## How to do Trachtenberg System Division

Now we come to the Trachtenberg System division method which is rather different than the other three methods we have looked at. The Trachtenberg system division has a couple of interesting points:

- No matter how large the divisor you will only ever divide by a single digit.
- Any multiplication done is always between two single digit numbers.

Another unique feature of the Trachtenberg System Division is the use of three terms; N, T and U which represent what part of the answer from a multiplication of two single digits we will use.

Whenever we multiply two single digits we mostly get a two digit result, in the cases we get a single digit result we put a leading zero as the tens digit to make it a two digit result.

In practice you do not need to put the leading zero but it helps with explaining the method and when first practicing it if you consciously imagine the leading zero in front of a single digit result.

Here are a few single digit multiplications and what we call the N, T and U of the results:

The layout of the equation and its workings are also different in the Trachtenberg System division. The division and the answer are all on the same line and below the dividend are two other lines; one we will call our "working" line which contains an intermediate number used to work out the value of the second line, our "partial dividend" line. The partial dividend is the number we will divide to get the next figure of our answer in the division.

The fully worked layout looks like this:

Lets have a look at the process for the Trachtenberg System division.

#### Divide 29064 by 84 using Trachtenberg System Division

We start off with setting out our equation as follows:

In the Trachtenberg System division we only look at the first digit of the divisor, in this case the 8 and we will divide by this.

When doing the division we need to remember these points:

- If the first figure of the dividend is smaller than the first figure of the divisor then use the first two figures of the dividend.
- If the second digit of the divisor is 8 or 9 then add 1 to the value of the first figure of the divisor and divide the dividend by this new figure.
- Always ignore any remainder in this step we are only interested in the whole number value from the division.

The dividend starts with a 2 but we cannot divide this into 8 so we include the next digit which gives us 29 which we can divide by 8. We copy the 29 down to our partial dividend row.

We divide the partial dividend by the first digit of the divisor

We write the 3 as the first digit of our answer.

Now we use the with the divisor and work out the N, T and U values.

With a two digit divisor the pattern is this:

The calculations we do are:

We want what is called the NT value which is the sum of the N value and the T value. In this case our N value is 24 and our T value is 1.

We subtract the NT number from our partial dividend.

We write the 4 up under the next digit of the dividend in the "working" row.

The 4 will become the tens digit of our "working number" and to complete the working number we will bring down the zero.

Our working number is 40 from which we subtract the U value, which is the unit value of the we did before and we write the result below the working value in the partial dividend row.

We divide the new partial dividend by the first digit of the divisor

We write the 4 as the second digit of our answer.

Now we use the with the divisor and work out the N, T and U values.

The calculations are:

We get the NT value by summing the N value and the T value.

We subtract the NT number from our partial dividend.

We write the 5 up under the next digit of the dividend in the "working" row.

The 5 will become the tens digit of our "working number" and to complete the working number we will bring down the 6.

Our working number is 56 from which we subtract the U value, which is the unit value of the we did before and we write the result below the working value in the partial dividend row.

We divide the new partial dividend by the first digit of the divisor

We write the 6 as the third digit of our answer.

The 346 is our tentative answer at this point. The remaining steps are to see if there is any remainder and to confirm the final result.

We use the with the divisor and work out the N, T and U values.

The calculations are:

We get the NT value by summing the N value and the T value.

We subtract the NT number from our partial dividend.

We can write the 0 up under the next digit of the dividend in the "working" row or just leave it blank. Then we bring down the 4.

Our working number is 4 from which we subtract the U value, which is the unit value of the we did before and we write the result below the working value in the partial dividend row.

The tells us two things; first that there is no remainder and secondly it confirms 346 as the final value.

#### Divide 26758 by 69 using Trachtenberg System Division

We setup our equation as follows:

The second digit of the divisor is a 9 so instead of using the 6 we will use 7 when dividing. The first digit of the dividend is 2 which is smaller than 7 so we will use 26. We copy the 26 down to the partial dividend line.

We divide the partial dividend by the 7 ignoring any decimal.

We write the 3 as the first digit of our answer.

We use the with the divisor and work out the N, T and U values.

The calculations are:

We get the NT value.

We subtract the NT number from our partial dividend.

We write the 6 up on the working line under the next digit of the dividend then we bring down the digit, 7, to form the working number.

Our working number is from which we subtract the U value, which is the unit value of the , and we write the result below the working value in the partial dividend row.

We divide the new partial dividend by the 7 ignoring any decimal.

We write the 8 as the second digit of our answer.

We use the with the divisor and work out the N, T and U values.

The calculations are:

We get the NT value.

We subtract the NT number from our partial dividend.

We write the 5 up on the working line under the next digit of the dividend then we bring down the digit, 5, to form the next working number.

Our working number is from which we subtract the U value, which is the unit value of the , and we write the result below the working value in the partial dividend row.

We divide the new partial dividend by 7 ignoring any decimal.

We write the 7 as the third digit of our answer.

We now have our preliminary answer the remaining steps are to check the remainder and validate the result.

We use the with the divisor and work out the N, T and U values.

The calculations are:

We get the NT value.

We subtract the NT number from our partial dividend.

We write the 5 up on the working line under the next digit of the dividend then we bring down the last digit, 8, to form the next working number.

From our working number, , we subtract the U value, which is the unit value of the , and we write the result below the working value in the partial dividend row.

So dividing 26758 by 69 is 387 with 55 remainder.

You can find more examples of the Trachtenberg System division in Fast Long Division also if you are a member you can download Trachtenberg System division worksheets.

Doug Edmunds says

August 6, 2018 at 11:36 pmSince you picked two divisions that are difficult to do using the Vedic ‘crowning gem’ a.k.a. ‘flag’ method, it would be fair to do the same problems using the Trachtenberg system. That would be more convincing that the Trachtenberg method does not run into the same complications.

Tony says

August 9, 2018 at 8:37 pmIt is true division is not always as straightforward as multiplication and there are times you may need to adjust the last calculated digit of your answer up or down using the Trachtenberg method.