Trachtenberg-System

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. The rest of this article presents some methods devised by Trachtenberg. These are for illustration only. To actually learn the method requires practice. Students begin learning the Trachtenberg system using multiplication algorithms. These initial algorithms are discussed first followed by a more general method for multiplication.

Beginning to Multiply Using the Trachtenberg Method

When performing any of these multiplication algorithms the following "steps" should be applied.

Write a zero before the number to be multiplied.

The answer must be found one digit at a time starting at the least significant digit ie the digit to the far right. Move left one digit at the time to work. The last calculation is on the leading zero of the multiplicand. ( In 0128 x 6, the 0128 is the multiplicand and the 6 is the multiplier. In 0382 x 28, the 0382 is the multiplicand and the 28 is the multiplier).

For each digit in the multiplicand, the digit immediately to the digit's right is this digit's neighbor. The rightmost digit's neighbor is zero because there is no neighbor.

The 'halve' operation has a particular meaning to the Trachtenberg system. If a digit is even take half this value. If a digit is odd, mentally subtract one before taking half the value. Alternatively half the odd digit may be taken discarding any decimal, fraction or remainder. For speed reasons people following the Trachtenberg system are encouraged to make this halving process instantaneous. Instead of thinking "half of seven is three and a half, so three" it's suggested that one thinks "seven, three". This speeds up calculation considerably.

The Trachtenberg system also uses something called the tens complement. In this same way the tables for subtracting digits from 10 or 9 are to be memorized. This requires subtracting digits from nine except the last digit to the right of a number which is subtracted from 10. The suggestion is to be able to look at the digits 0-9 and immediately know the result if this digit were subtracted from 9 or subtracted from 10.

Whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd.

There will be more discussion on the reasons and details of these methods below.

Multiplying by 10

It is easier to simply add a final zero to multiply by 10, however it is important to understand also how to multiply by 10 using a Trachtenberg type of technique. This technique will be used when Trachtenberg multiplies by 11, 12, 8, and 9.

We will use the multiplicand ab as an example. In the Trachtenberg method place a leading zero on the multiplicand. We will write the answer just below the multiplicand. Think of 0ab and the answer below it in columns. You will do a procedure to a digit in the multiplicand and this will give the digit in the answer directly below this digit in the multiplicand. The first column considered is that on the far right. In this case this column has a "b" in the multiplicand. The process proceeds from right to left.

 0ab x 10 =
 ab0

Rule: For each digit in the multiplicand take zero times that digit and add the neighbor to the right.

(Carry as you normally would in addition problems to the next column to the left if the value exceeds 9 ex if the answer is 10 write down the zero and carry the one in the next column to the left.)

In the first column on the right, start with zero x b. There is no neighbor to the right of b. Write down zero underneath the b in the answer.

In the second column from the right, use zero times a. The neighbor to the right is b. 0+b=b so write b in this position in the answer.

In the third column from the right, which is the leading zero, 0x0 x 0. The neighbor to the right is a. o+a=a , so write a as the answer in this position.

Notice we have made a big deal in that its zero times the digit first. Why? Later the multiplier for each digit may not be zero so we are making sure to be clear that for this case it is in fact zero.

Taking one times the digit in the multiplicand and also following the multiplication by 10 rule i. e. adding the number to the right would result in multiplying by (1+10). This would be 11 times the multiplicand.

Doubling each original digit then following the rule for multiplying by 10 would be equivalent to multiplying by (2+10). This would be 12 times the multiplicand.

It is possible to triple the factor and then follow the 10 rule to multiply by 13, and quadruple the factor and follow the 10 rule to multiply by 14.

Just realize that adding the neighboring digit to the right means adding 10 times your original multiplicand to your answer.

Multiplying by 11

Rule: Add the digit to its neighbor. (By "neighbor" we mean the digit on the right.)

Example: $3,425\times 11=37,675$

3 7 6 7 5

(= 0 + 3) (= 3 + 4) (= 4 + 2) (= 2 + 5) (= 5 + 0)

To illustrate:

$11=10+1$

Thus,

$3425\times 11=3425\times (10+1)$

$\Rightarrow 37675=34250+3425$

Multiplying by 12

Rule: to multiply by 12:
Starting from the rightmost digit, double each digit and add the neighbor. (The "neighbor" is the digit on the right.)

