Learn something new: the Trachtenberg system – mental arithmetic without the times tables

Take 3,425 × 11. Most of us would set out a calculation. But there is a rule that gives you the answer – 37,675 – without calculating in the ordinary sense: add each digit to the digit on its right.

That is the simplest rule in the Trachtenberg system, and a reasonable place to start. But the system is far larger than the eleven times table. It is a complete setup for arithmetic in your head, in which the times tables are replaced by short recipes you follow digit by digit.

How to read the rules

All the rules use the same machinery, and it is worth having that clear before looking at them one by one:

• Write a zero in front of the number. 344 becomes 0344. That zero is a position in its own right that you also work on, and it is the one that gives the answer its first digit.
• Work from right to left, one digit at a time.
• “The neighbour” is always the digit to the right of the one you are on. The rightmost digit has no neighbour – there you use 0.
• “Half” means half, rounded down. Half of 7 is 3.
• If the result has two digits, write down the last digit and carry the rest to the next position, exactly as in ordinary long multiplication.

The rules, digit by digit

For 2, 5, 6, 7, 11 and 12 the same rule applies throughout the number. For 3, 4, 8 and 9 the rule is slightly different on the rightmost digit and on the leading zero – those are written out separately.

• × 2 — Double the digit.
  46 × 2 = 92

• × 3 — Rightmost digit: double what it takes to reach 10, and add 5 if the digit is odd. Otherwise: double what it takes to reach 9, add 5 if the digit is odd, and add half the neighbour. On the leading zero: half the neighbour, minus 2.
  42 × 3 = 126

• × 4 — Rightmost digit: what it takes to reach 10, plus 5 if the digit is odd. Otherwise: what it takes to reach 9, plus 5 if the digit is odd, plus half the neighbour. On the leading zero: half the neighbour, minus 1.
  62 × 4 = 248

• × 5 — Half the neighbour, plus 5 if the digit itself is odd.
  426 × 5 = 2,130

• × 6 — The digit itself, plus half the neighbour, plus 5 if the digit is odd.
  344 × 6 = 2,064

• × 7 — Double the digit, add half the neighbour, and 5 if the digit is odd.
  34 × 7 = 238

• × 8 — Rightmost digit: double what it takes to reach 10. Otherwise: double what it takes to reach 9, and add the neighbour. On the leading zero: the neighbour, minus 2.
  456 × 8 = 3,648

• × 9 — Rightmost digit: what it takes to reach 10. Otherwise: what it takes to reach 9, plus the neighbour. On the leading zero: the neighbour, minus 1.
  8,432 × 9 = 75,888

• × 11 — The digit plus the neighbour.
  3,425 × 11 = 37,675

• × 12 — Double the digit, and add the neighbour.
  314 × 12 = 3,768

One calculation all the way through

The rules become clearer when you watch one of them run. Here is 344 × 6, with the leading zero in place, so 0344:

1. Rightmost 4, neighbour 0. Half of 0 is 0, and 4 is even. 4 + 0 = 4.
2. Next 4, neighbour 4. Half of 4 is 2. 4 + 2 = 6.
3. Then 3, neighbour 4. Half of 4 is 2, and 3 is odd, so we add 5. 3 + 2 + 5 = 10 → write 0, carry one.
4. The leading zero, neighbour 3. Half of 3 is 1. 0 + 1 = 1, plus the carry = 2.

Read from the top: 2,064. No times tables anywhere – only halving, addition and carrying.

When no single rule fits

The digit rules cover 2 to 12. For everything else the system has a general method, in which you work through the numbers in pairs and add up the products belonging to each place in the answer. It is harder to learn, but it takes any two numbers – and it is designed to keep as little as possible in your head as you go.

More than multiplication

Trachtenberg did not stop at multiplication. The system also has its own procedures for addition, division, squaring and square roots.

And it has a digit-sum check: you add up the digits in each number, reduce to a single digit, and compare with your answer treated the same way. If they do not match, you have made a mistake. But note that the reverse does not hold – the check catches most errors, not all. It is a test, not a proof.

Why does it work?

This is the part worth the most. The rules are not conjuring tricks, and they are not arbitrary. They fall out of the way the decimal system is put together. The rule for 9 is a good example: multiplying by 9 is multiplying by 10 and subtracting the number once – and that is precisely why the rule is about how much you are short of 9 and 10.

The man behind it

Jakow Trachtenberg was a Ukrainian-Jewish engineer and mathematician. He developed the system while held in a Nazi concentration camp – in his head, without pen or paper, to keep his mind working. The method was published in English in 1960 as The Trachtenberg Speed System of Basic Mathematics, translated by Ann Cutler and Rudolph McShane.

The content is a practical walk-through you can practise yourself. Sources are given in the episode.

Main topics

• The Trachtenberg system as a whole
• Every digit rule from 2 to 12, with examples
• The neighbour and halving – the basic ideas in the system
• The rules that add, and the ones that subtract
• The general method for large numbers
• Addition, division, squaring and square roots
• Digit sums – checking the answer without redoing the work
• Why the rules work
• Jakow Trachtenberg and the background to the system

Questions this episode answers

• What is the Trachtenberg system, and what does it cover?
• What is meant by “the neighbour”, and what does halving mean in the system?
• How does the rule go for each digit from 2 to 12?
• Why do the rules for 8 and 9 subtract, while most of the others add?
• Why are the rules for 3, 4, 8 and 9 different at the ends of the number?
• How does the system handle large numbers, where no single rule fits?
• Can you really calculate without knowing the times tables?
• How do you check that the answer is right?
• Who was Jakow Trachtenberg, and where did the system come from?

Key points

• The system is one short rule per digit, not one trick – it covers 2 to 12.
• Two ideas carry the whole thing: the neighbour to the right, and halving rounded down.
• Some rules add, while 3, 4, 8 and 9 are built on subtracting.
• The general method handles large numbers, where no single rule reaches.
• The system also covers addition, division, squaring and square roots.
• The digit-sum check catches most errors, but not all – it is a test, not a proof.
• The rules are not magic – they fall out of how the decimal system is built.