Trachtenberg system - High speed calculation
(This blog is a work in progress, which means that the content will be updated and changed frequently.)
The Trachtenberg system is a system of rapid mental calculation. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. It was developed by the Russian Jewish engineer Jakow Trachtenberg in order to keep his mind occupied while being held in a Nazi concentration camp. I won’t delve into the history much of this person, please feel free to check up the Wikipedia if you are interested.
While preparing for CAT I was browsing about speed calculation techniques and came across this supposedly legendary one. What intrigued me was the fact that of the very few persons who used it, gave rave reviews about the success and ease of use this method. So here I go, having gotten hold of the only book of this method I am starting my journey of this method. I will update and modify this blog as per my progress. Also note that most of the resources are taken from the book(Trachtenberg Speed System Of Basic Mathematics), please consider buying it if you find this technique interesting.
Basic Multiplication
Multiplication by 11
The simple rule goes like this “add the neighbor”, the demonstration will make things a lot easier:
Suppose we are to multiply 683 X 11:
Step 1: write the number as 0683 X 11
Step 2: Next go and add the neighbor, start from 3 and add the number right to it, so first case as there is nothing right to 3 the first number is 3
Step 3: 2nd number is 8+3=11, keep tens digit 1 in hand and put down 1 in the number so the number now reads 13 with 1 in hand.
Step 4: next 8+6=14 + 1 of hand=15, so number is 513 with again 1 in hand.
Step 5: 0+6=6+1=7, so the final solution is 7513.
As you must have understood by now the meaning of “add the neighbor”, you can try with bigger numbers if you want, but the result is always accurate.
Multiplication by 12
Similar to what we did in case of 11, the rule for 12 is: Double each number in turn and add its neighbor.
An example will demonstrate this better:
0 68942 0 X 12
Step 1: double 2 and add to 0 = 4
Step 2: double 4 and add to 2 = 10 → 0 will remain 1 will be in hand → 04
Step 3: double 9 and add to 4 = 22 + 1→ 3 will remain 2 will be in hand → 304
Step 4: double 8 and add to 9 = 25 + 2 → 7 will remain 2 will be in hand → 7304
Step 5: double 6 and add to 8 = 20 + 2 → 2 will remain 2 will be in hand → 27304
Step 6: double 0 and add to 6 = 6 + 2 → 8 will remain → 827304
There are also other methods for multiplying with all numbers from 1 to 9, but i don’t really feel that they are that necessary, so instead we will move on to multiplying bigger numbers.
There are 2 ways of doing it, the first method is called the direct method:
Let us multiply 123 X 56
Step 1: Here 123 is the multiplicand and 56 is the multiplier, The multiplier has 2 digits, so we write 2 zeroes in front of the multiplicand
00123 X 56
Step 2: 1 8 → 6 X 3 = 18
Step 3: 288 → 2 X6 + 3 X 5 + 1 = 28 , here we multiply the inside(2X6) and outside pairs(3X5) and add
00123 X 56
Step 4: 1888 → 1 X 6 + 2 X 5 + 2 = 18 , here we multiply the inside(1X6) and outside pairs(2X5) and add
Step 5: In the last step we add 1 X 5 + 1 = 6888 which is our final answer
Next suppose let us multiply 302 X 114
Now for one of the easiest methods to check if the product done is correct, this method does not necessarily belong to the Trachtenberg system, the rule this is:
The digit-sum of the product should be equal to the digit-sum of the product of the digit-sums of the multiplier and multiplicand.
204 X 31 = 6324
(Sum of digits of 204 = 6 X Sum of digits of 31 = 4) = Sum of digits of 6324 = 15
6 X 4 = 24 → 6 = 1+5 → 6, so the answer is correct.
The second method is called the unit ten method or the two finger method and it is more useful when the numbers are a bit big:
U
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T
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U
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T
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U
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U
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U
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U
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0
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0
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5
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2
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7
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3
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X
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5
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4
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2
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2
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unit digit of 3 X 4
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14
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8+5+1
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unit digit of 7 X 4
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unit digit of 3 X 5
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ten digit of 3 X 4
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17
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5+8+2+1+1
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unit digit of 7 X 5
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unit digit of 2 X 4
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ten digit of 7 X 4
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ten digit of 3 X 5
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4
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0+0+0+3+1
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unit digit of 5 X 4
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unit digit of 2 X 5
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ten digit of 2 X 4
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ten digit of 7 X 5
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8
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0+5+2+1
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unit digit of 0 X 4
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unit digit of 5 X 5
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ten digit of 5 X 4
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ten digit of 2 X 5
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2
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2
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unit digit of 0 X 4
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unit digit of 0 X 5
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ten digit of 0 X 4
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ten digit of 5 X 5
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Hence the answer of 54273 X 54 is 284742. It may seem a little complicated at first but once you get the hang of it, it is really fast. Now if you would have noted the above multiplication properly, you would have noticed that as we progress we take 2 rows of UT into account it is so because the multiplier contains 2 digits, now suppose what happens if the multiplier would have contained 3 digits, simple we would have taken into account 3 rows of UT, let us see another example on this:
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0
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0
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0
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2
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6
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3
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X
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1
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2
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7
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1
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Unit digit of 3 X7
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10
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Unit digit of 3 X2
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ten digit of 3 X7
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Unit digit of 6 X7
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14
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Unit digit of 6 X2
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ten digit of 3 X2
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ten digit of 6 X7
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Unit digit of 3 X1
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ten digit of 0X0
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Unit digit of 2 X7
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13
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Unit digit of 2 X2
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ten digit of 6 X2
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ten digit of 3 X1
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Unit digit of 6 X1
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ten digit of 2X7
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Unit digit of 0 X7
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3
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Unit digit of 0 X2
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ten digit of 2 X2
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ten digit of 6 X1
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Unit digit of 2 X1
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ten digit of 0X7
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Unit digit of 0 X7
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0
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Unit digit of 0 X2
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ten digit of 0 X2
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ten digit of 2 X1
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Unit digit of 0 X1
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ten digit of 0X7
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Unit digit of 0 X7
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3
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3
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4
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0
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1
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Final Answer
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Squaring of two digits number using Trachtenberg Method
For 2 digit numbers the Trachtenberg method is efficient but I personally feel for 3 digits the it is not worth the effort. So I will describe the method for 2 digits here:
Suppose we are to square 57, how do we do it:
572
Step 1: 5 * 5 (5 * 7)*2 7 * 7
Step 2: 25 70 49
Step 3: Then we collapse the number 2(5+7)(0+4)9
Step 4: 2+1 2 4 9 , here 1 is carried and added to 2
Step 5: 3249 which is the answer
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