If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the next operation. The remaining digit is one digit of the final result.

Example: $316\times 12$

Determine neighbors in the multiplicand 0316:

• digit 6 has no right neighbor
• digit 1 has neighbor 6
• digit 3 has neighbor 1
• digit 0 (the prefixed zero) has neighbor 3

${\begin{aligned}6\times 2&=12{\text{ (2 carry 1) }}\\1\times 2+6+1&=9\\3\times 2+1&=7\\0\times 2+3&=3\\0\times 2+0&=0\\[10pt]316\times 12&=3,792\end{aligned}}$

Multiplying by 5

• Rule: to multiply by 5: Take half of the neighbor, then, if the current digit is odd, add 5.

An odd number = (1 + an even number). 5 x an odd number = 5 x (1+ an even number ) = (5x1) + (5 x the even number) = 5 + (10 x (1/2) x an even number) .

An even number = (0 + an even number). 5 times an even number = 5 x ( 0 + an even number) = (5x0) + (5 x the even number) = 0 + (10 x (1/2) x the even number).

10 x (1/2 x an even number) means following the multiplication by 10 rule for 1/2 of each even number portion of each digit in the multiplicand. 10 times any number results in a zero in the ones place with a number carried to the column to the left. This is handled in the Trachtenberg method by waiting to deal with the even portion of a number until you move one column further left. Then you add half the neighbor to the right.

It seems to bother students to toss out some bit of a number as in discarding a remainder when taking half the number to the right. Remember however you counted that bit of a remainder when you added 5 for odd numbers. Discarding the remainder when you go on to the next column and divide the even portion of the right hand number by two only means not counting that additional one that makes the number odd in the right hand neighbor twice.

Example: 042×5=210

Half of 2's neighbor, the trailing zero, is 0.

Half of 4's neighbor is 1.

Half of the leading zero's neighbor is 2.

043×5 = 215

Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5.

Half of 4's neighbor is 1.

Half of the leading zero's neighbor is 2.

093×5=465

Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5.

Half of 9's neighbor is 1, plus 5 because 9 is odd, is 6.

Half of the leading zero's neighbor is 4.

Multiplying by 6

• Rule: to multiply by 6: Take the digit, add 5 if the digit is odd, then add half of the neighbor to each digit, .

6= 1 + 5

Use digit itself i. e. 1 times the digit, and add in the multiplication by 5 rule.

Example:

0357 × 6 = 2142

Working right to left,

7 has no neighbor, add 5 (since 7 is odd) = 12. Write 2, carry the 1.

5 + half of 7 (3) + 5 (since the starting digit 5 is odd) + 1 (carried) = 14. Write 4, carry the 1.

3 + half of 5 (2) + 5 (since 3 is odd) + 1 (carried) = 11. Write 1, carry 1.

0 + half of 3 (1) + 1 (carried) = 2. Write 2.

Multiplying by 7

7 = 2 + 5

Double each digit and also add in the multiplication by 5 rule.

Rule: to multiply by 7:

1. Double each digit.
2. Add half of its neighbor.
3. If the digit is odd, add 5.

Example: 0523 × 7 = 3,661.

3×2 + 0 + 5 = 11, 1.

2×2 + 1 + 1 = 6.

5×2 + 1 + 5 = 16, 6.

0×2 + 2 + 1 = 3.

3661.

Multiplying by 9

Multiplying by 9 is equivalent to multiplying by (-1 + 10).

This method also relies on the fact that adding a number to a problem then subtracting that same number results in no net change to the problem.

Lets multiply qrstx9 where q,r,s,and t are each digits in a 4 digit number. A leading zero is placed in front of qrst giving us 0qrst. Always remember to place a leading zero in front of the multiplicand (the number you are going to multiply).

0qrstx9 = (-1 x 0qrst)+ (10 x 0qrst)

How will you work with -1 x 0qrst? Try adding a multiple of 10 that is just larger than qrst ie a one followed by as many zeros as there are digits in your multiplicand excluding its leading zero. In this case this means we will use 10000 as qrst is a 4 digit number. If you add in 10000 this will change the problem unless you subtract this back off later.

