3

end

def test*with*mathn

require 'mathn'

assert_equal 0, 1/3

end

end

}}}

But it does fail:

{{{

~/projects/jruby ➔ ruby test*fixnum*math.rb

Loaded suite test*fixnum*math

Started

.F

Finished in 0.016003 seconds.

1) Failure:

test*with*mathn(TestFixnumMath) [test*fixnum*math.rb:11]:

expected but was

.

2 tests, 2 assertions, 1 failures, 0 errors

}}}

=end

### History

#### #1 Updated by Charles Nutter over 4 years ago

=begin

One correction; 'quo' is broken in that it returns 0.333333333... in the default case and "1/3" in the mathn/rational case. These values are also of different magnitude, since 1/3 cannot be represented exactly in a floating point number.

=end

#### #2 Updated by Charles Nutter over 4 years ago

=begin

On Sat, Sep 19, 2009 at 12:28 AM, Joel VanderWerf

vjoel@path.berkeley.edu wrote:

Perhaps it should be the responsibility of users of numeric operators to

#floor explicitly when that is the intent, rather than rely on the (mostly

standard, sometimes convenient, but questionable) 1/2==0 behavior. Doing so

would make it easier to adapt the code to float, rational, or other numeric

types.In your proposal, would Rational(1,3) be the preferred notation, since

1/3==0? Or would there be something else, 1//3 or ...?I've always thought of mathn as a kind of alternate ruby, not just another

core library, hence to be used with caution...

I think Brian Ford expressed what I feel best...there should always be

another method or operator. Using another operator or method is an

explicit "buy-in" by the user--rather than a potential (at some

undetermined time in the future) that everything you know about

integral division in your program changes wildly. It should not be

possible for any library to undermine the basic mathematical

expectations of my program. Doing so, or expecting the user to do

extra work to guarantee the *common case*, is a recipe for serious

failure.

- Charlie

=end

#### #3 Updated by Charles Nutter over 4 years ago

=begin

On Sat, Sep 19, 2009 at 1:01 AM, Charles Oliver Nutter

headius@headius.com wrote:

integral division in your program changes wildly. It should not be

possible for any library to undermine the basic mathematical

expectations of my program. Doing so, or expecting the user to do

extra work to guarantee thecommon case, is a recipe for serious

failure.

I will revise this slightly; it should not be possible for any *core
library* to undermine the basic mathematical expectations of my

program. There's a well-accepted assumption that third-party libraries

are not subject to the stricter requirements of a core set, and I

don't mean to say that it should not be possible to make these sorts

of intriguing changes. But it should not be standard practice for any

language runtime to violate key, core expectations of a programming

language like the results of integral division. So I understand the

utility and the intrigue, but I don't think it should be allowed in

the core libraries to go that far.

- Charlie

=end

#### #4 Updated by Brian Shirai over 4 years ago

=begin

Hi,

On Sat, Sep 19, 2009 at 1:04 AM, Charles Oliver Nutter

headius@headius.com wrote:

On Sat, Sep 19, 2009 at 12:28 AM, Joel VanderWerf

vjoel@path.berkeley.edu wrote:Perhaps it should be the responsibility of users of numeric operators to

#floor explicitly when that is the intent, rather than rely on the (mostly

standard, sometimes convenient, but questionable) 1/2==0 behavior. Doing so

would make it easier to adapt the code to float, rational, or other numeric

types.In your proposal, would Rational(1,3) be the preferred notation, since

1/3==0? Or would there be something else, 1//3 or ...?I've always thought of mathn as a kind of alternate ruby, not just another

core library, hence to be used with caution...I think Brian Ford expressed what I feel best...there should always be

another method or operator. Using another operator or method is an

explicit "buy-in" by the user--rather than a potential (at some

undetermined time in the future) that everything you know about

integral division in your program changes wildly. It should not be

possible for any library to undermine the basic mathematical

expectations of my program. Doing so, or expecting the user to do

extra work to guarantee thecommon case, is a recipe for serious

failure.

There are a number of issues combined here, but I think they generally

reduce to these:

- How do you model the abstractions that are number systems in the abstractions that are classes and methods.
Should the behavior of mathn be acceptable in the core language.

