User:Netherstar/Draft

The Trachtenberg System  is a method 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. Without pencil or paper he developed a system of easily memorized operations that allows one to perform mental arithmetic computations very quickly. These methods have been successfully taught to children who had consistently failed at arithmetic. Not only did their mathematical ability improve but as they became more proficient in handling numbers their confidence in all areas of study improved.

The rest of this article presents some of the methods devised by Trachtenberg. The most important algorithms are the ones for general multiplication, division and addition. Also, the system includes some specialized methods for multiplying by small numbers between 3 and 12 that is no more involved than doubling or halving a one digit number. The chapter on addition demonstrates an effective method of checking calculations known as Casting out nines.

General multiplication
The method for general multiplication is to achieve multiplication of a*b with low space complexity, i.e. as few temporary results as possible to be kept in memory. By solving the equation in columns instead of rows we can sum the intermediate steps without having to write them down for later reference. In general, for each position n in the final result, we sum for all i = 0 to n (the right-most digit of each number is position 0): Units digit of the product b(digit at i) x a(digit at(n-i)) + Tens digit of the product b(digit at i) x a(digit at (n-i-1)). 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.

Example: 123456 x 789

To find the first digit of the answer: The units digit of 9 x 6 = 4.

To find the second digit of the answer, start at the second digit of the multiplicand: The units digit of 9 x 5 + the tens digit of 9 x 6, plus The units digit of 8 x 6. The second digit of the answer is 8 and carry 1 to the third digit.

To find the fourth digit of the answer, start at the fourth digit of the multiplicand: The units digit of 9 x 3 + the tens digit of 9 x 4, plus The units digit of 8 x 4 + the tens digit of 8 x 5, plus The units digit of 7 x 5 + the tens digit of 7 x 6. 7 + 3 + 2 + 4 + 5 + 4 = 25 + 1 carried from the third digit. The fourth digit of the answer is 6 and carry 2 to the next digit.

Professor Trachtenberg called this the 2 Finger Method. The calculations for finding the fourth digit from the example above are illustrated at right.The arrow from the right-most digit of the multiplier will always point to the digit of the multiplicand directly above the digit of the answer you wish to find, with the other arrows each pointing one digit to the right. If an arrow points to a space with no digit there is no calculation for that arrow. As you solve for each digit you will move each of the arrows over the multiplicand one digit to the left.

General division
Division in the Trachtenberg System is done much the same as in multiplication but with subtraction instead of addition. Splitting the dividend into smaller Partial Dividends, then dividing this Partial Dividend by only the left-most digit of the divisor will provide the answer one digit at a time. As you solve each digit of the answer you then subtract Product Pairs (UT pairs) and also NT pairs (Number-Tens) from the Partial Dividend to find the next Partial Dividend. The Product Pairs are found between the digits of the answer so far and the divisor. If a subtraction results in a negative number you have to back up one digit and reduce that digit of the answer by one. With enough practice this method can be done in your head. In the diagram at right begin by dropping the first number of the dividend down to the Partial Dividend row. The left digit of the divisor, 3, goes into 8 twice so 2 is the first digit of the answer. Then move up and right from the 8 by subtracting the NT pair to get 2. Drop down the 2 from the dividend to get 22. Then move down by subtracting UT pairs to get 19. In other words, 8 minus (2 from the answer times 3 from the divisor) minus (tens digit of 2 times 1) is the 2 placed on the Working Row. Dropping down the next digit from the Dividend makes the number in the Working Row 22. Then 22 minus(units digit of 2 times 1) minus(tens digit of 2 times 6) is the 19 placed on the Working Dividend row.

General addition
A method of adding columns of numbers and accurately checking the result without repeating the first operation. An intermediate sum, in the form of two rows of digits, is produced. The answer is obtained by taking the sum of the intermediate results with an L-shaped algorithm. As a final step, the checking method that is advocated removes both the risk of repeating any original errors and allows the precise column in which an error occurs to be identified at once. It is based on a check (or digit) sums, such as Casting out nines.

For the procedure to be effective, the different operations used in each stage must be kept distinct, otherwise there is a risk of interference.

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

The answer must be found one digit at a time starting at the least significant digit and moving left. The last calculation is on the leading zero of the multiplicand.

Each digit has a neighbor, i.e., the digit on its right. The rightmost digit's neighbor is the trailing zero.

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.

And remember that whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd. This makes up for dropping 0.5 in the next digit's calculation.

Multiplying by 11
Rule:
 * 1) Add the neighbor.

