The Trachtenberg method: what it is and where it came from, the rules for multiplying by 11, 12, 6, 7, 5, 9, 8, 4 and 3 with worked examples, the two-finger method for any multiplier, why each rule works, who it suits (and who it does not), how it compares with Vedic maths, and a two-week plan to learn it

The Trachtenberg method is the most interesting of the speed-arithmetic systems, because of where it came from and because it does something the others do not: it replaces the multiplication tables with a small set of rules that work digit by digit. This page gives the rules for the multipliers a child actually uses, with an example and the reason for each, and says honestly who it suits.

Where it came from

Jakow Trachtenberg was a Russian-born engineer who was imprisoned in Nazi concentration camps in the 1940s. Without paper, he kept his mind alive by devising a system of mental arithmetic, and he survived to teach it in Switzerland after the war. The system was published in 1960 as The Trachtenberg Speed System of Basic Mathematics and has been in print since. Its claim is modest and true: with a rule for each multiplier, a student can multiply any number by any single digit quickly and without tables, and with one further method, by any number at all.

How the basic rules work

Every rule has the same shape. Write the number to be multiplied, imagine a zero in front of it, and work from right to left, one digit at a time. At each step you use only that digit and its right-hand neighbour (the digit to its right; the last digit has no neighbour, so its neighbour counts as zero). Some rules use "half" of a digit, which means the whole-number half, ignoring any fraction: half of 7 is 3, half of 4 is 2. Carries are added into the next step as usual.

The rules, with examples

Multiply by 11: add the neighbour. 4 2 3 × 11. From the right: 3 (no neighbour) → 3. Then 2 + 3 → 5. Then 4 + 2 → 6. Then the imagined 0 + 4 → 4. Answer: 4653. Why it works: 11 = 10 + 1, so each digit is shifted one place left and added to itself, which is exactly "digit plus right neighbour".

Multiply by 12: double the digit, add the neighbour. 4 2 3 × 12. 3 × 2 = 6 → 6. Then 2 × 2 + 3 = 7 → 7. Then 4 × 2 + 2 = 10 → 0, carry 1. Then 0 × 2 + 4 + 1 = 5 → 5. Answer: 5076. Why: 12 = 10 + 2.

Multiply by 6: add half the neighbour, and 5 more if the digit is odd. 4 2 3 × 6. 3 is odd: 3 + half of 0 + 5 = 8 → 8. Then 2 is even: 2 + half of 3 (which is 1) = 3 → 3. Then 4 is even: 4 + half of 2 = 5 → 5. Then 0 + half of 4 = 2 → 2. Answer: 2538. Why: 6 = 5 + 1, and multiplying by 5 is half of multiplying by 10, which shifts the neighbour's half across; the "5 if odd" handles the lost half.

Multiply by 7: double the digit, add half the neighbour, and 5 more if the digit is odd. 4 2 3 × 7. 3 odd: 6 + 0 + 5 = 11 → 1, carry 1. 2 even: 4 + 1 + carry 1 = 6 → 6. 4 even: 8 + 1 = 9 → 9. 0: 0 + 2 = 2 → 2. Answer: 2961. Why: 7 = 5 + 2.

Multiply by 5: half the neighbour, and 5 if the digit is odd. 4 2 3 × 5. 3 odd: half of 0 + 5 = 5 → 5. 2 even: half of 3 = 1 → 1. 4 even: half of 2 = 1 → 1. 0: half of 4 = 2 → 2. Answer: 2115. Why: 5 is half of 10.

Multiply by 9: subtract the digit from 10 (first step) or from 9 (middle steps), add the neighbour; the last step is the leading digit minus 1. 4 2 3 × 9. First: 10 − 3 = 7 → 7. Then 9 − 2 + 3 = 10 → 0, carry 1. Then 9 − 4 + 2 + 1 = 8 → 8. Last: 4 − 1 = 3 → 3. Answer: 3807. Why: 9 = 10 − 1, so the method is "subtract the number from ten times itself", done digit by digit with the complements.

Multiply by 8: subtract from 10 (first) or 9 (middle), double it, add the neighbour; the last step is the leading digit minus 2. 4 2 3 × 8. First: (10 − 3) × 2 = 14 → 4, carry 1. Then (9 − 2) × 2 + 3 + 1 = 18 → 8, carry 1. Then (9 − 4) × 2 + 2 + 1 = 13 → 3, carry 1. Last: 4 − 2 + 1 = 3 → 3. Answer: 3384. Why: 8 = 10 − 2.

