Trachtenberg system

The Trachtenberg System is a system of rapid mental calculation, somewhat similar to Vedic mathematics. It was developed by the Ukrainian engineer Jakow Trachtenberg in order to keep his mind occupied while being held in a Nazi concentration camp.

The system consists of a number of readily memorized patterns that allow one to perform arithmetic computations very quickly.

The rest of this article presents some of the methods devised by Trachtenberg. These are for illustration only. To actually learn the method requires practice and a more complete treatment.

The most important algorithms are the ones for general multiplication, division and addition. In addition, the method includes some specialized methods for multiplying small numbers between 5 and 13.

General multiplication

The method for general multiplication is a method to achieve multiplication of a*b with low space complexity, i.e. as few temporary results as possible to be kept in memory. This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplicands. This is a held as a temporary result. To find the next to last digit, we need everything that influences this digit: The temporary result, the last digit of a times the next-to-last digit of b, as well as the next-to-last digit of a times the last digit of b. This calculation is performed, and we have a temporary result that is correct in the final two digits.

In general, for each position n in the final result, we sum for all i:a(digit at i)*b(digit at(n-i)). Ordinary people can learn this algorithm and multiply 4 digit numbers in their heads, writing down the final result, with the last digit first.

Trachtenberg defined this algorithm with a kind of pairwise multiplication where 2 digits are multiplied by 1 digit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held.

General division

Builds upon the multiplication method

General addition

A method to sum columns of numbers. Creates an intermediate sum, in the form of two rows of digits. The final step is to sum the intermediate results with an L-shaped algorithm

Other multiplication algorithms

When performing any of these multiplication algorithms, the multiplier should have as many zeros prefixed to it as there are digits in the multiplicand. This will provide room for any carrying operations. For instance, when multiplying 366 × 7, add one zero to the front of 366 (write it "0366"); when multiplying 985 × 12, prefix two zeroes to 985 ("00985").

Each digit but the last, including the prefixed zeros, has a "neighbor", i.e., the digit on its right.

Multiplying by 12

Rule: to multiply by 12:
Starting from the rightmost digit,double each digit and add the neighbor. (By "neighbour" we mean the digit on the right.)

This gives one digit of the result. If the answer is greater than 1 digit simplycarry over the 1 or 2 to the next operation.
Example: 316 × 12 = 3,792:
In this example:
* the last digit 6 has no neighbor.
* the 6 is neighbor to the 1.
* the 1 is neighbor to the 3.
* the 3 is neighbor to the second prefixed zero.
* the second prefixed zero is neighbor to the first.
6 × 2 = 12 (2 carry 1)

1 × 2 + 6 + 1 = 9

3 × 2 + 1 = 7

0 × 2 + 3 = 3
0 × 2 + 0 = 0

Multiplying by 11

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

Example: 3,425 × 11 = 37,675

0 3 4 2 5 x 11=

3 7 6 7 5

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

Proof:
11=10+1

Thus,

3425 x 11 = 3425 x(10+1)
= 34250 + 3425

Multiplying by other numbers

The 'halve' operation has a particular meaning to the Trachtenberg system. It is intended to mean "half the digit, rounded down" but for speed reasons people following the Trachtenberg system are encouraged to make this halving process instantaneous. So 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.

In the same way the tables for subtracting digits from 10 or 9 are to be memorized.

Multiplying by 5

*Rule: to multiply by 5:
*#Take half of the neighbor
*#Add 5 if number is oddExample:42x5=210
4=2,2=1
43x5=2

Multiplying by 6

*Rule: to multiply by 6:
*#Add half of the neighbor to each digit.
*#If the starting digit is odd, add 5.

Example:
6 × 357 = 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

*Rule: to multiply by 7:
*#Double each digit.
*#Add half of its neighbor.
*#If the digit is odd, add 5.

Multiplying by 8

*Rule: to multiply by 8:
*#Subtract last digit from 10 and double
*#Subtract the other digits from 9 and double
*#Add result to the neighboring digit on the right.
*#For the last calculation (The leading Zero), subtract 2 from the neighbour.

Multiplying by 9

*Rule: to multiply by 9:
*#Subtract the last digit from 10.
*#For each of the remaining digits, subtract it from 9, and add its neighbor.
*#Finally, prepend one less than the first digit.

*Example: 2,130 × 9 = 19,170
*#10 - 0 = 10. Write 0; carry 1.
*#9 - 3 = 6; 6 + 0 + 1 (carried) = 7. Write 7.
*#9 - 1 = 8; 8 + 3 = 11. Write 1; carry 1.
*#9 - 2 = 7; 7 + 1 + 1 (carried) = 9. Write 9.
*#2 - 1 = 1. Write 1.

Book

"The Trachtenberg Speed System of Basic Mathematics"by Jakow Trachtenberg, A. Cutler (Translator), R. McShane (Translator), Rudolph Mcshane (Translator) was published by Doubleday and Company, Inc. Garden City, New York in 1960. [cite book
title= The Trachtenberg Speed System of Basic Mathematics
last=Trachtenberg
first=Jakow
coauthors(translator)=A. Cutler, R. McShane
year=1960
pages=270
publisher=Doubleday and Company, Inc.]

There are specific algebra rules for each of the above to be found in the complete book.

All of this information is from the original book.

The book's copyright is by Ann Cutler, a journalist in New York City at the time. The other person involved in the translation to English, Rudolph McShane, is a Mathematician who lived in New Orleans at the time of publication and also worked on restricted USA government projects. [All of this information is from a original book published and printed in 1960. The original book has seven full Chapters and is exactly 270 pages. The Chapter Titles are as follows (chapter sub-categories not listed with each, they are too numerous).] [Trachtenberg Speed System of Basic Mathematics
______________________________________________

Chapter 1 Tables or No Tables
Chapter 2 Rapid Multiplications by the Direct Method
Chapter 3 Speed Multiplication-"'TWO-FINGER" Method
Chapter 4 Addition And The Right Answer
Chapter 5 Division Speed And Accuracy
Chapter 6 Squares and Square Roots
Chapter 7 Algebraic Description of the Method

"The revolutionary new method for high-speed multiplication , division, addition, subtraction and square root."
Multiplication is done without multiplication tables"Can you multiply 5132437201 times 4522736502785 in seventy seconds?""One young boy (grammar school-no calculator) did--successfully--by using The Trachtenberg Speed System of Basic Mathematics"
Jakow Trachtenberg (its founder) escaped from Hitlers Germany from anactive institution toward the close of WWII. Professor Trachtenberg fledto Germany when the czarist regime was overthrown in his homeland of Zurich Switzerland and lived there peacefully until his mid-thirties whenhis anti-Hitler attitudes forced him to flee again. He was a fugitive andwhen captured spent a total of seven years in various concentration camps. It was during these years that Professor Tractenberg devised the system of speed mathematics. Most of his work was done without pen or paper. Therefore most of the techniques can be performed mentally.]

References