        10000     
-  1 x  0qrst    
+ 10 x  0qrst 
       -10000
      _______
        9 x 0qrst 

In the initial step, you can see why each digit in qrst is subtracted from 9 except the right most digit which is subtracted from 10. This process completes 10000 - (1xqrst). This step is called finding the "tens complement". This procedure is not unique to the Trachtenberg method. See "Method of Complements" in Wikipedia.

Adding the digit to the right in the multiplicand 0qrst is equivalent to multiplying by ten and adding this to your answer.

Finally when working under the leading zero of the factor 0qrst, DO NOT try to create some sort of tens complement for this zero. Remember this is just a place holder and not part of your original multiplicand. Add the neighbor ie q of qrst minus one. You must remember to use q as a neighbor, otherwise you havent completely multiplied qrst by 10. Subtracting 1 in this final step is the equivalent of subtracting 10000. You added 10000 to perform the problem. It must be removed or the problem will be incorrect.

Your digits in your answer should be (and please read these right to left):

0qrst x 9 = q-1___(9-q)+r___(9-r)+s____(9-s)+t____(10-t)

Editors note: Read the underlines as separating the different digits.

Rule:

1. Subtract the right-most digit from 10.
   1. Subtract the remaining digits from 9.
2. Add the neighbor.
3. For the leading zero, subtract 1 from the neighbor.

For rules 9, 8, 4, and 3 only the first digit ( the digit to the far right ) is subtracted from 10. After that each digit is subtracted from nine instead.

Example: 02,130 × 9 = 19,170

Working from right to left:

• (10 − 0) + 0 = 10. Write 0, carry 1.
• (9 − 3) + 0 + 1 (carried) = 7. Write 7.
• (9 − 1) + 3 = 11. Write 1, carry 1.
• (9 − 2) + 1 + 1 (carried) = 9. Write 9.
• 2 − 1 = 1. Write 1.

Multiplying by 8

Rule:

1. Subtract right-most digit from 10.
   1. Subtract the remaining digits from 9.
2. Double the result.
3. Add the neighbor.
4. For the leading zero, subtract 2 from the neighbor.

To understand what is going on here, understand that 8 = -2+10.

Again let's use qrst as an example 4 digit number.

 -  1 x  0qrst           
 -  1 x  0qrst    
 + 10 x  0qrst 
    __________
     8 x 0qrst 

How do you work with - 0qrst? You work with the tens complement. This means adding in a multiple of 10 with as many zeros as digits in your original multiplicand i. e. 4 for 0qrst. Since your working with -0qrst twice you must add this multiple of 10 twice. This is only to make the working easier. You must remove these later if your answer is to remain correct.

This is equivalent to:

 +       10000
 -  1 x  0qrst
 +       10000    
 -  1 x  0qrst    
 + 10 x  0qrst
        -20000           
       __________
        8 x 0qrst 

This simplifies to:

 2 x ( 10000 -  0qrst )
 + 10 x    0qrst
 - 20000
       __________
        8 x 0qrst 

So you find the 10s complement of a digit and double it. Add the neighbor to the right i. e. use the multiply by 10 procedure and add that. When working under the leading zero of the multiplicand, get the neighboring digit to the right in the multiplicand and subtract 2. This step subtracts the 20000 you added to work the problem. Use this neighbor to the right minus 2 as your left most digit in your answer.

Your digits in your answer should be (and please read these right to left):

0qrst x 8 = q-2 ____ {2 x (9-q)}+r ____ {2 x (9-r)}+s ____ {2 x (9-s)}+t ____ {2x(10-t)}

Editors note: Read the underlines as separating the different digits.

Example: 456 × 8 = 3648

Working from right to left:

• (10 − 6) × 2 + 0 = 8. Write 8.
• (9 − 5) × 2 + 6 = 14, Write 4, carry 1.
• (9 − 4) × 2 + 5 + 1 (carried) = 16. Write 6, carry 1.
• 4 − 2 + 1 (carried) = 3. Write 3.

