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)?

Context Context

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!

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007