We seem to think of the basic mathematical operations +, -, *, / as

being roughly equal. But of these four, division on the integers is

distinct. The set of integers is closed under addition, subtraction,

and multiplication. Given any two integers, you can add, subtract, or

multiply them and get an integer. But the result of dividing one

integer by another is not always an integer. The integers are not

closed under division. In mathematics, whether a set is closed under

an operation is a significant property.As such, there is nothing at all questionable about defining division

on the integers to be essentially floor(real(a)/real(b)) (where

real(x) returns the (mathematical) real number corresponding to the

value x, because the integers are embedded in the reals and the reals

are closed under division). You basically have five choices:floor(real(a)/real(b))

ceil(real(a)/real(b))

round(real(a)/real(b)) where round may use floor or ceil

real(a)/real(b)

raise an exception

In computer programming, there are a number of reasons for choosing 1,

2 or 3 but basically it is because that's the only way to get the

"closest" integer (i.e. define division in a way that the integers are

closed under the operation) . Convention has selected option 1.

Numerous algorithms are implemented with the assumption that integral

division is implement as option 1. It's not right or wrong, but the

convention has certain advantages. In programming, we are typically

implementing algorithms, not just "doing math" in some approximations

of these abstractions called number systems. Any system for doing math

takes serious steps to implement the real number system in as

mathematically correct form as possible.My contention that we should always have two operators for integral

division is a compromise between the need to implement algorithms and

the desire to have concise "operator" notation for doing more

math-oriented computation. Given that programming in Ruby is more

about algorithms than it is about doing math, it's unreasonable to

expect (a/b).floor instead of a / b. At the same time, math-oriented

programs are not going to be happy with a.quo b. The reasons for

options 1 and 4 above are not mutually exclusive nor can one override

the other.The mathn library is clearly exploiting an implementation detail. Were

Ruby implemented like Smalltalk (or Rubinius), mathn would have never

been written as it is. The fact that it is even possible to load mathn

results from the fact that countless critical algorithms in MRI are in

the walled garden of C code. That's not true for your Ruby programs.

Any algorithm you implement that relies on the very reasonable

assumption of integral division will be broken by mathn.You can say, "but mathn is in the standard library, you have to

require it to use it". But that ignores the fact that requiring the

library fundamentally changes assumptions that are at the very core of

writing algorithms. Essential computer programming and mathn can never

coexist without jumping through hoops.This is the point were some grandiose scheme like selector namespaces

are suggested. But I think the simple solution of two distinct

operators handily solves the problem of the messy facts of

mathematical number systems implemented in very untheoretic (i.e.

really real) silicon.As for which symbol to select, what about '/.' for real(a)/real(b).

Cheers,

Brian

=end

#### #5 Updated by Run Paint Run Run over 4 years ago

=begin

I agree that *mathn* constitutes a problem, but due to the means it employs rather than the end it achieves. In Ruby 1.8 the expectation is that '/' performs integer division, and having such a fundamental be subverted is untenable.

However, under Ruby 1.9, where Rational is a native class, it is altogether more reasonable that, by default, the '/' method coerce to Rational (as Lisp), preferably, or Float (as Perl and Python 3) when necessary.

GvR rationalizes (pun unintentional) his decision to have Python 3 perform mathematical division by default, in contrast to Python 2.0 which behaves as Ruby sans *mathn*, as follows:

"When you write a function implementing a numeric algorithm (for example, calculating the phase of the moon) you typically expect the arguments to be specified as floating point numbers. However, since Python doesn’t have type declarations, nothing is there to stop a caller from providing you with integer arguments. In a statically typed language, like C, the compiler will coerce the arguments to floats, but Python does no such thing – the algorithm is run with integer values until the wonders of mixed-mode arithmetic produce intermediate results that are floats."

