Addition

We will look at the Trachtenberg System addition method which uses a technique that allows you to add columns independently of each other. We also have a look at the error checking included in this addition using the digital root to check and double-check our results. We will also have a look at what to do if our tests do find a problem.

The rule for addition is:

 
What this means is as we are adding up the numbers down a column, when the running total becomes eleven or higher, we subtract eleven from the running total, make a mark next to the digit that we just added, then continue with the reduced running total. When we finish the column, we write down this running total below the column we just added up.

The mark can be anything you like, but usually, a small stroke, check mark or tick is best. Just make sure it is easy to spot when you need to come back and count up the marks.

As we are counting up to eleven, this does mean that sometimes we may end up with a ten in the running total for a column. We put the ten as the result, and we do not carry.

   

As we would with the standard addition, we write a line below the numbers we are about to add up to separate our total from the data. Below the line we will have two rows of new data; the first is the running total for each column, and the second row is the count of marks in each column. We then draw another line below these and calculate our total.

To calculate the total, we add up the running total and the ticks in this way. For the rightmost column we only add the running total and tick value together and write the result below For all other columns we add the running total, the tick count then we also add the tick count from the column to the right.

In the example below for the second column from the right we would add the following:

   

   

When adding up the total if we get a value larger than ten for the result, we would write down the units figure of the result and carry the tens figure as normal.

The Addition

Lets have a look at an example:

   

In this method of addition it does not matter which order the columns are added up so we will start on the left and work our way to the right.

   

We end up with as our running total for the first column and the number of marks is one.

   

   

For the second column we have:

   

We have as our running total and one mark.

   

For the third column we have:

   

We have as our running total and one mark.

   

For the last column we have:

   

We have as our running total and three marks.

   

 

Getting the Total

Now we have the data complete we now need to add up the running totals and the ticks. Before we do we will add an extra column of zeros on the left for the running total and the mark count. I will explain why later.
This time we do carry if a total is greater than nine, so we will start on the right-hand column.
For the right hand column we just add the running total and the tick count for this column.

   

   

Moving left to the next column we now add the tick count of the next column to the right to the sum of our running total and tick count for this column.

   

We write down the and carry the .

   

Moving left again to the third column we add the tick count of the second column to the sum of our running total and tick count for this column.

   

We write down the .

   

Moving left again to the fourth column we add the tick count of the third column to the sum of our running total and tick count for this column.

   

We write down the .

   

Now this is why we added the extra column of zeros on the left. As we are adding up in an L shape by including the mark count in the column to the right we need an extra column, so we don't finish too early. You could leave off the zeros as long as you remember to do the final count.

   

We write down the .

   

An there is our answer, we think. Now we need to check that our answer is correct.
 

Three parts of the Addition

Our addition is now made up of three parts; the first being the columns of figures we just added up, the second is the working table made up of the running totals and the mark counts, the third being the answer.

We use the columns of figures to get the working table then we use the working table to get the answer.
We now need to check our work on all three section and the steps are:

1. We find a check figure, a digital root, for each of the columns of figures.
2. We find a check figure, a digital root, for the working table.
3. We find a check figure, a digital root, for the answer.

 

Check figure for each Column

We find the check figure for each column using the casting out nines method.As we are only working on each column individually and separately it does not matter which order we do them. We will start on the left and work to the right in this instance.

The numbers from the first column on the left are: .
We ignore any or digits adding to which leaves us with:

   

So 7 is the check figure for the first column.

The numbers in the second column are: so our check figure is .

The numbers in the third column are: .
Which gives us

   

The check figure for the third column is .

The numbers in the fourth column are: .
Which gives us

   

The check figure for the fourth column is .

Our working should look something like this:

   

 

Check figure for the Working Table

To work out the check figure for each column of the working table we need to count the mark count twice when adding it to the running total.

As we are again only working on each column individually and separately it does not matter which order we do them. We will again start on the left and work to the right.

As this is relatively straight forward, I will summarize the working for each column from right to left.

   

We ignore the extra column of zeros we added.

   

We can see the check figures at the top of each column match the check figures at the bottom of each column. This means that our working table is correct, in other words we added all the columns up correctly.

If the check figures in one of the columns did not match up then, it would mean we made an error when adding up the numbers or when writing down the running total or mark count in that column only. The other columns would be correct so we would only need to recheck our count on that one column.

The error checking can save us time when we have made a mistake as normally you would have to repeat the whole addition again to check the result.
 

Check figure for the Answer

We now calculate a check figure for the answer using the casting out nines method.

   

We now do the same to one set of the previous check figures, we will do the bottom set this time.

   

   

Both check figures are the same so we have confirmed that our result is correct.

 

 
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