[Tutor] a puzzle about -3**2 vs (-3)**2

[Steven D'Aprano]

On Thu, Jul 30, 2015 at 05:58:27PM -0700, D Wyatt wrote:
> I just read in a book a little while ago that ** trumps a negative
> sign?  

Correct. Exponentiation has higher priority than subtraction, addition, 
multiplication and division:

2+3**2 => 11 not 25

10-3**2 => 1 not 49

2*3**2 => 18 not 36

100/5**2 => 4 not 400

As you have discovered, it also has higher priority than unary minus so 
that:

-3**2 => -(3**2) => -9 NOT (-3)**2 => 9

Mathematically, this is perfectly acceptable, and what we would 
normally expect. In algebra, if we write:

-x² 

we normally mean the negative of (x squared), not (negative x) squared, 
which would be just x². So Python here agrees with standard mathematical 
notation.


> I am struggling with the audacity of that as -1 is negative 1,
> NOT minus 1.  How can an arithmetic operation trump an attribute of a
> negative integer?  It truly makes no sense to me.  Thank you for any
> enlightenment you can provide.

Speaking as a maths tutor with about 20 years experience, and a B.Sc. 
with a major in mathematics, I'm not sure I understand what you are 
getting at. There is no mathematical difference between the inherent 
negativeness of -1 and the arithmetic operation - 1 (unary minus 
operator followed by 1).

Whichever way you treat it, we have to agree what it means. For example, 
2x means 2 multiplied by x; but 23 doesn't mean 2 multiplied by 3. It 
could if we wanted it to, but that would be inconvenient. Mathematicians 
could define -3² as (-3)² = 9 if they wanted to, but generally they 
don't, although there are exceptions. Consequently such expressions are 
ambiguous and are best avoided. Although -x² never means -x squared, 
it always means minus (x squared).

Python removes the ambiguity and has exponentiation always take priority 
over other arithmetic operators, regardless of whether you consider - a 
unary or binary operator, or an inherent part of the integer.

Although, for the record, the standard Python interpreter does NOT parse 
an expression like -123 as "negative 123", but as "unary minus 123". In 
practice this makes zero difference to the code.



-- 
Steve