Multiplying by nine in the Trachtenberg system is the first time we come across subtraction the digits from 9 or 10, as well as reducing the left-hand digit.
The Rules For Multiplying By Nine
- First step: subtract from 10
- Middle steps: subtract from 9 and add the neighbor.
- Last step: reduce the left hand digit of multiplicand by 1
I have included the small nines and ten in the following example, only to help remind you what we are doing. Normally you would simply remember these numbers and not write them down. Following the first two rules we are simply subtracting each digit of the multiplicand by the number written above it, then adding the neighbor.
The third rule means we would subtract 1 from the left hand digit, in this case we would subtract 1 from the 7.
An Example - Multiplying 7869 By 9
For our example we will look at 7869 multiplied by 9. We will deform our red box to include the number above the digit we are looking at so all the numbers we are looking at are in the now L shape.
Following rule 1 we subtract the 9 from 10 to give us 1. There is no neighbor so we write down the 1.
Next, moving one place to the left we now have 6 which we subtract from 9 giving us 3, then we add the neighbor, which is 9 giving us 12. We write down the 2 and carry the 1. We show this with a dot.
Moving left again we now have 8 which we subtract from 9, giving 1, we add the carry from the previous step to give 2, then we add the neighbour, 6 giving us a sum of 8. We write the 8 down.
Moving left we now have 7 which we take from 9, giving us 2, then we add 8, the neighbor, to give us 10. We write down the zero and put a dot to show the carry.
Now, at the zero, we are on the last step so we refer to rule 3, which tells us to subtract 1 from the neighbor, in this case, 7 giving us 6. We then add the carry to give us 7. We write down 7 and we have our answer of 70821.
These rules will work for all numbers, however when multiplying a single digit number by 9 you would simply omit rule 2 and only use rules 1 and 3.
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