10 Apr 03:32 2014

### why does a/(b/c) = a(c/b)?

Kragen Javier Sitaker <kragen <at> canonical.org>

2014-04-10 01:32:45 GMT

2014-04-10 01:32:45 GMT

My friend Santi asked me why we divide by a fraction by interchanging the numerator and the denominator and multiplying; that is, why a/(b/c) = a(c/b). I wasn’t quite sure how to answer, but after thinking about it, it turns out that there are many deep and fascinating answers that involve many aspects of the universe of mathematics. Here are three different answers. Sort of. Part of the problem is that it’s difficult to say what really counts as an explanation here, because, as Feynman explained in his famous BBC video on “Fucking magnets, how do they work?”, an explanation has to start with things that you already understand to be true. In cases like this, it’s really easy to fool yourself into thinking that you have an explanation, when all you really have is circular logic. Here, let me demonstrate. An answer based on group theory ------------------------------- A “group” is a set with an operation that have the four properties of closure, associativity, identity, and invertibility. Nonzero fractions, together with multiplication, are a group. It turns out that the divide-by-multiplying-upside-down thing isn’t limited to fractions at all; it’s a much more general property that applies to any group, including bizarre things like permutations under composition, three-dimensional rotations of polyhedra under composition, matrices under matrix multiplication, Gaussian integers under addition, bit strings of some fixed length under XOR, and integers under multiplication modulo a prime number! To explain, first I will explain the meaning of the group properties.(Continue reading)