"For everything except division, integers behave the same as the corresponding floating point numbers. For example, 1+1 equals 2 just as 1.0+1.0 equals 2.0, and so on. Therefore one can easily be misled to expect that numeric algorithms will behave regardless of whether they execute with integer or floating point arguments. However, when division is involved, and the possibility exists that both operands are integers, the numeric result is silently truncated, essentially inserting a potentially large error into the computation. Although one can write defensive code that coerces all arguments to floats upon entry, this is tedious, and it doesn’t enhance the readability or maintainability of the code. Plus, it prevents the same algorithm from being used with complex arguments (although that may be highly special cases)."

http://python-history.blogspot.com/2009/03/problem-with-integer-division.html

http://research.microsoft.com/en-us/um/people/daan/download/papers/divmodnote.pdf and http://python.org/dev/peps/pep-0238/ are also relevant.

=end

#### #6 Updated by Brian Shirai over 4 years ago

=begin

Hi,

On Sun, Sep 20, 2009 at 9:19 AM, Rick DeNatale rick.denatale@gmail.com wrote:

Actually in most languages which I've encountered, and that's quite a

few. Mixed mode arithmetic has been implemented by having some kind of

rules on how to 'overload' arithmetic operators based on the

arguments, not by having different operator syntax.And those rules are usually based on doing conversions only when

necessary so as to preserve what information can be preserved given

the arguments,So, for example

integer op integer - normally produces an integer for all of the

'big four' + - * /

integer op float - normally produces a float, as does float op integerAs new numeric types are added, in languages which either include them

inherently or allow them to be added, this pattern is usually

followed.

This is a distinctly different issue. Mixed-type arithmetic in Ruby is

handled by the #coerce protocol.

As for which symbol to select, what about '/.' for real(a)/real(b).

Well, first the problem we are talking about is Rationals, not Floats,

and second, what happens as more numeric classes are introduced.

The mathn library aliases Fixnum and Bignum #quo to #/. By default

#quo returns a Float. Rational redefines #quo to produce a Rational

rather than a Float.

But what class of object is not the point. It could be Complex. The

point is that integers are not closed under division so you *must*

choose one of the options if you expect a value when dividing any two

integers.

The division operation is so fundamental that assumptions about it

should not change under your feet. Having a separate operator that

returns a different type when integral division would be undefined

allows both normal algorithms and mathy stuff to coexist nicely. In my

algorithms, I *never* want my integral division to suddenly return

something non-integer. In my math codes, I almost never want my

quotient truncated or rounded.

Cheers,

Brian

=end

#### #7 Updated by Rick DeNatale over 4 years ago

=begin

On Sun, Sep 20, 2009 at 3:51 PM, brian ford brixen@gmail.com wrote:

Hi,

On Sun, Sep 20, 2009 at 9:19 AM, Rick DeNatale rick.denatale@gmail.com wrote:

Actually in most languages which I've encountered, and that's quite a

few. Mixed mode arithmetic has been implemented by having some kind of

rules on how to 'overload' arithmetic operators based on the

arguments, not by having different operator syntax.And those rules are usually based on doing conversions only when

necessary so as to preserve what information can be preserved given

the arguments,So, for example

integer op integer - normally produces an integer for all of the

'big four' + - * /

integer op float - normally produces a float, as does float op integerAs new numeric types are added, in languages which either include them

inherently or allow them to be added, this pattern is usually

followed.This is a distinctly different issue. Mixed-type arithmetic in Ruby is

handled by the #coerce protocol.

Not sure why it's distinctly different, what happens when a new

numeric class is introduced, e.g. Rational, is what we seem to be

talking about.

And #coerce is just an implemention detail whose motivation seems to

be in line with what I'm saying.

As for which symbol to select, what about '/.' for real(a)/real(b).

Well, first the problem we are talking about is Rationals, not Floats,

and second, what happens as more numeric classes are introduced.The mathn library aliases Fixnum and Bignum #quo to #/. By default

#quo returns a Float. Rational redefines #quo to produce a Rational

rather than a Float.But what class of object is not the point. It could be Complex. The

point is that integers are not closed under division so youmust

choose one of the options if you expect a value when dividing any two

integers.

Right, and Ruby like most other languages made a choice to use integer

division, rather than converting.

Smalltalk made another choice, to return a Fraction when dividing two integers.

In both cases, the / operator is effectively overloaded, and can

return other kinds of numbers given different pairs of arguments.

