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 Alexander
Aitken was the greatest mathematician of his era and possessed an
astonishing computational brain that could complete challenges that today
are reserved for the most complex computers. As one of the most
remarkable mathematical brains of all time, Aitken could recite Pi to 707
decimal places, multiply two nine digit numbers in his head in 30 seconds,
and render fractions to 26 decimal places in under five seconds.
Phenomenal
Memory and Brain Aitken's phenomenal skill in mental arithmetic made him the greatest mental calculator for whom there is any reliable record. In psychological tests in Britain in the 1920s he took thirty seconds to multiply 987,654,321 by 123,456,789 and produce the correct answer: 121,932,631,112,635,269. Asked to render the fraction 4/47 as a decimal, he waited four seconds and answered: "Point 08510638297872340425531914 – and that’s as far as I can carry it."
Legacy
of Knowledge His main mathematical interests were in Actuarial Mathematics, Linear Algebra, Numerical Methods and Statistics. Econometricians have benefited especially from his applications of matrix algebra to problems in numerical analysis, as well as his statistical contributions to the theory of linear models. Dr. David Giles, currently teaching in the Department of Economics at University of Victoria (Canada) writes that, "Every student of Econometrics must be indebted to Alexander Aitken". Econometricians also use the Generalised Least Squares ("Aitken") estimator when this model has a non-standard error covariance matrix. He published on such topics as
symmetric groups, invariants, the solution of linear and polynomial
equations, eigenvalue problems, and computational algorithms. The
statistics paper "On the Estimation of Statistical Parameters"
(co-authored with H.C. Silverstone, 1942) has been most influential and
his books, "Determinants and Matrices", and "Canonical
Matrices" (with H.W. Turnbull) are classics in their field. |
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Numerical
Epiphany Aitken was born in Dunedin, New Zealand on 1 April, 1895, and attended Otago Boys' High School from 1908 to 1912. He was not a child prodigy: arithmetic bored him and he did badly in it at school until he was nearly 14 years old. Then he experienced some sort of snumeric epiphany, during a mathematical lesson by a good teacher at Otago Boys. After that the whole subject came into focus and he became absorbed by figures. Aitken writes: "Only at the age of 15 did I feel that I might develop a real power and for some years about that time, without telling anyone, I practised mental calculation from memory like a Brahmin Yogi, a little extra here, a little extra there, until gradually what had been difficult at first became easier and easier." He had the distinction of gaining first place in the nationwide University Scholarship Examination of 1912. He then studied at the University of Otago in 1913, 1914, and 1918, but his studies were cut short by active service during the First World War. He enlisted in 1915 and left New Zealand with the Sixth Reinforcements and served with the Otago Infantry. He served in the Gallipoli landing, Egypt, and in France where he was wounded in the Battle of the Somme in France. After three months in hospital he was sent back to New Zealand in 1917. His wartime experiences were to haunt him for the rest of his life. During the war he astounded his
fellow soldiers by his ability to memorise, amongst other things, the
numbers on their rifles. While serving in the Otago Company at
Armentières, the platoon book was destroyed. Aitken recited the names and
numbers of all members of his platoon. |
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Horror of
War His memoir, "Gallipoli to the Somme: Recollections of a New Zealand Infantryman" (Oxford University Press, 1963), is regarded as one of the most moving accounts of the appalling reality of life in the trenches during the Great War. The infantrymen were very much the spearhead in the battles and young Aitken saw at close quarters the grimness of trench fighting. Long afterwards he described vividly one bombardment in France when shells crumpled into the frontline, with red flames shooting up almost continuously, and it seemed impossible that anyone could survive. He saw in one low dugout four signallers dead at their telephones without a mark on them. They had been killed by shock waves.
His account of fighting as an ANZAC speaks with a human voice that astutely articulates not merely the rational absurdity of war, but also the tightrope between emotion and detachment that a solider had to negotiate:
Following his return to New Zealand and recovery, he achieved, in 1918, First Class Honours in Latin and French and (remarkably) Second Class Honours in Mathematics. Aitken followed his original intention and became a school teacher at his old school, Otago Boys' High School. His mathematical genius bubbled under the surface and, encouraged by the new professor of mathematics at Otago University, Aitken gained a post-graduate scholarship which took him to Edinburgh University, Scotland in 1923. He studied for a Ph.D. at Edinburgh under Sir E T Whittaker, one of the most distinguished of British mathematicians. Aitken’s Ph.D. thesis on the smoothing of data was considered so outstanding that he was awarded a D.Sc.
