No, 2 times something is greater than something. Something over something is 1.

If we change the division axiom to be piecewise with an exception only for infinity, we could claim that any problem involving division of a symbol is unsolvable because the symbol could be infinity.
This is incorrect:
x / 2 is unsolvable because x could be infinity
x / 2 > x / 3 (where x > 0; Z+) is indeterminate because if x is infinity, then they are equal.

assert 1 / 0 != 2 / 0
assert 2*inf > inf
assert inf / inf == 1

I should have said capricious (not specious). I'm again replying to the main thread because this is relevant: there would need to be changes to tests in order to return (scalar times) infinity instead of ZeroDivisionError.

We should not discard the scalar in scalar*infinity expressions.

On Sun, Oct 11, 2020, 5:18 PM Chris Angelico <rosuav@gmail.com> wrote:
On Mon, Oct 12, 2020 at 8:07 AM Wes Turner <wes.turner@gmail.com> wrote:
>
> So you're arguing that the scalar is irrelevant?
> That `2*inf == inf`?
>
> I disagree because:
> ```2*inf > inf```

On what basis? If you start by assuming that infinity is a number,
then sure, you're going to deduce that double it must be a greater
number. But you're just concluding your own assumption, not proving
anything.

> And:
>
> ```# Given that:
> inf / inf = 1

Is that the case?

>>> from math import inf
>>> inf / inf
nan

> # When we solve for symbol x:
> 2*inf*x = inf
> 2*x = 1
> x = 1/2
>
> 2*inf*x = inf
> inf*x = inf
> x = 1
>
> #  I think it's specious to argue that there are infinity solutions; that axioms of symbolic mathematics do not apply because infinity
> ```

Once again, you start by assuming that infinity is a number, and that
you can divide by it (which is what happens when you "solve for x" by
removing the infinities). You can't prove something by first assuming
it.

"Infinity" isn't a number. In the IEEE 754 system, it is a value, but
it's still not a number (although it's distinct from Not A Number,
just to confuse everyone). In mathematics, it's definitely not an
actual number or value.

ChrisA
______________________________

On Sun, Oct 11, 2020 at 5:04 PM Wes Turner <wes.turner@gmail.com> wrote:
So you're arguing that the scalar is irrelevant?
That `2*inf == inf`?

I disagree because:
```2*inf > inf```

And:

