## Example: Silly Definition of Multiplication |

Why is multiplication of two tuples |

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-zeroaexistsbsuch thata*b=1

This axiom implies the *cancellation property*:

ifac = bcandcis not zero thena = b

But if we define *(x,y)*(u,v) = (xu, yv)* we have, with
*c = (0,1) (0,0)*:

(1,0)*c = (0,0)*cyet(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!