Re: division

[jenyan1 <jenyan1@xxxxxxxxxxx>]

Charles,

David Welch's definition of division is not quite right. What he said was
that the integers are a euclidean domain (a technicality which we can
ignore for the moment).

Suppose Z = {integers}= {...-2,-1,0,1,2,3,...}

then multiplication by a non zero number is a "map" from Z to a subset of
Z. Best to look at a concrete example, say multiplication by 5.

 {integers} * 5 = { ... -10, -5, 0, 5, 10, 15,...}

The set on the right hand side (call it 5Z) is in one to one
correspondence with Z. That is for every element x in Z there corresponds
_exactly_ one element x*5 in 5Z. so the map

                x --> x*5

pairs every element of Z with exactly one element of 5Z. Such a pairing or
correspondence is called a bijection.

To say that the elements of 5Z are divisible by 5 means that the
correspondence can be reversed

                5*x --> x

eg              15 --> 3, 20 --> 4 etc.

reversing the correspondence is just "division by 5".

If you try and do a similar thing with the element 0 in stead of 5 you run
into trouble since zero "collapses everything to a point".

 {integers} * 0 = {0}

Notice that now you cannot establish a correspondence which associates
every element of Z with _exactly_ one element of {0}. The thing we called
Z is an infinite set while {0} has only one element. This is the reason
why division by zero does not make sense in the integers.

        x->x*5  this map is bijective so reversible.

        x->x*0   map is not bijective so not reversible.

Notice that say 2,3,4,6,... are not divisible by 5 in the integers. But
they are divisible by 5 in the rationals, since 1/5 is a rational number
but not an integer. This is a technicality which we can ignore for the
moment (for everyday purposes it's not necessary to be precise about
whether you are working in the integers, rationals or the real numbers).

But you can generalise the above argument about division by zero to
rationals etc...

Last remark: you can create other "number systems" where elements other
than zero do not behave well with respect to division.


On Fri, 20 Apr 2001, Charles Brown wrote:

>
> CB: For someone who is starting out in arithmetic, they can add zero,
subtract zero, multiply by zero, but not divide by zero.  I guess in that
way zero is sort of a big hole from the beginning. But aren't holes
paradoxes ? I understood someone to be saying paradoxes aren't really very
significant in mathematics and its development.