The division operation is so fundamental that assumptions about it

should not change under your feet. Having a separate operator that

returns a different type when integral division would be undefined

allows both normal algorithms and mathy stuff to coexist nicely. In my

algorithms, Ineverwant my integral division to suddenly return

something non-integer. In my math codes, I almost never want my

quotient truncated or rounded.

Yes, I agree that I don't want the rules to change under my feet. I

want a / b to give me the same integer as Ruby 1.8 sans mathn gives me

when a and b are integers, and I expect 1 / 1.2 to give me the same

float etc. I'm not sure I see the need for additional operators, but

that's a side issue.

Run Paint Run suggested that 1.9 SHOULD produce a Rational or maybe a

float as the result of dividing two integers, because "that what Guido

would do."

The brutal facts are that there's is lots of code written in Ruby, and

lots of that code uses integer divide, and would be broken if this

change were made, it would be the same as silently including mathn in

every existing ruby program, which seems like a bad idea.

Guess what! I did some experimentation with irb1.9 and was pleasantly

surprised to find that 1.9 seems to be doing quite the opposite, it

acts just like the "thought experiment" proposal I suggested here.

$ irb1.9

irb(main):001:0> Rational

=> Rational

irb(main):002:0> 1/2

=> 0

irb(main):003:0> Rational(1)

=> Rational(1, 1)

irb(main):004:0> 1.to_r

=> Rational(1, 1)

Which I guess indicates that "that's what Matz would do."

--

Rick DeNatale

Blog: http://talklikeaduck.denhaven2.com/

Twitter: http://twitter.com/RickDeNatale

WWR: http://www.workingwithrails.com/person/9021-rick-denatale

LinkedIn: http://www.linkedin.com/in/rickdenatale

=end

#### #8 Updated by Run Paint Run Run over 4 years ago

=begin

Run Paint Run suggested that 1.9 SHOULD produce a Rational or maybe a

float as the result of dividing two integers, because "that what Guido

would do."

He said, of course, no such thing. I suggest that when your strawman necessitates the sensationalist mis-characterization of another's position that amateur rhetoric may not be your calling. In fact, your pastiche is not even internally consistent because GvR did not advocate rational results.

I was simply illustrating that other, similar languages have faced this issue, and so providing a justification for, and the results of, their decisions.

The brutal facts are that there's is lots of code written in Ruby, and

lots of that code uses integer divide, and would be broken if this

change were made, it would be the same as silently including mathn in

every existing ruby program, which seems like a bad idea.

The same argument can always be rallied to support inertia. As we progress toward Ruby 2.0 it behooves us to revisit our prior decisions and consider whether they remain defensible in hindsight. Further, your analogy is flawed: it would not at all "be the same as silently including mathn in every existing ruby program", because neither would such a change in language semantics be ushered in "silently", nor does *mathn* perform only this function.

Which I guess indicates that "that's what Matz would do."

The existence of an "infallible designer" would obviate this very bug tracker, as every aspect of the language could be reasoned so. It indicates what the current behavior is, nothing more.

As to the matter at hand, Brian's solution seems eminently reasonable for 1.8 at least; the desired behavior of *mathn* and '/' under 1.9 is perhaps a separate issue.

=end

#### #10 Updated by Rick DeNatale over 4 years ago

=begin

On Sun, Sep 20, 2009 at 3:29 AM, brian ford brixen@gmail.com wrote:

Hi,

On Sat, Sep 19, 2009 at 1:04 AM, Charles Oliver Nutter

headius@headius.com wrote:On Sat, Sep 19, 2009 at 12:28 AM, Joel VanderWerf

vjoel@path.berkeley.edu wrote:Perhaps it should be the responsibility of users of numeric operators to