Multiply by 4: subtract from 10 (first) or 9 (middle), add half the neighbour, and 5 if the digit is odd; the last step is half the leading digit minus 1. 4 2 3 × 4. First: 3 odd: 10 − 3 + 0 + 5 = 12 → 2, carry 1. Then 2 even: 9 − 2 + half of 3 + 1 = 9 → 9. Then 4 even: 9 − 4 + half of 2 = 6 → 6. Last: half of 4 − 1 = 1 → 1. Answer: 1692. Why: 4 = 5 − 1, done with complements.

Multiply by 3: subtract from 10 (first) or 9 (middle), double it, add half the neighbour, and 5 if the digit is odd; the last step is half the leading digit minus 2. 4 2 3 × 3. First: (10 − 3) × 2 + 0 + 5 = 19 → 9, carry 1. Then (9 − 2) × 2 + half of 3 + 1 = 16 → 6, carry 1. Then (9 − 4) × 2 + half of 2 + 1 = 12 → 2, carry 1. Last: half of 4 − 2 + 1 = 1 → 1. Answer: 1269. Why: 3 = 5 − 2.

Multiply by 2: double each digit, which needs no rule, and by 10, which adds a zero. The system also gives a rule for 1 (the number itself) for completeness.

Check every answer the ordinary way while learning; 423 × 7 by the school method is 2961, and so on. The point of the first fortnight is to make the rules automatic, not to trust them blindly.

The two-finger method for any multiplier

For multiplying by a number of more than one digit, Trachtenberg gives a general method that uses pairs of digits: for each position in the answer, you add the "units" products and "tens" products of certain digit pairs, working across the multiplier. It is essentially long multiplication done without writing the intermediate rows, and it is where the system is slower to learn and less obviously better than the Vedic "vertically and crosswise" method for two-digit by two-digit. For most children the basic rules above are the useful part of the system.

Why each rule works, in one line

Every rule is the multiplier rewritten in terms of 10 and 5. Eleven is 10 + 1, twelve is 10 + 2, six is 5 + 1, seven is 5 + 2, five is half of 10, nine is 10 − 1, eight is 10 − 2, four is 5 − 1, three is 5 − 2. Multiplying by 10 shifts the digit left (so the neighbour appears), multiplying by 5 shifts half of it, and subtracting uses the complements from 9 and 10. A child who is shown this learns some algebra and some place value without noticing, and it is the strongest reason to teach the method with the "why" rather than as a bag of tricks.

Who it suits, and who it does not

It suits a child of about 9 or older with secure place value and tables, who likes a routine and a challenge, and especially one who finds mental arithmetic slow and wants to be fast. The rules for 11 and 12 take an hour and give an immediate win. Adults who want to recover mental arithmetic also take to it.

It does not suit a child whose place value or tables are not yet secure, because every rule assumes both and the rules become a substitute for understanding; nor a child who needs the general school method for an exam this year, because exams mark the written method and the Trachtenberg working looks like nothing a marker recognises. It is a supplement to school maths, taught alongside it, never instead.

Trachtenberg against Vedic maths and the school method

The same multiplications done by both speed systems, and which a 9-year-old should learn first, are in Trachtenberg method vs Vedic maths; the Vedic system itself is introduced in introduction to Vedic mathematics.

A two-week plan

• Days 1 and 2: multiply by 11. Ten three-digit numbers a day, checked by the school method.
• Days 3 and 4: multiply by 12.
• Days 5 to 7: 6, 7 and 5, which share the "half the neighbour, 5 if odd" idea.
• Days 8 to 10: 9 and 8, which share the complements idea.
• Days 11 and 12: 4 and 3.
• Days 13 and 14: mixed practice, twenty a day, timed; the "why" of each rule said aloud once.

Ten minutes a day, always with the check, always alongside school homework.

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics: the school's own curriculum first, and the Trachtenberg rules taught alongside it once place value and tables are secure, with the reason for each rule shown so that it is mathematics and not a trick. For Grade 3 to 8, CBSE Class 3 to 8 and KS2. The free 30-minute first class is a real lesson on whatever the child is doing in school this week.

Questions parents ask

See how we teach this, 1-on-1 online →