Expert: David Hemmer - 5/2/2007

Question
Can you give me any examples of mathematical entities that do not obey the associative law?

(A*B)*C  is not the same as  A*(B*C)
(A+B)+C  is not the same as  A+(B+C)

Thank you for your opinions.
Jeff  

Answer
You question is a little  misphrased, it's not a question of which "entities" obey the associative law, it's what "operations". The same entities may have many different operations.

You already know one such operation, it's called subtraction!  (A-B)-C is not the same as A-(B-C).
You say "subtraction is not associative" not that "numbers are not associative".


But other than that, there are not too many operations in mathematics which are not associative, and nonassociative algebras are strange things. One famous such set is called the Octonions, also called the Cayley numbers. These are a certain set with a nonassociate multiplication defined on them.  You can read about these very interesting numbers here:

http://en.wikipedia.org/wiki/Octonion