Multiply by TwelveWhen multiplying by twelve in the Trachtenberg system we add an additional step to the rule used when multiplying by eleven. We now first double the digit value before adding the neighbor.

The Rule For Multiplying By Twelve

  1. Double each number in turn and add its neighbor.

When looking at a digit what we do is add the number to itself before we add its neighbor.The Red BoxIn the number above we are looking at the 5 and its neighbor is 6. So we first add 5 to itself, 5 + 5 = 10 and then we add the 6, giving us 16.

An Example - Multiplying 3524 By 12


If you have not already read multiplying by eleven then please have a quick look here. This will explain the setup of the calculation which we will use below.

In this example we will multiply 3524 by 12.TS_M12_E1_SetupWe will use the red box again to help us follow along.

Step 1
We have 4 as the first digit, which we double, giving us 8, there is no neighbor to add.
Multiplying by Twelve Example 1 Step 1
We write down the 8 under the 4.

Step 2
We move the red square one place to the left.
We have 2, which we double to give us 4 then we add its neighbor, which is 4 to give us 8.Multiplying by Twelve Example 1 Step 2We write this 8 down under the 2.

Step 3
We move the red square one place to the left.
We have 5, which we double, giving us 10, then we add its neighbour, which is 2, giving us 12.Multiplying by Twelve Example 1 Step 3We write 2 down below the 5 and we carry the 1, which we show by putting a dot above the 2 we just wrote down.
Note: as it is possible that we may have to carry a 1 or 2 depending on the number being multiplied, feel free to write a small 1 or 2 above the digit in the product to keep track of how much you are carrying.

Step 4
Again, moving the red square one place to the left.
We have 3, which doubles gives us 6 then adding the 5 we get 11, plus the 1 carried gives us 12.Multiplying by Twelve Example 1 Step 4Write the 2 down under the 3 and put a dot above it to show the carry.

Last Step
Moving the red square for the last time we have zero, which doubled is still zero, then we add 3 giving us 3, plus the 1 carried gives 4.Multiplying by Twelve Example 1 Last StepWrite the 4 down below the leading zero and we have our answer of 42288.

Please feel free to leave a comment if you found this interesting.