Ok, so let’s modify Trachtenberg for people that calculate from left to right.

For example multiplication with 8. This one has the most rules.

These are the rules for right to left:

- First figure: subtract from ten and double.
- Middle figures: subtract from nine and double what you get, then add the neighbor.
- Left-hand figure: subtract two from the left-hand figure of the long number.

Let’s see how and why this works and for simplicity’s sake, let’s take a 2 digit number:

If we have a 2 digit number with the first digit ‘a’ and the second ‘b’, we can write this as:

a | b

where the | separates the digits.

If we multiply a|b with 8, using the rules above we get:

a-2 | 2(9-a)+b | 2(10-b)

If ‘a-2’ now denotes the hundreds, etc. we can also write this as:

100(a-2) + 10(2(9-a)+b) + 2(10-b) =

100a - 200 + 180 -20a +10b + 20 -2b =

80a +8b = 8(10a+b) =

8(a|b)

So yes, the rules indeed will multiply by 8.

Let’s check with an actual 2 digit number:

78 * 8 =160;

7-2 | 2*2+8 | 2(10-8) =

5 | 12 | 4 = (take care of the carry)

624

11-proofing:

First the starting point. 78 - 77 = 1. 1 * 8 = 8.

Now the answer: 624 - 550 = 74. 74 - 66 = 8. Alternatively: 4-2+6 = 8.

Both end in 8, so the answer is probably correct.

Some caution.

1: The Trachtenberg rules are great for digits which are close to 10.

With digits close to zero the rules still work, however imho they don’t speed up the calculation anymore. This is true whether you calculate from left to right or the other way round. An extreme example is multiplying 11 by 8:

11 * 8 =

-1 | 2 * 8 + 1 | 2 * 9 =

-1 | 17 | 18 = (now the 2 carries)

0 | 8 | 8 =

88

2: the system is designed for calculating from right to left as it keeps the amount of digits in short term memory as small as possible.

It is designed to put digit by digit on paper.

789 * 8 =

Following the steps we get for the units digit: 2(10-9) = 2.

tens: 2(9-8)+9 = 11 = 1 (carry the 1)

hundreds: 1+2(9-7)+8 = 13 = 3 (carry the 1)

thousands: 1+7-2 = 6

Result: 6312

3: The system is designed for multiplication with numbers until 12.

Above 12 Trachtenberg switches to what he calls Direct Multiplication. This is the exact same as criss-cross multiplication.

In those cases, add Arthur Benjamins system:

24 * 36 is a lot of work doing criss-cross.

However; this is where Benjamin shines:

24 * 36 = 20 * 40 + 4 * 16 = 800 + 64 =

864.

11 proof:

24-22 = 2 . 36 - 33 = 3. 2*3 = 6

864 - 770 = 94. 94 - 88 = 6. Alternatively: 4-6+8 = 6.

Both end in 6; answer is right.

My personal thoughts. The book states that the system is great for people with calculation problems. Being able to just work out a difficult calculation digit by digit can indeed give a student the confidence it needs to overcome this.

So this is a great thing.