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 engineer Jakow Trachtenberg in order to keep his mind occupied while being in a Nazi concentration camp. The rest of this article presents some methods devised by Trachtenberg. Some of the algorithms Trachtenberg developed are ones for general multiplication, division and addition. Also, the Trachtenberg system includes some specialised methods for multiplying small numbers between 5 and 13 (but shown here is 2 -12). The section on addition demonstrates an effective method of checking calculations that can also be applied to multiplication. General multiplication, Trachtenberg math theory. The method for general multiplication is a method to achieve multiplications formula_1 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 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  formula_2 times the next-to-last digit of formula_3, as well as the next-to-last digit of formula_2 times the last digit of formula_3. This calculation is performed, and we have a temporary result that is correct in the final two digits. In general, for each position formula_6 in the final result, we sum for all formula_7: People can learn this algorithm and thus multiply four digit numbers in their head – writing down only the final result. They would write it out starting with the rightmost digit and finishing with the leftmost. Trachtenberg defined this algorithm with a kind  of pairwise multiplication where two digits are multiplied by one 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: formula_9 To find the first (rightmost) digit of the answer, start at the first digit of the multiplicand: To find the second digit of the answer, start at the second digit of the multiplicand: To find the third digit of the answer, start at the third digit of the multiplicand: To find the fourth digit of the answer, start at the fourth  digit of the multiplicand: Continue with the same method to obtain the remaining digits. 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 nine 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. Each arrow head points to a UT Pair, or Product Pair. The vertical arrow points to the product where we will get the Units digit, and the sloping  arrow points to the product where we will get the Tens digits of the Product Pair. 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 until all of the arrows point to prefixed zeros. 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. 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 both removes the risk of repeating any original errors and identifies the precise column in which an error occurs all at once. It is based on check (or digit) sums, such as the nines-remainder method. 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. In this same way the tables for subtracting digits from 10 or 9 are to be memorized. And 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 2. Rule: Multiply each digit  by 2 (with carrying). Example: 8624 x 2 Working from left to right: Example:76892 x 2 Working from left to right: Multiplying by 3. Rule: Example: 492 × 3 = 1476 Working from right to left: Multiplying by 4. Rule: Example: 346×4 = 1384 Working from right to left: Example: Multiplying by 5. 42×5=210 Multiplying by 6. Rule: Example: 357 × 6 = 2142 Working right to left: Multiplying by 7. Rule: Example: 523 × 7 = 3,661 Working from right to left: Multiplying by 8. Rule: Example: 456 × 8 = 3648 Working from right to left: Multiplying by  9. Rules: For rules 9, 8, 4, and 3 only the first digit is subtracted from 10. After that each digit is subtracted from nine instead. Example: 2,130 × 9 = 19,170 Working from right to left: Multiplying by 11. Rule: Example: formula_36 To illustrate: Thus, 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: formula_39 Determine neighbors in the multiplicand 0316: Publications. 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 does not specify, despite the chapter on algebraic description. In popular culture. The 2017 American film "Gifted" revolves around a child prodigy who at the age of 7 impresses her teacher by doing calculations in her head using the Trachtenberg  system. Other systems. There are many other methods of calculation in mental mathematics. The list below shows a few other methods of calculating, though they may not be entirely mental. Software. Following are known programs and sources available as teaching tools Web iPhone Android