The Trachtenberg system

The Trachtenberg system – mental arithmetic without the times tables

A method for mental arithmetic in which the times tables are replaced by one short rule per digit. The whole Trachtenberg system walked through, with the rule and an example for every digit from 2 to 12 – and how to check the answer.

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What this covers

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

What this covers (cont.)

Why the rules work
Jakow Trachtenberg and the background to the system

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:

The machinery that recurs in every rule

Write a zero in front344 becomes 0344. The zero is a position of its own that you also calculate on, and it is what gives the answer its first digit.
Start at the far rightOne digit at a time, moving left. Never two in your head at once.
Find the neighbourThe neighbour is always the digit to the right of the one you are on. The rightmost has none – there you use 0.
Apply the rule for the multiplierDoubling, halving, or “what you are short of 9 or 10”. Half always means rounded down: half of 7 is 3.
Write the last digit, carry the restIf the answer is two digits, the carry goes on to the next position – exactly as in ordinary long multiplication.

All ten rules below use precisely this setup. The only thing that changes from multiplier to multiplier is what you do in the fourth step.

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.

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:

344 × 6, position by position

Position	Digit	Neighbour	Calculation	Write
1 (rightmost)	4	none → 0	4 + 0, even	4
2	4	4	4 + 2, even	6
3	3	4	3 + 2 + 5, odd → 10	0, carry one
4 (leading zero)	0	3	0 + 1 + carry	2

The rule for 6 is “the digit itself, plus half the neighbour, plus 5 if the digit is odd”. Read from the bottom up, the right-hand column gives 2,064. The largest thing you have had to hold in your head along the way is the number 10.

When no single rule fits

More than multiplication

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

The digit-sum check

Add up the digits in each number344 gives 3 + 4 + 4 = 11, and 11 gives 1 + 1 = 2. The multiplier 6 is already a single digit.
Do the same with the two check numbers2 × 6 = 12, and 12 gives 1 + 2 = 3.
Treat your answer the same way2,064 gives 2 + 0 + 6 + 4 = 12, which gives 3. The same number – the answer passes the check.

The check is quick enough to be worth doing every time. But it catches most errors, not all: swap two digits in the answer and the digit sum is unchanged. It is a test, not a proof.

Why does it work?

The man behind it

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.

Key points (cont.)

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.

Questions this 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?

Questions this answers (cont.)

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?

Spørretimen

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