Explanation of the Trachtenberg System


I hear this a lot.
The Trachtenberg System is presented without an explanation why it works.

In this post, I will analyse his system and show how and why it works.

His system is a way of making sure that you keep as low an amount of digits in your short term memory as possible. He does this by presenting just a few simple operations that you only do on one digit and it’s right neighbour.

The operations are:

  • Subtraction from 10 & 9
  • Adding 5 when neighbour odd
  • Doubling
  • Halving (and dropping halves). For example; halve of 7 becomes 3, not 3.5.
  • Subtracting or adding 1 or 2 from the leftmost digit.

With these operations he can multiply any digit with another in a multiplication.
In a complex multiplication, for example a 3X3, he uses this system and combines it with cross multiplication in order to generate each digit one by one (and maybe a carry).

In multiplication, he calculates from right to left.

He imagines each number to have a leading zero, so 123 in his mind is presented as 0123.
The system automatically shifts digits. We will show this below.

I will show how he calculates and why it works. First; he converts numbers between 1 and 12 to operations on 10, 1 and 2, using only division by 2 and adding or subtracting 1 or 2.

I’ll just present his operations here:

these are the rules where he uses halves:
3 = 5 - 2 = 10/2 - 2
4 = 5 - 1 = 10/2 - 1
5 = 5 = 10/2
6 = 5 + 1 = 10/2 + 1
7 = 5 + 2 = 10/2 + 2

here no halves are used:
8 = 10 - 2
9 = 10 - 1
10 = 10
11 = 10 + 1
12 = 10 + 2

By thus reducing all digits to a few simple operations he makes multiplication easier.


Example

Instead of directly multiplying by for example 6, he first multiplies by 10, then divides by 2 and finally he adds the number to the result.
As you can see above, this leads to the same result.
It’s the simplicity of the system combined with the fact that short term memory needs less digits that makes this system still fast and interesting.

Without explaining the system yet, I’ll just show an example of multiplication by 6, by using these simple operations:

34 * 6 = 204.

Recall that 6 = 5 + 1 = 10/2 + 1
Multiplying by 6 is the same as first multiplication by 10, then halving the result and finally adding it to the halve.

Here goes:
1: Multiply by 10: 34*10 = 340
2: Halve: 340/2 = 170
3: Adding the number (34) to the result: 170+34 =204
Done.

In mental calculation, adding a zero is not needed if you start to add the 3 (in 34) to the 7 (of 17).
This means just one step less:
1: 34/2 =17
2: Add 34 by starting with the 3 of 34 and the 7 of 17:

17
\nobreakspace\nobreakspace34
___ +
204

Instead of 3 steps I only have 2 steps:
1: Halving 34
2: Adding 34 (one digit shifted).


Another example

Multiplying by 3.
124 * 3 = 372

Recall that 3 = 5 - 2 = 10/2 - 2
This translates to halving and then subtracting twice the number after a digit shift.

I’ll shift the digit by multiplying by 10:
1: 124 * 10 = 1240
2: 1240/2 = 620
3: 620 - 2*124 = 620 - 248 = 372


Adding of 5’s

In all rules for the numbers 3 - 7 or the numbers where he uses halves there is a rule for adding 5 when odd. I’ll show where the 5 comes in.

43 * 3=129
Recall that multiplying by 3 translates to halving and then subtracting twice the number after a digit shift.

I’ll shift the digit by multiplying by 10:
1: 43 * 10 = 430 (times 10)
2: 430/2 = 215 (halving)
3: 215 - 2 * 43 = 215 - 86 = 129 (subtract twice the number)

(Mentally, just calculate without a decimal point:
1: 43/2 = 21.5 => 215 (just forget the decimal point)
2: 215 - 2 * 43 = 215 - 86 = 129)


Using the system

Let’s do the same calculation using the Trachtenberg System.

Here are the rules from the Wikipedia page:
1: Subtract the rightmost digit from 10.
2: Subtract the remaining digits from 9.
3: Double the result.
4: Add half of the neighbour to the right, plus 5 if the digit is odd.
5: For the leading zero, subtract 2 from half of the neighbour.

Recall we are still multiplying by 43 or with its leading zero 043.
Digit per digit and from right to left:

1: 10-3 = 7. 7 * 2 = 14. Digit (7) is odd, so add 5: 14 + 5 = 19. Write 9, carry 1.
2: 9-4=5. 5 * 2 = 10. Add half of it’s neighbour 3 (1). 10+1 = 11. Add the carry of step 1: 11 +1= 12. Write 2, carry 1.
3: Leading zero. Halve of 4 is 2. 2-2 = 0. Write down the carry: 1.

Answer: 129.


The complement (rules 1 & 2)

Look again at rules 1 & 2 of the five Trachtenberg rules:
1: Subtract the rightmost digit from 10.
2: Subtract the remaining digits from 9.

These two rules generate the complement of the digits we are working with.
Adding the complement of a number is the same as subtracting.

Example

The complement of 43 if 57.
100 - 43 = 0 + 57

The subtraction (-43) became an addition (+57).


Rule 3:

If the number we work with is x, then doubling the complement (rule 3) takes care of subtracting twice the number.


Rules 4 & 5:

Rules 4 and 5 take care of halving and using the ‘neighbour to the right’ takes care of multiplication by 10 or the digit shift I am talking about.


Complete

We have done al steps for the multiplication:
10x/2-2x = 3x


Correcting the complement:

The last part of rule 5 - subtracting 2 - takes care of a side effect of the complement.

If we have 2 digits, then first subtracting from 10 and the second from 9 results in -x + 100

example
The complement of 43 if 57. If we need to work out 100 - 43 and substitute - 43 for + 57 we get 100 + 57 = 157

The answer is 100 too large. Instead of -x we now have - x + 100.

In the case of the rules for multiplication by 3 we even double this result.
- x + 100 doubled becomes - 2x + 200.
Subtracting 2 from the last digit takes care of this.

If we work with 3 digits then the 200 would have been 2000 because of the extra digit. We would subtract the 2000 in the last step, so no matter how many digits, these always cancel each other out.

And that is why rule 5 is for the leading zero, subtract 2 from half of the neighbour.


We are done!

Now all five rules should be clear. Try to reread the rules for the numbers 3 to 9:

All parts of the rules for 3-9 should now be clear.
If not, just ask me.

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Little Trivia

In the movie “Gifted” from 2017, the main character uses the Trachtenberg system in order to do a quick calculation in elementary school.

That scene was a bit disappointing to me because the main character was supposed to be a math genius but she couldn’t even do a simple calculation in her head without the Trachtenberg system.

@Kinma how do you thik which is the best way to solve imo math question
also what is the best books for math olympiad ?

Awesome! Thanks for this!