Multiplying by Eleven
In looking at the Trachtenberg system  we will start with multiplying by eleven. The rule here is very simple and gives us a very fast way to multiply any number by eleven.

The Rules For Multiplying By Eleven

  1. Put the last number of the multiplicand down as the right hand figure of the answer.
  2. Add each successive number of the multiplicand to its neighbor on the right.
  3. The first number of the multiplicand becomes the left hand number of the answer.

Actually we can replace these three rules with a single rule.

  1. Add the neighbor

An Example - Multiplying 3456 by 11

For this example we will look at the following multiplication:
Multiplying by Eleven Example 1

Setting up the Calculation
The first thing we will do is write in a zero at the front of the multiplicand and then draw a line under it. We draw the line as we will be writing the answer directly below the multiplicand as we calculate it.
Multiplying by Eleven Example 1 Setup

First Step
We draw in our red box to show you where we are (you don't need to draw in the box when doing this yourself). On the left side of the box we have 6, the right hand side of the box is empty. We have no neighbor to add to the six so our first digit is 6.
Multiplying by Eleven Example 1 Step 1

Second Step
Now we move our box to the left by one digit. Now we have 5 and its neighbor is 6, 5 plus 6 adds up to 11, so we write 1 and we carry 1.
Multiplying by Eleven Example 1 Step 2
To show we are carrying one I will put a dot above the digit in the product we just wrote down. Doing the calculation mentally you would just remember you are carrying 1.

Third Step
This time we have 4 and its neighbor is 5, which adds up to 9, to which we add 1 carried over in the previous step, giving us a total of 10.
Multiplying by Eleven Example 1 Step 3
So we write the 0 and carry the 1. Again we put a dot above the zero to show we are carrying 1.

Fourth Step
Moving the box to the left again we now have 3 and its neighbor is 4, which add up to 7, to which we add the 1 carried over before giving us a total of 8.
Multiplying by Eleven Example 1 Step 4
We write the 8 down.

Last Step
Moving the box one digit to the left again there is 0 and its neighbor is 3, which adds up to 3.
Multiplying by Eleven Example 1 Last Step
So we write 3 into the product giving us the answer of 38016.

Note: If we didn't have the zero in front we may have stopped at the 3 and we would have ended up with the wrong answer.
This concept will work for all numbers multiplied by 11, from single digit numbers up to numbers of any size.

Multiplying 2 Digit Numbers by 11

Although for two digit numbers the process is the same as above there I just want to emphasis how easy multiplying by eleven is using this method and you can easily do this in your head.
When multiplying a two digit number by eleven you simply add the two digits then put the sum in between the two digits.
Multiplying a Two Digit Number by Eleven Example 1

To multiply 24 by 11 you simply add 2 and 4 to get 6 then place the 6 between the 2 and the 4 giving you the answer 264. Simple isn't it?

This works if the two digits add up to less than ten. If they add up to more than ten you simply put the unit value between the two digits then carry the one to the digit on the left.
Multiply a Two digit Number by Eleven Example 2
To multiply 39 by 11 you add 3 and 9 together to get 12, then you put the 2 between the 3 and 9 then change the 3 to 4 because we need to add 1 for the carry.

Multiplying Single Digit Numbers by 11

Lets have a quick look at a very simple single digit example, 6 x 11,  just to show the method above still works.
We place the box over the 6, so we have 6 and it has no neighbor, giving us 6, so we write this down.
Multiplying by Eleven Example 2 Step 1
Moving the box to the left we have the zero and the 6 in the box. Now we have zero and we add its neighbor which is 6, this gives us 6, so we write 6 giving us our answer of 66.
Multiplying by Eleven Example 2 Step 2
This method works for all numbers multiplied by eleven.

If you have any comments or questions please leave a comment below.