On 4 September 2024 03:44:42 BST, Kwankyu Lee <ekwan...@gmail.com> wrote:
>
>
>On Wednesday, September 4, 2024 at 3:05:19 AM UTC+9 dim...@gmail.com wrote:
>
>On Tue, Sep 3, 2024 at 5:08 PM Kwankyu Lee <ekwa...@gmail.com> wrote:
>>
>>
>> That in Python one has non-real value for (-1)**(1/3) is
>> two things: 1/3 is actually a float, and absence of typing.
>> These are artifacts of the programming language, and make little sense
>> mathematically.
>>
>>
>> Do you mean that the value of (-1)**(1/3) is arbitrarily chosen,
>regardless of the mathematical value of (-1)^(1/3)?
>
>no, I mean that
>
>1) 1/3 gets converted to a float, and then you cannot escape dealing
>with (-1)**(0.333333...3), which is
>not equal to ((-1)**(0.333333...3))**3, unlike exact (-1)**(1/3).
>>>> a=(-1)**(1/3); a
>(0.5000000000000001+0.8660254037844386j)
>>>> a**3
>(-1+3.885780586188048e-16j)
>
>2) Python used to be untyped, and still is, to an extent, so returning
>a complex number instead of real wasn't
>such a big deal.
>
>
>I agree that python language could not define (-1)**(1/3) as the real cube
>root of -1.
>
>Hence it chose (-1)**(1/3) to return the complex principal root of -1, yes
>by computing exp(log(x)*y) with branch cut negative axis, *which makes a
>perfect mathematical sense.*
>
>Now back in sage, sage chose to be compatible with python's behavior in
>x^(1/3) for whatever x that represents a number in the complex field, say x
>= -1, RR(-1), CC(-1), QQbar(-1). For x = ZZ(-1), QQ(-1), they chose to
>return symbolic expression (-1)^(1/3), which again converted to the same
>value with QQbar(-1)^(1/3). For x = RBF(-1), it chose to raise an error.
>For x = RDF(-1), it chose to return "nan". All these suggest to sage users
>that x^(1/3) means the principal cube root of the number x.
and for numpy datatype users, too, it's natural to get NaN, because numpy
returns NaN.
For mathematicians who do real algebra and real algebraic geometry, it's
extremely unfortunate when
reality is not respected by functions which, mathematically, can respect
reality. For them your "perfect mathematical sense" is not perfect at all.
>
>Now AA(-1)^(1/3) returns -1. So AA made a different choice. There is no
>mathematical definition that applies to AA(-1)^(1/n). Is it n-th root in
>AA? Then look AA(-1)^(1/4) = 0.7071067811865475? + 0.7071067811865475?*I .
>Is it the principal root? Then look AA(-1)^(1/3) = -1. It is AA that is an
>artifact of mathematical programming language.
>
>I think that only valid argument for AA(-1)^(1/3) = -1 is that it is the
>status quo. It is just human that likes what she/he used to. Removing
>inconsistency is for the future.
>
>
>Kwankyu
>
>
>
>
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