## The Trachtenberg System of Speed Arithmetic

While he was in a German concentration camp, during World War II, Jakow Trachtenberg invented his system of speed arithmetic. His methods can be divided into two parts, (1) multiplication by small numbers (2 through 12), and (2) addition, multiplication, and division as other speed arithmetic experts do it (see Speed Arithmetic). All of this can be very helpful for any student, and has been very successfully used by students who have difficulty with numbers. In my opinion, the first part, multiplication by small numbers, is successful mainly because it is fun, not because it is a particularly fast method. I will show how Trachtenberg multiplied by 7, below. The one major exception to my observation (that these are not particularly fast) is multiplication by 11. For 11, Trachtenberg's method is a dream. You, and everyone else, should consider using it.

Multiplication by 11: Trachtenberg did this from right to left. It can be done from left to right. This is slightly harder (sometimes), but has the benefit of producing the answer from left to right, just as you would read off the answer. And so, it can be slightly faster, for that reason. Speed arithmetic experts do most of their arithmetic (including multiplying large numbers by large numbers) from left to right. I'll start with a simple example (with no carries):

4253 x 11 = 46783

Starting at the right, we write down the first digit (3). Then we add the first to the next digit (5) and write that down (8). Then we add the second digit to the third, etc. Finally we write down the leftmost (last) digit (4). In most cases, we have to deal with carries:

4683 x 11 = 51513

Starting at the right, we write down the first digit (3). Then we add the first pair of digits (8+3=11) and write down the right digit of this sum (1), and carry the one. We add the next pair of digits to the carry (6+8+1=15) and write down the right digit of this sum (5), and carry the one, etc. Finally, we write down the leftmost digit plus the carry.

A popular English language book on this subject is The Trachtenberg Speed System of Basic Mathematics translated and adapted by Ann Cutler and Rudolph McShane. Click on the name of the book (above) to go to that book on amazon.com.

Multiplication by 7: Going from right to left, we use this rule: Double each number and add half the neighbor (digit to the right, dropping any fraction); add 5 if the number (not the neighbor) is odd. And of course, we have to deal with carries:

3852 x 7 = 26964

Starting at the right (2), we double the first number (it has no neighbor) and write down the rightmost digit of that (4) and we have no carry. Then we double the next number (2x5=10), add five (+5=15), and add half the neighbor (+1=16), and write down the right digit (6) of that and carry the 1. Then we double the next number (2x8=16), and add half the neighbor (+2=18), and add the carry (+1=19). Then we double the next number (2x3=6), add five (+5=11), add half the neighbor (+4=15), and add the carry (+1=16). Now we double a zero off to the left of our 3852 (Trachtenberg wrote the zero out there: 03852) and add half the neighbor (0+1=1), and add the carry (+1=2). And we have our answer.

Notice that the carries are smaller than they were in normal multiplication by 7. The above rule is not simple, but once mastered, it is easy to use. It should be about as fast as multiplying normally (which requires memorizing the multiplication table). Multiplication by other small numbers (3 through 12) uses similar rules.