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From
Calculator to Contributor In 1925 he was appointed to Edinburgh where he spent the rest of his life. His initial position was Lecturer in Statistics and Mathematical Economics. After encounters with mechanical calculating machines that made his extraordinary mental powers unnecessary, Aitken’s interests matured. At 28 he was at the peak of his calculating abilities, but from lightening fast calculations his focus began to shift to the theoretical. In describing his period of recovery from a small operation in 1934, Aitken writes
He became a Reader in Statistics in 1936, the year he was elected a Fellow of the Royal Society. Ten years later, in 1946, at the age of 51, he was appointed to Whittaker's chair in Mathematics. Aitken is without a doubt a hero on the edge, but his advantage was not just metaphorical – at Edinburgh he was mentor to a number of renowned mathematicians including one of the Twentieth Century's leading geometers, one William Leonard Edge. Appointed Fellow of the Royal Society in 1936, Aitken won the Royal Society of Edinburgh’s prestigious Gunning Victoria Jubilee Prize in 1953 for original work in Physical Mathematics, as well as honorary degrees from the University of Edinburgh and the (then) University of New Zealand. Aitken wrote several books:
"The Theory of Canonical Matrices" (1932) was written jointly
with H.W. Turnbull, (the University of St. Andrew’s mathematician who
made an extensive and noted contribution to the study of algebraic
invariants and quadrants). With D.E. Rutherford he was editor of a series
of University Mathematical Texts which have enduringly remained on
required reading lists for mathematics students world-wide. His
solo-authored papers, "Determinants and Matrices" (1939) and
"Statistical Mathematics" (1939) formed pivotal contributions to
their field. |
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Diverse Talent and Brilliant
Teacher Alexander Aitken was a man of great and disparate talents. A champion high-jumper into his late twenties, he was also a writer and a poet. He was devoted to music and was regarded as a fine violinist and viola player, as well as being an occasional composer. Aitken explained his arithmetic technique as dividing numbers into sets of five and "submitting them to German waltz time." The Edinburgh University psychologist Dr. Ian Hunter, who studied Aitken in the 1920’s, noted that numbers appeared to him as a tune, "like a Bach Fugue". Dr. Hunter reported that his calculating actions, before analytic or biological interpretation, appeared as reflexive and automatic as those of a boxer or an expert typist. Alexander Aitken is remembered as a warm and gentle man, and a brilliant lecturer. A student at Edinburgh in the early 1960's recalls:
While impressing students, the rational stretch of Aitken’s mind would not have made him friends with casinos or lotteries - another idiosyncratic memory of Aitken’s ‘stories’, as part of his lectures on probability, was a rather stern warning about the evils and foolishness of gambling. It is believed that extraordinary acts of genius in computation, such as those performed by Aitken and also exhibited in autistic people, come from an obsessional, almost morbid concentration. Aitken makes it sound simple, stating that he breaks problems down into the simplest elements. "If I see a motor car with registration number 731, I cannot help observing that it is 17 times 43. Sometimes I find myself squaring the numbers on the lapels of the bus conductors."
To think that everyone can
achieve such feats as Aitken by ‘thinking harder’ is a little
misleading. Aitken’s innate advantage in performing the
unbelievable computations was his mind-boggling memory. When he examined the pages of a mathematical journal he only had to scan it page
by page, turning the pages over at a rate at which the ordinary reader
would record only the first dozen lines or so. Subsequent discussion made
it apparent that he had registered all the material – as he often
claimed, he never forgot what he had once seen. Despite the incredulity
and admiration he fostered, he was modestly pragmatic about his own
abilities as a mathematician: |
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"Familiarity with
numbers acquired by innate faculty sharpened by assiduous practice does
give insight into the profounder theorems of algebra and
analysis." |
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Haunting
