The Trachtenberg method is a system to perform arithmetic operations, mainly multiplication, in an easy and fast way, once its rules are known and mastered.
It was devised by the Russian-born engineer Jakow Trachtenberg (1888-1953) when he was a prisoner of the Nazis in a concentration camp, as a form of distraction to maintain sanity while continuing in captivity..
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The advantage of this method is that in order to perform multiplication it is not necessary to memorize the multiplication tables, at least in part, it is only enough to know how to count and add, as well as to divide a digit by two.
The downside is that there is no universal rule for multiplying by any number, rather the rule varies according to the multiplier. However, the patterns are not difficult to memorize and in principle allow operations to be carried out without the aid of paper and pencil..
Throughout this article we will focus on the rules for multiplying quickly.
To apply the method it is necessary to know the rules, that is why we are going to present them one by one and with examples:
-To multiply any number by 10, simply add a zero to the right. For example: 52 x 10 = 520.
-A zero is added to the beginning and end of the figure.
-Each digit is added with its neighbor to the right and the result is placed below the corresponding digit of the original figure.
-If the result exceeds nine, then the unit is noted and a dot is placed on it to remember that we have a unit that will be added in the sum of the next figure with its neighbor on the right.
Multiply 673179 by 11
06731790 x 11 =
--
= 7404969
The steps required to arrive at this result, illustrated by colors, are as follows:
-The 1 of the unit of the multiplier (11) was multiplied by the 9 of the multiplier (06731790) and 0 was added to it. The unit digit of the result was obtained: 9.
-Then we multiply 1 by 7 and add nine to 16 and we carry 1, we place the ten digit: 6.
-After multiplying 1 by 1, adding the neighbor on the right 7 plus 1 that he carried gives as a result 9 for the hundred.
-The next figure is obtained by multiplying 1 by 3 plus the neighbor 1, it is 4 for the thousands digit.
-Multiply 1 by 7 and add the neighbor 3 resulting in 10, place the zero (0) as a ten thousand digit and takes one.
-Then 1 times 6 plus neighbor 7 results 13 plus a 1 that had results 14, the 4 as a hundred-thousand digit and takes 1.
-Finally, 1 is multiplied by the zero that was added at the beginning, giving zero plus the neighbor 6 plus one that was carried. It turns out finally 7 for the digit corresponding to the millions.
To multiply any number by 12:
-A zero is added at the beginning and another zero at the end of the figure to be multiplied.
-Each digit of the figure to be multiplied is doubled and added with its neighbor on the right.
-If the sum exceeds 10, one unit is added to the next duplication operation and sum with the neighbor.
Multiply 63247 by 12
0632470 x 12 =
-
758964
The details to arrive at this result, strictly following the stated rules, are shown in the following figure:
The method of multiplication by 12 can be extended to multiplication by 13, 14 through 19 simply by changing the rule of doubling by tripling for the case of thirteen, quadrupling for the case of 14 and so on until reaching 19.
-Add zeros to the beginning and end of the figure to multiply by 6.
-Add half of its neighbor to the right to each digit, but if the digit is odd add 5 additionally.
-Add zeros to the beginning and end of the number to multiply.
-Double each digit and add the lower whole half of the neighbor, but if the digit is odd additionally add 5.
-Multiply 3412 by 7
-The result is 23884. To apply the rules, it is recommended to first recognize the odd digits and place a small 5 above them to remember to add this figure to the result..
-Add zeros to the beginning and end of the number to multiply.
-Place under each digit the lower whole half of the neighbor to the right, but if the digit is odd, add additionally 5.
Multiply 256413 by 5
-A zero is added at the beginning and another at the end of the figure to be multiplied by nine.
-The first digit to the right is obtained by subtracting the corresponding digit of the number to multiply from 10.
-Then the next digit is subtracted from 9 and the neighbor is added.
-The previous step is repeated until we reach the zero of the multiplicand, where we subtract 1 from the neighbor and the result is copied below zero..
Multiply 8769 by 9:
087690 x 9 =
--
78921
Operations
10 - 9 = 1
(9-6) + 9 = 1two (the two and takes 1)
(9-7) + 1 + 6 =9
(9-8) +7 =8
(8-1) = 7
-Add zeros to the beginning and end of the number to multiply.
-For the first digit from the right subtract from 10 and the result is doubled.
-For the following digits subtract from 9, the result is doubled and the neighbor is added.
-When reaching zero, subtract 2 from the neighbor on the right.
-Multiply 789 by 8
-Add zeros to the right and left of the multiplicand.
-Subtract the corresponding digit of the unit from 10 by adding 5 if it is an odd digit.
-Subtract from 9 in the form each digit of the multiplicand, adding half of the neighbor to the right and if it is an odd digit add 5 additionally.
-When reaching the zero of the beginning of the multiplicand, place half of the neighbor minus one.
Multiply 365187 x 4
-Add zero to each end of the multiplicand.
-Subtract 10 minus the ones digit and add 5 if it is an odd digit.
-For the other digits, subtract 9, double the result, add half the neighbor and add 5 if it is odd..
-When you reach the zero of the header, place the lower integer half of the neighbor minus 2.
Multiply 2588 by 3
-Add zeros at the ends and double each digit, if it exceeds 10 add one to the next.
Multiply 2374 by 2
023740 x 2
04748
The rules listed above apply, but the results are run to the left by the number of places corresponding to tens, hundreds, and so on. Let's look at the following example:
Multiply 37654 by 498
0376540 x 498
301232 ruler for 8
338886 rule for 9
150616 ruler for 4
18751692 final sum
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