Example: Silly Definition of Multiplication
Why is multiplication of two tuples (x,y) and (u,v) not simply defined as (x,y) * (u,v) = (x u, y v)?
If we did define this "silly" multiplication, the resulting system would no longer be a field! Recall the last axiom for a field. It says, in part:
There is a multiplicative inverse, i.e. for all non-zero a exists b such that a*b=1
This axiom implies the cancellation property:
if ac = bc and c is not zero then a = b
But if we define (x,y)*(u,v) = (xu, yv) we have, with c = (0,1) (0,0):
(1,0)*c = (0,0)*c yet (1,0) (0,0)
Phrased more succinctly: if we defined multiplication via the above "silly" multiplication, and we insisted on the field axioms, then we could prove that any number is zero! That's indeed silly!