When I was taught basic arithmetic, I was taught about cancellation as a technique for simplifying problems. Later I was taught to think of it as combining with an inverse of what's being cancelled; I have since come full circle and regard it as a primitive property, in its own right, of the things to which it can be applied. It is the essential value of monic relations; and it (together with completion) provides a clearer description of group theory.
I'm pleasantly surprised to find that Persian polymath Muḥammad ibn
Mūsā al-Khwārizmī appears to have taken a similar view.
His Compendious
Book on Calculation by Completion and Balancing
is the (truly
revolutionary) book that gave us the word algebra
, as الجبر (al-jabr); and my
impression is that he was using this word to mean roughly what we do
by cancelling
, which I find particularly pleasing. He also seems to have
preferred dealing with positive values, to the exclusion of negatives, as do
I.
His methods for dealing with quadratic equations begin with grouping all
terms in each power of the root on one side of the equation, the completion
and balancing
of the title, perhaps better translated as balancing and
restoring
(if I understand Wikipedia's article aright). First, put all
terms on the side where they're positive, then cancel the smaller from the
larger where terms of the same kind appear on opposite sides of the equation.
As he seems to be using الجبر for the latter, I
tentatively suppose cancellation is the true origin of algebra :-)
Having re-cast the axioms of group theory in terms of cancellation and a form of completion (where, given one value and wanting another, a third is found to combine with the given to obtain the wanted), motivated by a desire for axioms that are reasonably independent, I found that cancellation proves the more powerful of the two, when considered on its own. I have thus re-cast ring theory in terms that don't require completeness for the addition; this enables me to prove that natural arithmetic supports prime factorisation, without having to embed the naturals in the integers to do it (i.e. without needing negatives).
Written by Eddy.