Trachtenberg Method: The Speed Math System Explained

The Trachtenberg system is a method of fast arithmetic in which you multiply by working along a number from right to left and writing one digit of the answer at a time. The multipliers 5 through 9, 11 and 12 each have a short rule of their own, and a general method handles the rest, so there are no rows of partial products to write down and add up at the end.

This guide explains the rules for multiplying by 11, 12 and 5 through 9, and the general method for any two numbers. Each rule has a worked example and a short drill right after it, so you can try the rule while it is fresh.

Who Was Jakow Trachtenberg?#

Jakow Trachtenberg was born in Odessa in 1888 and worked as an engineer. During the Second World War he was held in prison camps, and he worked out his system of arithmetic there.

In 1945 he reached Switzerland, and he went on to found an institute in Zurich, where he taught his method to children and adults. Ann Cutler and Rudolph McShane translated and adapted his system into English as The Trachtenberg Speed System of Basic Mathematics, which Doubleday published in 1960. The book covers multiplication, addition, division, and squares and square roots. This guide covers the multiplication rules, which are where most people start.

How the Rules Work#

Every rule uses the same few ideas.

• The number is the number you are multiplying, such as 4,786 when you multiply it by 11.
• The neighbor of a digit is the digit just to its right. The last digit of the number has no neighbor.
• Picture a zero written in front of the number. The last step of every rule is worked under that zero in front.
• You work under each digit in turn, from right to left, starting with the last digit, and write one digit of the answer under each one.
• When a step adds up to 10 or more, write its last digit and carry the rest into the next step, just as in ordinary addition. Under the zero in front, write the whole total.

The rules for 5 through 9, 11 and 12 need no times tables, only small sums, doubling and taking half. The general method does use the times tables, since it multiplies pairs of digits. When a rule says to take half of an odd neighbor, drop the half that is left over, so half of 7 is 3.

Multiplying by 11#

To multiply by 11, add each digit to its neighbor.

Here is 4,786 × 11 = 52,646, worked one step at a time.

1. Under the 6: 6. Write 6.
2. Under the 8: 8 + 6 = 14. Write 4, carry 1.
3. Under the 7: 7 + 8 + 1 = 16 (1 carried). Write 6, carry 1.
4. Under the 4: 4 + 7 + 1 = 12 (1 carried). Write 2, carry 1.
5. Under the zero in front: 4 + 1 = 5 (1 carried). Write 5.

Reading the written digits from left to right gives 52,646.

Multiplying by 12#

To multiply by 12, double each digit and add its neighbor. It is the rule for 11 with one change, since 12 is one more than 11.

Here is 3,571 × 12 = 42,852.

1. Under the 1: 1×2 = 2. Write 2.
2. Under the 7: 7×2 + 1 = 15. Write 5, carry 1.
3. Under the 5: 5×2 + 7 + 1 = 18 (1 carried). Write 8, carry 1.
4. Under the 3: 3×2 + 5 + 1 = 12 (1 carried). Write 2, carry 1.
5. Under the zero in front: 3 + 1 = 4 (1 carried). Write 4.

Multiplying by 6#

To multiply by 6, add half the neighbor to each digit, and add 5 more if the digit is odd. Under the zero in front, that leaves just half of the first digit.

Here is 2,358 × 6 = 14,148.

1. Under the 8: 8. Write 8.
2. Under the 5: 5 + 4 + 5 = 14 (4 is half of 8 and 5 because 5 is odd). Write 4, carry 1.
3. Under the 3: 3 + 2 + 5 + 1 = 11 (2 is half of 5, 5 because 3 is odd and 1 carried). Write 1, carry 1.
4. Under the 2: 2 + 1 + 1 = 4 (1 is half of 3 and 1 carried). Write 4.
5. Under the zero in front: 1 (1 is half of 2). Write 1.

Multiplying by 7#

The rule for 7 is the rule for 6 with the digit doubled. Double each digit, add half the neighbor, and add 5 more if the digit is odd.

Here is 2,413 × 7 = 16,891.

1. Under the 3: 3×2 + 5 = 11 (5 because 3 is odd). Write 1, carry 1.
2. Under the 1: 1×2 + 1 + 5 + 1 = 9 (1 is half of 3, 5 because 1 is odd and 1 carried). Write 9.
3. Under the 4: 4×2 + 0 = 8 (0 is half of 1). Write 8.
4. Under the 2: 2×2 + 2 = 6 (2 is half of 4). Write 6.
5. Under the zero in front: 1 (1 is half of 2). Write 1.

Multiplying by 5#

The rule for 5 drops the digit altogether. Under each digit, take half the neighbor, and add 5 if the digit is odd.

Here is 3,746 × 5 = 18,730.

1. Under the 6: 0 (nothing to add). Write 0.
2. Under the 4: 3 (3 is half of 6). Write 3.
3. Under the 7: 2 + 5 = 7 (2 is half of 4 and 5 because 7 is odd). Write 7.
4. Under the 3: 3 + 5 = 8 (3 is half of 7 and 5 because 3 is odd). Write 8.
5. Under the zero in front: 1 (1 is half of 3). Write 1.