#floor explicitly when that is the intent, rather than rely on the (mostly

standard, sometimes convenient, but questionable) 1/2==0 behavior. Doing so

would make it easier to adapt the code to float, rational, or other numeric

types.In your proposal, would Rational(1,3) be the preferred notation, since

1/3==0? Or would there be something else, 1//3 or ...?I've always thought of mathn as a kind of alternate ruby, not just another

core library, hence to be used with caution...I think Brian Ford expressed what I feel best...there should always be

another method or operator. Using another operator or method is an

explicit "buy-in" by the user--rather than a potential (at some

undetermined time in the future) that everything you know about

integral division in your program changes wildly. It should not be

possible for any library to undermine the basic mathematical

expectations of my program. Doing so, or expecting the user to do

extra work to guarantee thecommon case, is a recipe for serious

failure.There are a number of issues combined here, but I think they generally

reduce to these:

- How do you model the abstractions that are number systems in the abstractions that are classes and methods.
- Should the behavior of mathn be acceptable in the core language.
We seem to think of the basic mathematical operations +, -, *, / as

being roughly equal. But of these four, division on the integers is

distinct. The set of integers is closed under addition, subtraction,

and multiplication. Given any two integers, you can add, subtract, or

multiply them and get an integer. But the result of dividing one

integer by another is not always an integer. The integers are not

closed under division. In mathematics, whether a set is closed under

an operation is a significant property.As such, there is nothing at all questionable about defining division

on the integers to be essentially floor(real(a)/real(b)) (where

real(x) returns the (mathematical) real number corresponding to the

value x, because the integers are embedded in the reals and the reals

are closed under division). You basically have five choices:

- floor(real(a)/real(b))
- ceil(real(a)/real(b))
- round(real(a)/real(b)) where round may use floor or ceil
- real(a)/real(b)
- raise an exception
In computer programming, there are a number of reasons for choosing 1,

2 or 3 but basically it is because that's the only way to get the

"closest" integer (i.e. define division in a way that the integers are

closed under the operation) . Convention has selected option 1.

Numerous algorithms are implemented with the assumption that integral

division is implement as option 1. It's not right or wrong, but the

convention has certain advantages. In programming, we are typically

implementing algorithms, not just "doing math" in some approximations

of these abstractions called number systems. Any system for doing math

takes serious steps to implement the real number system in as

mathematically correct form as possible.My contention that we should always have two operators for integral

division is a compromise between the need to implement algorithms and

the desire to have concise "operator" notation for doing more

math-oriented computation. Given that programming in Ruby is more

about algorithms than it is about doing math, it's unreasonable to

expect (a/b).floor instead of a / b. At the same time, math-oriented

programs are not going to be happy with a.quo b. The reasons for

options 1 and 4 above are not mutually exclusive nor can one override

the other.

Actually in most languages which I've encountered, and that's quite a

few. Mixed mode arithmetic has been implemented by having some kind of

rules on how to 'overload' arithmetic operators based on the

arguments, not by having different operator syntax.

And those rules are usually based on doing conversions only when

necessary so as to preserve what information can be preserved given

the arguments,

So, for example

integer op integer - normally produces an integer for all of the

'big four' + - * /

integer op float - normally produces a float, as does float op integer

As new numeric types are added, in languages which either include them

inherently or allow them to be added, this pattern is usually

followed.

Smalltalk has the concept of generality of a number class. More

general classes can represent more numbers, albeit with some potential

for adding 'fuzziness' in the standard image Floats are the most

general, then Fractions, then equally LargePositiveIntegers and

LargeNegativeIntegers (which together serve the same role as Bignum in

Ruby), then SmallInteger (Ruby's Fixnum).

The mathn library is clearly exploiting an implementation detail. Were

Ruby implemented like Smalltalk (or Rubinius), mathn would have never

been written as it is. The fact that it is even possible to load mathn

results from the fact that countless critical algorithms in MRI are in

the walled garden of C code. That's not true for your Ruby programs.

Any algorithm you implement that relies on the very reasonable

assumption of integral division will be broken by mathn.

The problem with mathn is that it introduces new numeric types, and

also changes the behavior of the existing types, particularly integer,

so that when mathn is included

integer / integer produces a rational if the result can't be

reduced to an integer.

This is at odds with most languages and, as Charles points out, it

effectively changes the 'rules of physics' for other code which is

likely unaware that mathn has been introduced.

In Smalltalk, there is a standard Fraction class and integer division

does in fact return a Fraction rather than an Integer. But that's

known and expected by Smalltalk programmers.

