When exactly is XY not equal to YX ?

My answer to a question on Quora: When exactly is xy not equal to yx?

The expression \(xy\) can be used in a variety of situation for different kinds of mathematical objects (not only to numbers!) and operations on them, and in many (if not in most) situations \(xy\) is not equal \(yx\).

Let \(x\) and \(y\) be two processes or operations and \(xy\) is the outcome of their consecutive application: first \(x\), then \(y\). A kindergarten level “real life” example:

  • \(x\) is putting a sock on the left foot and \(y\) is putting a sock on the right foot; very obviously, the order of operation does not matter, \(xy = yx\).
  • \(x\) is putting a sock on the left foot and \(y\) is putting a boot on the same foot. You would perhaps agree that \(xy \ne yx\).

In geometry, the result of composition (that is, consecutive application) of rotations and other geometric transformations in the space almost always depends on the order in which they are performed. These rotations and their consecutive execution are described as matrices (certain tables of numbers) and their multiplication (defined by some specific rules) – and, as a consequence, for multiplication of matrices, in most cases, the result depends on the order of multiplicands, \(xy \ne yx\).

In real life, time is the principal source of non-permutability (or non-commutativity, in mathematical parlance) of events. By certain age, you start to understand, that there were things that you had to have done 20 years ago, not today or tomorrow.

Another nasty property of time: you can re-use space (say, empty a cupboard and fill it again), but cannot re-use time.

Both of these principles apply to learning mathematics: certain things have to be mastered at a certain age, and in specific order. Learning mathematics is growing neuron connections in one’s brain; like in growing a garden, processes are not freely permutable, and, in many cases, cannot be reversed and done again.