Example: 3,425 × 11 = 37,675

Working right to left:
 * 5 + 0 = 5. Write 5.
 * 2 + 5 = 7. Write 7.
 * 4 + 2 = 6. Write 6.
 * 3 + 4 = 7. Write 7.
 * 0 + 3 = 3. Write 3.

Multiplying by 12
Rule:
 * 1) Double each digit.
 * 2) Add the neighbor.

Example: 316 × 12 = 3792

Prefix two zeros to the multiplicand because the multiplier 12 has two digits → 00316 × 12

Working right to left:
 * (6 × 2) + 0 = 12. Write 2, carry 1.
 * (1 × 2) + 6 + 1 (carried) = 9. Write 9.
 * (3 × 2) + 1 = 7. Write 7.
 * (0 × 2) + 3 = 3. Write 3.

Multiplying by 6
Rule:
 * 1) Add half of the neighbor, plus 5 if the digit is odd.

Example: 347 × 6 = 2082

Working right to left:
 * 7 + half of 0 (0) + 5 (since 7 is odd) = 12. Write 2, carry 1.
 * 4 + half of 7 (3) + 1 (carried) = 8. Write 8.
 * 3 + half of 4 (2) + 5 (since 3 is odd) = 10. Write 0, carry 1.
 * 0 + half of 3 (1) + 1 (carried) = 2. Write 2.

Multiplying by 7
Rule:
 * 1) Double each digit.
 * 2) Add half of the neighbor, plus 5 if the digit is odd.

Example: 523 x 7 = 3,661.

Working right to left:
 * (3 x 2) + Half of 0 (0) + 5 (since 3 is odd) = 11. Write 1, carry 1.
 * (2 x 2) + Half of 3 (1) + 1 (carried) = 6. Write 6.
 * (5 x 2) + Half of 2 (1) + 5 (since 5 is odd) = 16. Write 6.
 * (0 x 2) + Half of 5 (2) + 1 (carried) = 3.

Multiplying by 9
Rule:
 * 1) Subtract the right-most digit from 10.
 * 2) Subtract the remaining digits from 9.
 * 3) Add the neighbor.
 * 4) For the leading zero, subtract 1 from the nieghbor.

Example: 2,130 × 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.
 * 2) Subtract the remaining digits from 9.
 * 3) Double the result.
 * 4) Add the neighbor.
 * 5) For the leading zero, subtract 2 from the neighbour.

Example: 456 x 8 = 3648

Working from right to left:
 * (10 - 6) x 2 + 0 = 8. Write 8.
 * (9 - 5) x 2 + 6 = 14, Write 4, carry 1.
 * (9 - 4) x 2 + 5 + 1 (carried) = 16. Write 6, carry 1.
 * 4 - 2 + 1 (carried) = 3. Write 3.

Multiplying by 4
Rule:
 * 1) Subtract the right-most digit from 10.
 * 2) Subtract the remaining digits from 9.
 * 3) Add half of the neighbor, plus 5 if the digit is odd.
 * 4) 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
Rule:
 * 1) Subtract the rightmost digit from 10.
 * 2) Subtract the remaining digits from 9.
 * 3) Double the result.
 * 4) Add half of the neighbor, plus 5 if the digit is odd.
 * 5) For the leading zero, subtract 2 from half of the neighbor.

Example: 492 x 3 = 1476

Working from right to left:
 * (10 - 2) x 2 + Half of 0 (0) = 16. Write 6, carry 1.
 * (9 - 9) x 2 + Half of 2 (1) + 5 (since 9 is odd) + 1 (carried) = 7. Write 7.
 * (9 - 4) x 2 + Half of 9 (4) = 14. Write 4, carry 1.
 * Half of 4 (2) - 2 + 1 (carried) = 1. Write 1.

Multiplying by 5
Rule:
 * 1) Take half of the neighbor, plus 5 if current digit is odd

Example: 42 x 5 = 210 Example: 743 x 5 = 3715
 * 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.
 * Half of 3's neighbor is 0, + 5 (since 3 is odd), is 5.
 * Half of 4's neighbor is 1.
 * Half of 7's neighbor is 2, + 5 (since 7 is odd), is 7.
 * Half of the leading zero's nieghbor is 3.

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. The book contains specific algebraic explanations for each of the above operations.

Most of the information in this article is from the original book.

The algorithms/operations for multiplication etc. can be expressed in other more compact ways that the book doesn't specify, despite the chapter on algebraic description.

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