# Re: [Scheme-reports] Definition of expt when the base is exact 0 (was Re: John Cowan's votes and rationales on the seventh ballot)

```On Wed, Sep 19, 2012 at 4:17 PM, Mark H Weaver <mhw@x> wrote:
> Hi Alex,
>
> On 09/19/2012 01:40 AM, Alex Shinn wrote:
>>
>> On Wed, Sep 19, 2012 at 1:47 PM, Mark H Weaver<mhw@x>  wrote:
>>>
>>> On 09/18/2012 11:40 PM, Alex Shinn wrote:
>>>>
>>>> I prefer my earlier wording:
>>>>
>>>>        The value of 0^z is 1 if (zero? z), 0 if (real-part z)
>>>>        is positive, and an error otherwise.  Similarly for 0.0^z,
>>>>        with inexact results.
>>>
>>>
>>>
>>> This wording has serious flaws.  It suggests that for the cases specified
>>> above, the exactness of the result depends only on the exactness of the
>>> base.  For example, it suggests that (expt 0 0.0) =>  1 and (expt 0.0 0)
>>> =>
>>> 1.0.
>>
>>
>> If you think the wording is bad, it would be nice if you
>> suggested an alternate wording.  As it is, this is more
>> clear than R5RS or R6RS, and I was hoping to avoid
>> listing out all 12 cases of signs and exactness.
>
>
> I will work on a proposed wording.  It needn't be that complicated.
>
>> I was trying to clean up the simplification "usual rules
>> of inexact contagion" but we can put that back in,
>> so that (expt 0 0.0) =>  1.0.
>>
>>> On the contrary, (expt<anything>  0) should yield an exact 1
>>
>>
>> I disagree, but this is just a convention, and there is
>> disagreement in the community about whether even
>> (* 0 x) is allowed to return exact 0 for inexact x.
>
>
> I explained the rationale for (expt <anything> 0) => 1 in my reply to John
> Cowan.
>
>>> Furthermore, I challenge anyone to justify (= 1 (expt 0 0.0)), or for
>>> that
>>> matter the claim that (expt 0<non-integer>) is well-defined.  I'm not
>>> aware
>>> of any established definition for 'expt' that can justify these claims.
>>
>>
>> (expt 0 1/2) =>  (sqrt 0) =>  0, what's the problem there?
>
>
> That's a specific example that could perhaps be extended to positive
> rational exponents, but I'm questioning the validity of the general rule
> that (expt 0 <non-integer>) is well-defined for an arbitrary exponent with
> positive real part.  Consider (expt 0 pi) for example, or (expt 0 1.3+1.4i).

Rational exponents are defined in terms of roots.  Real exponents
are defined in terms of limits using rational exponents.  Complex
exponents are defined in terms of real exponents using the
formula I gave.

--
Alex

>> More generally, there is a well-established definition
>> of complex exponentiation, a simple explanation of
>> which is in the wikipedia article on the topic.
>>
>> Given complex numbers w and z with:
>>
>>    w = re^iθ  in polar notation
>>    z = c + di
>>
>> then
>>
>>    w^z = (r^c e^-dθ) [cod(d log(r) + cθ) + i sin(d log(r) + cθ)]
>>
>> Now when w is 0, then r is 0 and the factor r^c is
>> only defined when c (i.e. (real-part z)) is positive.
>
>
> log(r) is undefined when r is exact 0.  Furthermore, you're begging the
> question here.  The problem is that you've started with a definition of
> complex expt in terms of real expt.
>
> We need to start with the conventional definitions of (expt w z), of which
> there are two:
>
> 1. If z is an (exact?) non-negative integer, then (expt w z) may be
>    defined as (* w w w w ...) with 'z' copies of 'w'.
>
> 2. (expt w z) => (exp (* z (log w))).
>
> The second definition can be used when 'w' is an _inexact_ zero, because in
> that case a limit is used to evaluate (log w), which results in -inf.0+??i,
> where the imaginary part is (angle w).  Note that there are potentially
> eight different kinds of inexact zeroes: +0.0, -0.0, +0.0i, -0.0i,
> +0.0+0.0i, +0.0-0.0i, -0.0+0.0i, and -0.0-0.0i.
>
> When 'w' is an _exact_ zero, then the second definition cannot be used at
> all, because a limit is no longer appropriate, and indeed (log 0) is an
> error.  Therefore, we must rely on the first definition, which is only
> applicable when the exponent is an (exact?) integer.
>
>    Regards,
>      Mark

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