Prof. Jakow Trachtenberg was a Jewish native of Russia who fled to Germany when the czarist regime was overthrown, but later his anti-Hitler attitudes ended with him spending seven years in concentration camps. It was during that time that he developed his system of mathematics in order to keep his mind occupied. Since most of his work was done without pen or paper, the resulting techniques can be often be performed mentally.
His method of addition utilizes the principle: "Never count higher than eleven." Counting up or down a column, when the total exceeds 11, subtract 11 and put a tick mark at that point. At the end of the column, write down your final total, and under it write the number of tick marks. Then add these two rows adding the running total, the tick total for that column, plus the number of ticks from the column to the right (the first column doesn't have a column to its right, of course). Write down the last digit and carry the other digit, if any, working right-to-left. As a final step, assume a column of zeros on the left end, so the only things to add are really the number of ticks from the previous column, plus and figure being carried.
Example
4 6 5 9
6 7' 5 8'
9' 7 6' 7'
0 8' 6' 9'
6' 0 9 1
----------
3 6 9 1 running totals
2 2 2 3 ticks
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2 8 '1 '4 4 total
This is a particularly technique, and is traditionally done by starting at the righthand digit of the multiplicand (the number being multiplied by 11) and moving left. With the addition calculation we assumed a column of zero on the left, and here we assume a leading zero at the left of the multiplicand.
We can also assume the rightmost digit has a trailing zero as a neighbor.
The basic
rule for multiplication by 11 is simply to add each digit to its neighbor (the figure on its right) plus any carry.
Example
783152 x 11 = 8614672
The steps are:
1. Add 2 (the rightmost digit)) to its imaginary neighbor of 0, and write down 2.
2. Add 5 (the second digit) to its neighbor (2), and write down 7.
3. Add 1 (the third digit) to its neighbor (5), and write down 6.
4. Add 3 (the fourth digit) to its neighbor (1), and write down 4.
5. Add 8 (the fifth digit) to its neighbor (3), which gives 11, so write down 1 and carry 1.
6. Add 7 (the sixth digit) to its neighbor (8) plus the carried 1, giving 16, so write down 6 and carry 1.
7. Assume the leading 0 and add its neighbor (7) plus the carry, and write down 8
8. Pat yourself on the back.
With practice, you can do this procedure working right to left (you need to foresee the carries), which is the method that practitioners of speed mental arithmetic use (working right to left, rather than left to right as we do when calculating with quill and parchment).