I first saw Arthur Benjamin on British TV, about 25 years ago, when he appeared on the Paul Daniels magic show, instantly squaring large numbers in his head. His very engaging personality, and his undoubted arithmetical genius, immediately won the hearts of the audience, including mine. But what of this book, co-authored with Michael Schermer?
The 230 pages, and eight chapters, cover the themes of mental addition, subtraction, multiplication and division, and the very useful art of ‘guesstimation’. There is also a chapter on techniques for memorising numbers, starting with mnemonics for recalling the value of π to varying degrees of accuracy.
For example, one method of simple addition would be: 67 +28 = 87 +8 = 95, which relies on the associative property for (N, +). And the ‘trick’ for squaring numbers is shown to depend upon the algebraic result x2 – y2 = (x + y)(x – y), as follows:
1932 = (193 + 7) x (193 – 7) + 72 = (200 x 186) + 72 = 37 200 + 72 = 37 249.
In fact, the most appealing aspect of the book is that the so-called ‘magic’ nature of such methods is demystified by clear explanations that invoke the use of the algebraic properties of the natural numbers, and even modular arithmetic (for the analysis of divisibility tests).
Speaking of divisibility, most people know that a number is a multiple of 3 if its digital root is 3, 6 or 9. But there are lesser -known short-cuts to ascertain divisibility by other numbers, such as 11 or 7; here is the rule to determine whether 11 is a factor of a particular number:
A number is divisible by 11 if you arrive at 0 or a multiple of 11 when you alternately add and subtract the digits of that number
For example, 8,492 is divisible by 11, because 8 – 4 + 9 – 2 = 11. Voilà! Take a bow on national television!
Of course, ‘mental math’ usually refers to mental arithmetic, and the contents of the book conform to this limited connotation. However, there is scope to include geometrical ideas and exercises on spatial thinking, and there is also room for the inclusion of verbally given problems of a logical nature. In fact, I think it was Michael Atiyah who said that, when he was working on a problem, his method for doing so, was to wander from room to room, or stroll round the garden, mulling over various possibilities until he was ready to put pen to paper. But Atiyah’s mind would have been engaged upon a particular aspect of topological K-theory, rather than squaring numbers in his head.
The truth is, of course, that all mathematics is ‘mental’, and some people rarely resort to pencil and paper methods. One such person of my acquaintance was a stevedore, whom I met on when I worked on the Liverpool docks. He could mentally solve most of the problems on ‘Permutations and Combinations’, in Durrell’s Advanced Algebra. Equipped only with the most basic level of elementary education, he had informally acquired this expertise by means of his pre-occupation with gambling (cards, dice games, horse racing and so on).
Interestingly, many children also develop their own methods, and keep them hidden from disapproving teachers, and there are various research findings that reveal the extent and variety of such phenomena. Amongst many such instances, I remember an 8-year old girl, whose teacher found to be problematic in many ways. Amongst other ‘tricks’, this 8-year old had devised the following method for halving any natural number, which she carried out a lightning speed:
5678 → 2½ 3 3½ 4 → 2839
But, for some time now, many of the techniques explained in this book have been included in the sections on mental mathematics in the various primary mathematics curricula; and this book illustrates such processes in a most entertaining manner. As such, I would seriously recommend it for use in the training of mathematics teachers, to equip them with alternative approaches to the teaching of arithmetic, and hopefully encourage children to recognise mathematic as a creative activity.
Peter Ruane, although having taught mathematics at all levels, from kindergarten to undergraduate mathematics, has comparatively moderate ability with respect to mental arithmetic.