Trachtenberg Speed System

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The Trachtenberg Speed System of Basic Mathematics

This constitutes a system of performing high-speed multiplication, division, addition, subtraction, and square root, all in one's head. The details are given in this reference:

Ann Cutler and Rudolph McShane, The Trachtenberg Speed System of Basic Mathematics, (Garden City, NY: Doubleday & Company, Inc., 1960); reprint (Westport, CT: Greenwood Publishing Group, 1983) [specific reference here].

While it is impossible to capture the contents of this 270-page book in one Web page, below you will find a part of the content of this system. For information about the life of Professor Jakow Trachtenberg, Founder of the Mathematical Institute in Zurich, see Trachtenberg Speed Math and Trachtenberg Math.

Multiplication

Finding the product of multiplicands of any length by small multipliers is accomplished using a set of rules. One works from right to left from digit to digit of the multiplicand, writing down the digits of the product. The rules use only addition, subtraction, doubling, and halving.

Here are the rules:

Multiplier  Rules
0Zero times any number at all is zero.
1Copy down the multiplicand unchanged.
2Double each digit of the multiplicand.
3
 First step: subtract from 10 and double, and add 5 if the number is odd. Middle steps: subtract from 9 and double, and add half the neighbor, plus 5 if the number is odd. Last step: take half the lefthand digit of the multiplicand and reduce by 2.
4
 First step: subtract from 10, and add 5 if the number is odd. Middle steps: subtract from 9 and add half the neighbor, plus 5 if the number is odd. Last step: take half the lefthand digit of the multiplicand and reduce by 1.
5Use half the neighbor, plus 5 if the number is odd.
6Use the number plus half the neighbor, plus five if the number is odd.
7Use double the number plus half the neighbor, plus five if the number is odd.
8
 First step: subtract from 10 and double. Middle steps: subtract from 9, double, and add the neighbor. Last step: Reduce the lefthand digit of the multiplicand by 2.
9
 First step: subtract from 10. Middle steps: subtract from 9 and add the neighbor. Last step: reduce the lefthand digit of the multiplicand by 1.
10Use the neighbor.
11Add the neighbor to the number.
12Double the number and add the neighbor.

When the sum exceeds 10, write down the last digit, and carry the first digit. If you are taking half an odd number, use the integer quotient and ignore the remainder of 1.

Examples:

1. Multiply 901247 by 2.

0 + 7 * 2 = 14. Write down 4, carry 1.
1 + 4 * 2 = 9. Write down 9.
0 + 2 * 2 = 4. Write down 4.
0 + 1 * 2 = 2. Write down 2.
0 + 0 * 2 = 0. Write down 0.
0 + 9 * 2 = 18. Write down 8, carry 1.
1 + 0 * 2 = 1. Write down 1.

Answer: 901247 * 2 = 1802494.

2. Multiply 901247 by 3.

0 + (10 - 7) * 2 = 6. 6 + 5 = 11. Write down 1, carry 1.
1 + (9 - 4) * 2 = 11. 11 + [7/2] + 0 = 14. Write down 4, carry 1.
1 + (9 - 2) * 2 = 15. 15 + [4/2] + 0 = 17. Write down 7, carry 1.
1 + (9 - 1) * 2 = 17. 17 + [2/2] + 5 = 23. Write down 3, carry 2.
2 + (9 - 0) * 2 = 20. 20 + [1/2] + 0 = 20. Write down 0, carry 2.
2 + (9 - 9) * 2 = 2. 2 + [0/2] + 5 = 7. Write down 7.
[9/2] - 2 = 2. Write down 2.

Answer: 901247 * 3 = 2703741.

These rules allow one to dispense with memorizing multiplication tables, if that is desired. Even better, it gives a way to help memorize them, by allowing one to work out the answer by rule if one cannot remember it by rote.

Once the multiplication tables are memorized, using multipliers of two digits is learned. Finally, using multipliers of any length is learned by the "two-finger" method.

The method advocated here involves the following principle: "Never count higher than eleven." If any running column total exceeds 11, subtract 11 and put a tick mark in that column. When you reach the bottom, write down the running total, and under it write the number of tick marks. Now add these two rows using the strange rule of adding the two numbers in any column and the neighbor tick number. Write down the last digit and carry the other digit, if any, working right-to-left.

Example

```      3  6  8  9
7' 5' 8'
9' 6  6  7'  column
1  0  6' 4   of
6  4' 9' 8'  figures
----------
8  1  1  3   running totals
1  2  3  3   ticks
----------
2 '1  6  7  6   sum
```
This technique works on arbitrarily long columns of figures, and the columns can be dealt with in any order desired, except at the very last step.

Division and Square Roots

The methods for these operations are too long to discuss here, although less complicated than the traditional ones and more adapted for speed and mental calculation.

Checking

One aspect of calculation which is advocated throughout this book is the importance of checking the correctness of one's work. The method of Casting Out Nines to check a result is used. For more about casting out nines, see the Dr. Math archives:

- Robert L. Ward, for the Math Forum

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