Re: [MATHEDCC] (fwd) Re: Confusion about Negatives

Kirby Urner (urner@alumni.Princeton.EDU)
Thu, 18 Nov 1999 20:34:10 GMT

>>There is no commonly use algebraic structure (algebra, field, monoid,
>>ring) for which this can cause confusion. Since subtraction is
>>nonassociative, symbols like a-b-c are ambiguous and should be
>>avoided, notwithstanding the normal custom towards
>>left-associativity.
>
> a - b - c =3D a + (-b) + (-c)=20
>
>Usually this operation is defined to be associative, no? (but
>maybe not commutative -- Abelian groups and rings only). It's=20
>multiplication where associativity goes out the window a lot=20
>of the time (matrices, quaternions and the like).
>
>Reference: http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/vector.html

Sigh. I flubbed this big time.

a-b-c is indeed ambiguous. 5-(3-2)=3D4 but (5-3)-2=3D0. If I
rewrite it as 5+(-3)+(-2), I'm taking out the ambiguity,=20
because addition is associative.

=46urthermore, it's not associativity that goes out the window
with matrices and quaternions, but commutativity -- although
we do have some algebras which aren't associative either
(e.g. Cayley Numbers of dimension 8).

My apologies for being sloppy.

Kirby

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