A step under the rule for 5 never adds up to 10 or more, so there is never anything to carry.

Multiplying by 9#

The rule for 9 has three parts, one for each end of the number and one for the digits in between.

• Under the last digit, subtract it from 10.
• Under every other digit, subtract it from 9 and add the neighbor.
• Under the zero in front, take the first digit of the number and subtract 1.

Here is 6,384 × 9 = 57,456.

1. Under the 4: 10 − 4 = 6. Write 6.
2. Under the 8: 9 − 8 + 4 = 5. Write 5.
3. Under the 3: 9 − 3 + 8 = 14. Write 4, carry 1.
4. Under the 6: 9 − 6 + 3 + 1 = 7 (1 carried). Write 7.
5. Under the zero in front: 6 − 1 = 5. Write 5.

Multiplying by 8#

The rule for 8 is the rule for 9 with the subtraction doubled.

• Under the last digit, subtract it from 10 and double the result.
• Under every other digit, subtract it from 9, double the result and add the neighbor.
• Under the zero in front, take the first digit of the number and subtract 2.

Here is 4,215 × 8 = 33,720.

1. Under the 5: (10 − 5)×2 = 10. Write 0, carry 1.
2. Under the 1: (9 − 1)×2 + 5 + 1 = 22 (1 carried). Write 2, carry 2.
3. Under the 2: (9 − 2)×2 + 1 + 2 = 17 (2 carried). Write 7, carry 1.
4. Under the 4: (9 − 4)×2 + 2 + 1 = 13 (1 carried). Write 3, carry 1.
5. Under the zero in front: 4 − 2 + 1 = 3 (1 carried). Write 3.

The General Method#

For a multiplier with no rule of its own, the book teaches a direct way to multiply any two numbers that still writes the answer one digit at a time from the right. Each digit of the answer comes from adding up the products of the pairs of digits that land in its place, plus the carry. For two numbers of two digits each, there are three steps.

1. Multiply the two last digits.
2. Multiply each number’s last digit by the other number’s first digit, crosswise, and add the two products.
3. Multiply the two first digits.

Here is 47 × 36 = 1,692.

1. Ones: 7×6 = 42. Write 2, carry 4.
2. Tens: 7×3 + 4×6 + 4 = 49 (4 carried). Write 9, carry 4.
3. Hundreds: 4×3 + 4 = 16 (4 carried). Write 16.

Longer numbers work the same way, with more pairs to add up in the middle places. Since only the carry passes from one step to the next, nothing else has to be held in your head.

Which Rule When#

The book also has rules for 3 and 4, which mix the halving of the rules for 5, 6 and 7 with the subtracting of the rules for 8 and 9. Multiplying by 2 or 10 needs no rule at all. For a multiplier such as 15, use the general method, or split the multiplier into ones you have rules for. Since 15 is 10 plus 5, 3,746 × 15 is 3,746 × 10 plus 3,746 × 5, and 37,460 + 18,730 = 56,190.

Checking Your Answer#

The book checks answers with digit sums, the old check that is also known as casting out nines. Add up the digits of each number, and keep adding until one digit is left. The digit sums of the two numbers, multiplied and reduced the same way, must match the digit sum of the answer.

For 4,786 × 11 = 52,646, the digits of 4,786 add up to 4 + 7 + 8 + 6 = 25, and 2 + 5 = 7. The digits of 11 add up to 1 + 1 = 2. Then 7 × 2 = 14, and 1 + 4 = 5. The digits of the answer add up to 5 + 2 + 6 + 4 + 6 = 23, and 2 + 3 = 5, which matches. A mismatch means there is a mistake somewhere. A match is a good sign, though a few mistakes slip through, such as two digits written in the wrong order.

Trachtenberg, the Major System and the Mental Abacus#

The Trachtenberg system is one of several ways to get better with numbers, and each one does a different job.

The Major System and the Trachtenberg system fit together well. The Major System is a memory technique, not a way to calculate, but a calculator who has to hold a number for a while can turn it into a picture and come back to it. The mental abacus is usually learned after months of practice on a real abacus, while the Trachtenberg rules can be tried in an afternoon. The general method above follows the same crosswise pattern that Vedic mathematics calls “vertically and crosswise.” For more ways to calculate in your head, see our guide to mental math.

Games for Mental Math Practice#

Frequently Asked Questions#

Sources#

• Trachtenberg, J. (1960). The Trachtenberg Speed System of Basic Mathematics (A. Cutler & R. McShane, Trans. and adapted). Doubleday.
• Petrov, I. (2023, July 2). Вычисляя Трахтенберга [Calculating Trachtenberg]. Radio Svoboda. In Russian, a historian’s account of Trachtenberg’s life from Swiss, Austrian and Russian archives.