Aitken's trick for calculating decimal expansions of rationals

Aitken's Method

Professor Aitken was both a mental calculator and a master mathematician, and a detailed account of his mental math abilities is found in An exceptional talent for calculative thinking, (IML Hunter, 1962)

I'd just like to sketch out one amazing trick, which is illustrated thus:


The proof is  $x = 5/23 = 15/69$, then, $x = (15 + x) /70$, so the trick is to calculate $15/70$ by short division, to generate the digits of $x$ one at a time, then immediately feed to the top to continue. A kind of "just-in-time" algorithm!




The proof is basically this: $x = 5/23 = 435/2001$, then $x = (434.999\cdots - x)/2000$, so the trick is to calculate $434.999\cdots/2000$ by short division, generate the first 3 digits of $x = 0.217\cdots$, then immediately subtract those three digits from the top to continue with $434. 217\cdots$, and so on.

The 999... was for the sake of doing the subtraction correctly.

The feeling of calculating

He is the opposite of synthesia:

He does not calculate by manipulating experienced representations of numbers which have any distinct degree of sensory realism. There is certainly no visual imaging, nor does there seem to be any auditory or kinaesthetic imaging (all three modes of imaging are familiar to him in other, non-calculative contexts). He reports that he could calculate in visual or auditory terms but that this would greatly slow him down.

There seems to be a bunch of mental subprocesses, only some accessible for verbal report:

During calculation, he is not an automaton lacking self-awareness. ‘I must be relaxed, yet possessed, in order to calculate well. I believe that conscious and subconscious activities are conspiring or in rapid alternation. I seem to move on several different levels. And last of all, when the result is complete, I return to the normal level of ordinary social contact.’

System 1 does most of the work:

For example, with this thinker, as with many other people, 12 is the immediated product of 3 and 4: but unlike most people, the transition from ‘9 times 12,345’ to ‘ 111,105’ is also immediate for this thinker... Other leaps concern procedural judgements, that is, diagnosing what method is best to use in calculation. These high-level procedural diagnoses derive from a breadth of past experience which is fully comparable to (and possibly in excess of) that which lies behind the so-called position sense of the chess-master, or the swiftly impressionistic diagnoses made by some experienced physicians, or the intuitive snap judgements made by experts in many fields of science, art, and commerce.

And there is a feeling of "something wrong", presumably from a sanity check module.

Yet other leaps occur between attaining an answer and recognizing either that it is correct or that something is wrong. ‘I find that if I do not doubt the result of a calculation, it is usually always correct ; but if I have any residual doubt, then some correction is usually required. ’

Thinking vs recalling

There was an interesting comment on the nature of thinking vs recalling.

When he attains the answer to a problem with rapidity, good timing and a feeling of ‘all correct’, then he cannot easily say whether he calculated this answer or recalled it-especially if he definitely knows that he has done this particular calculation before.

 These two activities are not quite distinct, and apparently people use a system 1 vs system 2 way to tell which is which. System 1? Probably recalling/intuition. System 2? Thinking. For Aitken, system 1 has reached such a level of sophistication that something as hard as 9 times 12,345 = 111005 is system 1, and makes him unsure if it is really recalling or generated "on the spot".

The way his System 1 developed was gradual, as usual: things started out being handled by System 2 before gradually sinking into the thoughtless depths of System 1:

Although Prof. Aitken cannot recall with certainty the details of his early calculative development, there was clearly that cumulative, hierarchically organized progression which so pervasively characterizes the acquisition of any comprehensive skill, e.g. learning to use a verbal language, to use telegraphic language, to typewrite, to play chess. ‘In the process of constantly extending my ability, during the period when I was doing that, I had to think very much of what I was doing. I found that the gains were cumulative and, so to speak, stratified, in the sense that they formed a deposit sinking deeper and deeper into the subconscious and forming a kind of potential upon which, in certain states, I made drafts at astonishing speed.’

And really, it's a matter of space/time tradeoff. Keep a big lookup table, or calculate anew? Or perhaps store compact intermediate results from which the full result could be generated on the spot? Cognitive science, psychology, and computer science merges at the interface of mental computation...