```# Given that:
inf / inf = 1

# When we solve for symbol x:
2*inf*x = inf
2*x = 1
x = 1/2

2*inf*x = inf
inf*x = inf
x = 1

#  I think it's specious to argue that there are infinity solutions; that axioms of symbolic mathematics do not apply because infinity
```

This is relevant to the (now-forked) main thread if the plan is to return inf/-inf/+inf instead of raising ZeroDivisionError; so I'm replying to the main thread.

On Sun, Oct 11, 2020, 4:10 PM Chris Angelico <rosuav@gmail.com> wrote:
On Mon, Oct 12, 2020 at 5:06 AM Wes Turner <wes.turner@gmail.com> wrote:
>
> SymPy ComplexInfinity, 1/0 < 2/0, *tests* for symbolic results
>
> FWIW, SymPy (a CAS: Computer Algebra System) has Infinity, NegativeInfinity, ComplexInfinity.
>
> Regarding a symbolic result for 1/0:
>
> If 1/0 is infinity (because 0 goes into 1 infinity times),
> is 2/0 2*inifnity (because 0 goes into 2 2 times more than into 1)
>

If you try to treat "infinity" as an actual number, you're inevitably
going to run into paradoxes. Consider instead: 1/x tends towards +∞ as
x tends towards 0 (if x starts out positive), therefore we consider
that 1/0 is +∞. By that logic, the limit of 2/0 is the exact same
thing. It's still not a perfect system, and division by zero is always
going to cause problems, but it's far less paradoxical if you don't
try to treat 2/0 as different from 1/0 :)

BTW, you're technically correct, in that 2/0 would be the same as 2 *
(whatever 1/0 is), but that's because 2*x tends towards +∞ as x tends
towards +∞, meaning that 2*∞ is also ∞.

ChrisA

On Sun, Oct 11, 2020 at 2:03 PM Wes Turner <wes.turner@gmail.com> wrote:
SymPy ComplexInfinity, 1/0 < 2/0, *tests* for symbolic results

FWIW, SymPy (a CAS: Computer Algebra System) has Infinity, NegativeInfinity, ComplexInfinity.

Regarding a symbolic result for 1/0:

If 1/0 is infinity (because 0 goes into 1 infinity times),
is 2/0 2*inifnity (because 0 goes into 2 2 times more than into 1)

A proper CAS really is advisable. FWIU, different CAS have different outputs for the above problem (most just disregard the scalar because it's infinity so who care if that cancels out later).

Where are the existing test cases for arithemetic calculations with (scalar times) IEEE-754 int, +inf, or -inf as the output?

On Tue, Sep 15, 2020 at 1:54 AM David Mertz <mertz@gnosis.cx> wrote:
Thanks so much Ben for documenting all these examples. I've been frustrated by the inconsistencies, but hasn't realized all of those you note.

It would be a breaking change, but I'd really vastly prefer if almost all of those OverflowErrors and others were simply infinities. That's much closer to the spirit of IEEE-754.

The tricky case is 1./0. Division is such an ordinary operation, and it's so easy to get zero in a variable accidentally. That one still feels like an exception, but yes 1/1e-323 vs. 1/1e-324 would them remain a sore spot.

Likewise, a bunch of operations really should be NaN that are exceptions now.

On Mon, Sep 14, 2020, 5:26 PM Ben Rudiak-Gould <benrudiak@gmail.com> wrote:
On Mon, Sep 14, 2020 at 9:36 AM Stephen J. Turnbull <turnbull.stephen.fw@u.tsukuba.ac.jp> wrote:
Christopher Barker writes:
> IEEE 754 is a very practical standard -- it was well designed, and is
> widely used and successful. It is not perfect, and in certain use cases, it
> may not be the best choice. But it's a really good idea to keep to that
> standard by default.

I feel the same way; I really wish Python was better about following IEEE 754.

I agree, but Python doesn't.  It raises on some infs (generally
speaking, true infinities), and returns inf on others (generally
speaking, overflows).

It seems to be very inconsistent. From testing just now:

* math.lgamma(0) raises "ValueError: math domain error"

* math.exp(1000) raises "OverflowError: math range error"

* math.e ** 1000 raises "OverflowError: (34, 'Result too large')"

* (math.e ** 500) * (math.e ** 500) returns inf

* sum([1e308, 1e308]) returns inf

* math.fsum([1e308, 1e308]) raises "OverflowError: intermediate overflow in fsum"

* math.fsum([1e308, inf, 1e308]) returns inf

* math.fsum([inf, 1e308, 1e308]) raises "OverflowError: intermediate overflow in fsum"

* float('1e999') returns inf

* float.fromhex('1p1024') raises "OverflowError: hexadecimal value too large to represent as a float"

I get the impression that little planning has gone into this. There's no consistency in the OverflowError messages. 1./0. raises ZeroDivisionError which isn't a subclass of OverflowError. lgamma(0) raises a ValueError, which isn't even a subclass of ArithmeticError. The function has a pole at 0 with a well-defined two-sided limit of +inf. If it isn't going to return +inf then it ought to raise ZeroDivisionError, which should obviously be a subclass of OverflowError.

Because of the inconsistent handling of overflow, many functions aren't even monotonic. exp(2*x) returns a float for x <= 709.782712893384, raises OverflowError for 709.782712893384 < x <= 8.98846567431158e+307, and returns a float for x > 8.98846567431158e+307.

1./0. is not a true infinity. It's the reciprocal of a number that may have underflowed to zero. It's totally inconsistent to return inf for 1/1e-323 and raise an exception for 1/1e-324, as Python does.

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