You can say, "but mathn is in the standard library, you have to

require it to use it". But that ignores the fact that requiring the

library fundamentally changes assumptions that are at the very core of

writing algorithms. Essential computer programming and mathn can never

coexist without jumping through hoops.

Yes this is a problem IMHO. The difference between Ruby and Smalltalk

here is that one language starts out including Rationals/Fractions,

and the other treats them as an optional add on which, when added,

changes the rules.

This is the point were some grandiose scheme like selector namespaces

are suggested. But I think the simple solution of two distinct

operators handily solves the problem of the messy facts of

mathematical number systems implemented in very untheoretic (i.e.

really real) silicon.As for which symbol to select, what about '/.' for real(a)/real(b).

Well, first the problem we are talking about is Rationals, not Floats,

and second, what happens as more numeric classes are introduced.

Another alternative would be to change mathn (or maybe make a new

alternative mathn for compatibility for programs already using mathn)

which

1. Left 1 / 2 as producing the Integer 1 2. Allowed explicit instantiation of Rationals * Rational.new(1,2) # i.e. make the new method public. * Change Object#Rational to always return a Rational for an

integer argument, with a denominator of 1.

* Integer#to*rational which could be implemented as:
class Integer
def to*rational

Rational(self)

end

end

Then rational arithmetic could be implemented so that

5 / 3 => 1 5.to_rational / 3 => 5 / 3 5 / 3.to_rational => 5 / 3 (5.to_r / 3).to_i => 1

Which would be in-line with standard arithmetic in Ruby IMHO.

Note: it might be more ruby-like to name the coercion method to*r
instead of to*rational, but that might be confused by some as meaning

something else like to real, although I don't really thing that that

would be that much of an issue.

--

Rick DeNatale

Blog: http://talklikeaduck.denhaven2.com/

Twitter: http://twitter.com/RickDeNatale

WWR: http://www.workingwithrails.com/person/9021-rick-denatale

LinkedIn: http://www.linkedin.com/in/rickdenatale

=end

#### #12 Updated by Akira Tanaka almost 3 years ago

**Project**changed from*Ruby*to*ruby-trunk*

#### #13 Updated by Charles Nutter over 1 year ago

This is still an issue. Requiring a standard library should never change the result of a Fixnum operation. The current behavior is a terrible bug.

#### #14 Updated by Marc-Andre Lafortune over 1 year ago

**Description**updated (diff)**Category**set to*lib*

It does create some problems in real apps (e.g. https://github.com/rails/rails/pull/8222 )

The problem is compatibility. I always thought that it was the intent of `mathn`

to change the semantics of `/`

so that the same operations would have different meanings. For example `Matrix[[1]] / 2`

returns different results depending on the presence of `mathn`

or not.

Avoiding the problem is trivial: when you mean "division with truncation to integer", use `Integer#div`

instead of `Integer#/`

.

`div`

has the advantage of being crystal clear, since `foo.div(bar)`

always truncates down to an integer, while `foo / bar`

can return any of Integer, Rational, Float, BigDecimal, Complex, Matrix, Vector, etc., depending on the classes of foo, bar (and if mathn is loaded or not)

What I'm trying to say is that, given the choices made in the past, using `/`

instead of `div`

can be seen as a user mistake, although it clearly is an easy one to make. Ideally, library/framework authors would be aware of this and reserve `Integer#/`

for mathematical libraries and use `div`

where appropriate.

I don't see how we can change the current behavior, but adding a note in the doc for `Integer#/`

about this could not hurt.

#### #15 Updated by Charles Nutter over 1 year ago

I sympathize with the desire to avoid breaking backward compatibility, but the idea that "10 / 2" is the wrong way and you should instead use "10.div(2)" is pretty anti-Ruby. This is terribly surprising and certainly not what any user would expect. I would suggest that a new method be added to Integer for the variable behavior...something like #ratio or #fraction.

Another example of why this sucks, from a Ruby implementer perspective...

In JRuby, the basic math operations are recognize and optimized for better performance...all except #/ because of the mathn problem. Silly, isn't it?