Memories The root of Aitken’s genius was also his curse. Aitken’s memories of the war did not fade and his horrific recollections of the battle of the Somme lived with him as real as the day he experienced them. He wrote of them near the end of his life, aged 68 in "Gallipoli to the Somme". It is believed these hauntings contributed to the ill health he suffered, and eventually led to his death in Edinburgh, Scotland, on 3 November, 1967, aged 72. He was acclaimed by his peers as the greatest living mathematician and regarded by his mentor, the pioneering Cambridge and Edinburgh mathematician Sir E.T. Whittaker, as the greatest mathematician since Arthur Cayley. Cayley was a Cambridge mathematician who played a leading role in founding the modern British school of pure mathematics. He was the first to formally develop matrices and made a vast contribution to mathematics. Cayley’s name is given to the operations tables in number systems. He was the recipient of nearly every academic distinction that can be conferred upon an eminent man of science: honorary degrees from most notable universities, election as fellow or foreign corresponding member of the academies of several countries, and the Copley Medal in 1883 from the Royal Society of London. Alexander Aitken was born in 1895, the year that Arthur Cayley died. He was a man whose world was filled by numbers. For such a rare person, whose extraordinary mind and memory was predicated on an ability to elucidate numeric progressions, the comparison with Cayley has a sure poetic symmetry. The annual student award given
by the New Zealand Mathematics Society, the ‘Aitken Prize’ is named in
Alexander Aitken’s honour, and in 1995, at Otago University, a conference was held in his honour, to commemorate the centenary of his birth. |
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Do it
Yourself
The following is extracted from the diverting site Mental Muscles (http://mentalmuscles.com/memory2.html), created by two cognitive scientists from the US, who run through some ‘toning exercises’ for the brain. Their (and no doubt Aitken’s) maxim is "It is not enough to have a good mind. The main thing is to use it well." (Rene Descartes, 1596-1650). Mental
Muscles: The line of poetry 'They passed the pleiades and the planets seven' - mysteries in the minds of the ancients -- Sabbath or the seventh day – religious observance of Sunday -- 7 in contrast with 13 and with 3 in superstition -- 7 as a recurring decimal .142857 which, multiplied by 123456, gives the same numbers in cyclic order -- a poem on numbers by Binyon, seen in a review lately – I could quote from it. (Hunter 1977:163) Could you learn to perform mental calculations as quickly and flawlessly as Aitken? For example, could you learn to multiply 2 three-digit numbers - say, 123 by 456 -- in two seconds? We return to Aitken's own remarkably self-aware account of his method in a passage from a biographical essay, and you can decide for yourself: I do this in two moves: I see at once that 123 times 45 is 5535 and that 123 times 6 is 738; I hardly have to think. Then 55350 plus 738 gives 56088. Even at the moment of registering 56088, I have checked it by dividing by 8, so 7011, and this by 9, 779. I recognize 779 as 41 by 19; and 41 by 3 is 123, 19 by 24 is 456. A check you see; and it passes by in about one second. (Hunter 1978:341) C) How many trials would it take you to memorize this list of 25 words? When he had the entire list read aloud to him at a rate of one word per second, Aitken required four trials; the first trial, he got 12 correct, the second trial 14 correct, the third trial 23 correct, and the fourth trial all correct. HEAD, GREEN, WATER, SING, DEAD D) How many trials would it take you to memorize a list of 16 three-digit numbers? When the following list was read aloud to him at a rate of one word every two seconds, Aitken needed four trials. In the first trial, he got 6 correct, the second trial 10 correct, the third trial 14 correct, the fourth trial all correct. 194, 503, 876, 327 |
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References:
Books: P.C. Fenton (1995), To Catch the Spirit: the memoir of A.C. Aitken with a biographical introduction, University of Otago Press, Dunedin, Information on the web: Excellent biography of Aitken, from the University of St. Andrews,
Scotland: "Conference pays tribute to Edinburgh Scholar", University of
Edinburgh press release to acknowledge the 1995 conference at Otago
University held to honour Aitken. On mental arithmetic and calculation: On matrices and determinants: From a site "Great Mathematicians", University of Santiago,
Chile, (in Spanish): Books and Articles Tee, G. J. (1981) "Two New Zealand Mathematicians", in J. H. Crossley (ed.), Proceedings of the First Australian Conference on the History of Mathematics, Monash University, pp. 182-199. Tee, G. J. (1988) "Mathematics in the Pacific Basin", British Journal of the History of Science, vol. 21, pp. 401-417. Talk given to the British
Society for the History of Mathematics: |
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Aitken | Alda | Alley
| Atack | Batten | Bowen |
Britten |
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He had such a prodigious memory
that he could store away solutions to problems and then call on them from
his brain when needed. He could do a series of calculations by mental
arithmetic and hold the answers to all of them in his head long enough to
bring them all together at the end and have the final answer.
The movie Pi about another mind whose world was filled with
numbers, is a ‘mathematical thriller’ about a person who descends into
madness trying to find the secret to the patterns beneath the chaos to
unearth the secrets of the share-market. There is an excellent visual
imagining (on the front page of the site) of just how astounding Aitken’s
reciting of Pi was.