Multiplying by 4

4 = -1 + 5

-1 times a number means in the Trachtenberg method a tens complement is used.  
5 times the number should suggest to you the multiply by 5s rule. 

 -1 X 0qrst
 +5 x 0qrst
 ___________
 4 x 0qrst

Your working this as

      10000
 -    0qrst
 +5 x 0qrst
 -    10000
 _______________
  4 x  0qrst  

The 10000 - 0qrst is the tens complement. Subtract each digit from nine unless its the digit to the far right. For the digit to the far right subtract it from 10. The leading zero in 0qrst is not used to find the tens complement. In other words, this is the same tens complement procedure you have been using.

Add to your answer i. e. to your tens complement, the multiplication by 5 procedure. Remember you add 5 if the number is odd and zero if even. Add half the number to the right discarding any remainder or fraction.

While working under the leading zero of your factor decrease your answer for half the neighbor by 1. You added 10000 to make the work easier. You must subtract that back out at the end or you will have changed the problem and end up with a wrong answer.

Your digits in your answer should be (and please read these right to left):

0qrst x 4 = (q-1) __(9-q) +5 if q is odd + (r/2) __ (9-r) +5 if r is odd + (s/2) __ (9-s) +5 if s is odd + (t/2) __ (10-t)+ 5 if t is odd

Remember in set of equations for the digits in this example when a number divided by 2 is written, discard any remainder or fraction.

Rule:

1. Subtract the right-most digit from 10.
   1. Subtract the remaining digits from 9.
2. Add half of the neighbor, plus 5 if the digit is odd.
3. For the leading 0, subtract 1 from half of the neighbor.

Example: 346 * 4 = 1384

Working from right to left:

• (10 − 6) + Half of 0 (0) = 4. Write 4.
• (9 − 4) + Half of 6 (3) = 8. Write 8.
• (9 − 3) + Half of 4 (2) + 5 (since 3 is odd) = 13. Write 3, carry 1.
• Half of 3 (1) − 1 + 1 (carried) = 1. Write 1.

Multiplying by 3

The general idea for multiplying by 3 is that 3 = -2 + 5.

 - 2 x   0qrst 
 + 5 x   0qrst 
 __________________
  3  x   0qrst

Work this as follows:

 2 x ( 10000
     - 0qrst)
+ 5 x 0qrst
-     20000
__________________
  3  x   0qrst

Find the tens complement and double it. Then follow the 5s rule. Remember that doubling the tens complement means you added 10000 twice to make the math easier. You must subtract that when calculating from the leading zero in 0qrst. This means you must subtract 2 from that leading number in your answer before you record it. This is usually expressed as taking 2 from half the neighbor to the right when calculating that last digit.

Your digits in your answer should be (and please read these right to left):

0qrst x 3 = (q-2) __2x(9-q)+5 if q is odd +(r/2) __ 2x(9-r) +5 if r is odd+(s/2) __ 2x(9-s)+5 if s is odd + (t/2) __ 2x(10-t)+ 5 if t is odd

Remember in set of equations for the digits in this example when a number divided by 2 is written, discard any remainder or fraction.

Rule:

1. Subtract the rightmost digit from 10.
   1. Subtract the remaining digits from 9.
2. Double the result.
3. Add half of the neighbor, plus 5 if the digit is odd.
4. For the leading zero, subtract 2 from half of the neighbor.

Example: 492 × 3 = 1476

Working from right to left:

• (10 − 2) × 2 + Half of 0 (0) = 16. Write 6, carry 1.
• (9 − 9) × 2 + Half of 2 (1) + 5 (since 9 is odd) + 1 (carried) = 7. Write 7.
• (9 − 4) × 2 + Half of 9 (4) = 14. Write 4, carry 1.
• Half of 4 (2) − 2 + 1 (carried) = 1. Write 1.

Multiplying by 2

• Rule: to multiply by 2, double each digit.

Further Note: If you really understand why you are doing the steps, you can create new procedures for yourself such as for multiplying by 15 by using the 10 rule plus the 5 rule ie add 5 if the number is odd and add one and a half times the neighbor discarding any fraction or remainder.