Thursday 7 August 2014


One of the books turned up by the latest invasion of the long-term storage attic was The Trachtenberg Speed System for Basic Mathematics translated and adapted by Ann Cutler and Rudolf McShane. You can get your own copy for £3+p&p. It has an extraordinary back-story. Jakow Trachtenberg was a Russian who shares my birthday (d.o.b 17 Jun 1888), and no, I am old but I don't share his birth year. I think he was a Russian although he was born in Odessa in the Crimea, a birth place and approximate time he shares with John Beshoff the chipper. So if Ukraine wants to claim him, please do so in the comments. In my piece about Beshoff, I suggested that Primo Levi couldn't have written his best book without being a chemist.  I might have added that his writing was reforged in the ovens of the holocaust - who knows but if he hadn't done time in Auschwitz he might have stayed as a thoughtful and competent chemist and never written his haunting and compelling books.  It is almost certainly true that if Trachtenberg hadn't spent a desperate time in prisons and concentration camps through the whole of WWII, he would never have written his book.

He was imprisoned because he wasn't going to have any of that bystander effect: when he saw things he disagreed with, he spoke out.  As a pacifist, he had much to disagree with as he lived through the February and October Revolutions in 1917; and then as an exile in Berlin (1919-1934), Vienna (1934-1938) and Yugoslavia.  He was able to nip across borders, often at the last minute, because he had assets and married well. In the infinitely corruptible system that ran the Third Reich, if you had money you had a far far better chance than the dispossessed. But der Mann caught up with him eventually and he spent many long years in concentration camps as a political prisoner. His aristocratic wife managed to stay outside the wire and she eventually winkled him out of his last camp and across the border to Switzerland.

In the camps he had few assets and certainly not the infinite supply of pencils and paper that a mathematician needs to think deep thoughts. To keep himself sane, he developed a system of mental arithmetic which enabled him to carry out any arithmetic operation in his head. This Trachtenberg System was, after the war, expanded into a book and used in an Institute in Zurich to empower people by polishing their math skills. He found that this could have astonishing knock-on consequences for competence in other areas and self-esteem.

In my first job after my PhD, my core teaching assignment was teaching lab practicals in genetics to large classes of first year under-graduates.  Tuesday and Thursday afternoons I had 60 youngsters under my care with a handful of post-graduate teaching assistants.  One experiment required each pair of students to count the pattern of white and black spores in the ordered asci of the ascomyscete fungus Sordaria fimicola.  You can get no useful information from counting a single ascus but if you count many you have data. The way we organised it was that each pair of students would get some data by counting and then post their results on the central black-board where they would be tallied up for a class result - that could be compared to the result from the Tuesday class.  Calculators existed in those days but it was still a chore to punch in the subtotals.  One year I had an extraordinary asset in a chap called Rob Harper who would stare at the board, set his jaw, make a faint humming sound and reel off the column totals.  Of course we verified the result with the calculator but it turned out, after a discrepancy, that he was correct and the chap on the calculator was wrong.

Trachtenberg loved numbers, they were his pals. And he used to talk to his friends in the camps and came up with nifty ways in which their natural relationships helped him carry out arithmetic operations on them.  All in his head because paper was a currency in the camps and couldn't be wasted on diaries, or sums or stuff not essential for survival.  I had hoped that I wouldn't have to explain how the Trachtenberg system works - I'd just point you at a brilliant youtube vid.  But no - all the top-hitting Trachtenberg  videos are earnest, didactic, boring and/or presented in a nearly impenetrable accent. Here's a taster: the rule for multiplying a big number by 6 just using your ability to add one digit numbers in your head.
Rule for 6x: to each "number" (digit to operate upon) add half the "neighbour" (the digit to the right); if the number is odd add another 5. If the neighbour is odd discard the fractional part when you half it: half of 5 is 2, half of 7 is 3 etc.
The challenge: 0443052 x 6
01.  0443052
           2 First number (2) has no neighbour: pop it down

02.  0443052
         '15 is odd: 5 + 5 + half neighbour (1): 5+5+1=11  enter 1 the ' is a "carry"

03.  0443052
         312 0 plus half of 5 (2) and add the carry: 0+2+1=3

04.  0443052
        8312 3 is odd so add 5, neighbour is 0: 3+5+0=8

05.  0443052
       58512 4 plus half of 3 (1): 4+1=5

06.  0443052
      658312 4 plus half of 4 (2) so 4+2=6

Last 0443052
     2658312 0 plus half of 4 (2) so 0+2=2
Check the answer on your calculator.  Trachtenberg has similar systems for multiplying by 11, 7, 5, 4. etc. doing 718 x 592 and then moves on to square roots and internal consistency checks, like casting out 9s. It's fun, you should check it out; when armageddon starts next week, there won't be no calculators, there will just be the sound of a banjo.

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