Trachtenberg Direct Multiplication

The direct multiplication method is a condensation of the multiplication process. We will learn how to multiply any number by any number no matter how long they are and immediately write down the answer.

Leading Zeroes


When using the direct multiplication method we will put leading zeroes in front of the multiplicand so that we do not stop the multiplication too early and get the wrong answer. Not all of the zeroes will always be used but it is a good habit to always use them. How many leading zeroes we put depends on the multiplier, the General rule is:

When multiplying by a multiplier of any length, put as many zeros before the multiplicand as there are digits in the multiplier.

 

Number Pairs


Trachtenberg Direct Multiplication Inner and Outer Pairs

The direct multiplication method involves multiplying pairs of numbers, a pair being made up of one number of the multiplicand and one number from the multiplier. The pair involving the right-hand digit of the multiplier is always known as the outer pair. The outer pair of numbers always goes from the right-hand digit of the multiplier to the digit of the multiplicand above the blank space where we will write the next digit of the answer. The result of the multiplication of the outer pair is added to the result of the multiplication of any and all inner-pairs.

The total number of pairs involved is limited by the number of digits of the multiplier, assuming the multiplier is always the smaller of the two numbers. The maximum number of number pairs is equal to the number of digits of the multiplier.

When multiplying the first step is always multiplying the right-hand digit of the multiplicand by the right-hand digit of the multiplier. In the first step, the outer pair is the only pair used.

Two Digit Multiplier


When we have two-digit multipliers, apart from the first step where we only use one pair of numbers, we use two pairs of numbers to calculate our answer.

We will look at 32 times 14 and we write out the equation on one line. As we have a two-digit multiplier we put two leading zeros on the multiplicand. The answer will be written directly beneath the multiplicand.

Trachtenberg Direct Multiplication-Two Digit Multiplier-1

First step: In this example we multiply the 2 of the 32 with the 4 of 14.

Trachtenberg Direct Multiplication-Two Digit Multiplier-2

    \[   2 \times 4 = 8 \]

We put down the answer, 8.

Second step: The outer pair now goes from the 4 of the 14 to the 3 of 32. The inner pair goes from the 1 of 14 to the 2 of 32.

Trachtenberg Direct Multiplication-Two Digit Multiplier-3

We multiply each pair.

    \begin{equation*}   \begin{split}    3\times 4 &= 12\\    2\times 1 &= 2   \end{split} \end{equation*}

Now we add the results.

    \[   12 + 2 = 14 \]

we write the 4 and carry the 1.

Third Step: The outer pair now goes from the 4 of the 14 to the 0 before the 3 of the 32. The inner pair goes from the 1 of 14 to the 3 of 32.

Trachtenberg Direct Multiplication-Two Digit Multiplier-4

We multiply each pair.

    \begin{equation*}   \begin{split}    0\times 4 &= 0\\    3\times 1 &= 3   \end{split} \end{equation*}

then we add the results.

    \[     0 + 3 = 3 \]

We need to add the carried 1.

    \[ 	3 + 1 = 4 \]

In this case, we do not need the second zero and we can stop here. we would only need the second zero if we had to carry in the third step but we only had a 4.

trachtenberg-direct-multiplication-two-digit-multiplier-5

The answer is 448.

Three Digit Multiplier


When we have three digit multipliers the first step we only use one pair of numbers, the second step we use two pairs of numbers and for the remaining steps we use three pairs of numbers to calculate our result.

We will look at 456 times 123 and we write out the equation on one line. As we have a three digit multiplier we put three leading zeros on the multiplicand. The answer will be written directly beneath the multiplicand.

Trachtenberg Direct Multiplication-Three Digit Multiplier-1

First step: We multiply the 6 from 456 with the 3 of 123.

Trachtenberg Direct Multiplication-Three Digit Multiplier-2

    \[   6 \times 3 = 18 \]

The result is 18 so we write 8 and carry 1.

Second Step: The outer pair goes from the 3 of 123 to the 5 of 456, the inner pair goes from the 2 of 123 to the 6 of 456.

Trachtenberg Direct Multiplication-Three Digit Multiplier-3

We multiply each of the pairs.

    \begin{equation*}   \begin{split}    5\times 3 &= 15\\    6\times 2 &= 12   \end{split} \end{equation*}

We add the results together and then add the carried 1.

    \begin{equation*}   \begin{split}    15 + 12 &= 27\\    27 + 1 &= 28   \end{split} \end{equation*}

We write the 8 and carry the 2.

Third Step: The outer pair goes from the 3 of 123 to the 4 of 456, the middle pair goes from the 2 of 123 to the 5 of 456 and the inner pair goes from the 1 of 123 to the 6 of 456.

Trachtenberg Direct Multiplication-Three Digit Multiplier-4

We multiply each of the pairs.

    \begin{equation*}   \begin{split}    4\times 3 &= 12\\    5\times 2 &= 10\\    6\times 1 &= 6   \end{split} \end{equation*}

We add the results together and then add the carried 2.

    \begin{equation*}   \begin{split}    12 + 10 &= 22\\    22 + 6 &= 28\\    28 + 2 &= 30   \end{split} \end{equation*}

We write the 0 and carry the 3.

For the remaining steps we continue to use the three pairs which all originate from the multiplier, the 123 and in each successive step they connect to the next digit on the left in the multiplier until we reach the leftmost zero.

Fourth Step: The outer pair goes from the 3 of 123 to the first zero to the left of the 4 of 456.

Trachtenberg Direct Multiplication-Three Digit Multiplier-5

We multiply each of the pairs.

    \begin{equation*}   \begin{split}    0\times 3 &= 0\\    4\times 2 &= 8\\    5\times 1 &= 5   \end{split} \end{equation*}

We add the results together and then add the carried 3.

    \begin{equation*}   \begin{split}    0 + 8 &= 8\\    8 + 5 &= 13\\    13 + 3 &= 16   \end{split} \end{equation*}

We write the 6 and carry the 1.

Final Step: The outer pair goes from the 3 of 123 to the second zero to the left of the 4 of 456.

Trachtenberg Direct Multiplication-Three Digit Multiplier-6

We multiply each of the pairs.

    \begin{equation*}   \begin{split}    0\times 3 &= 0\\    0\times 2 &= 0\\    4\times 1 &= 4   \end{split} \end{equation*}

We add the results together and then add the carried 1.

    \begin{equation*}   \begin{split}    0 + 0 &= 0\\    0 + 4 &= 4\\    4 + 1 &= 5   \end{split} \end{equation*}

We write the 5.

We do not need to do the next step because we did not have any carry from this step. When we reach the last zero on the left of the multiplicand all digits of the multiplier are multiplied by zero which equals zero. This is a placeholder for any carry from this step.

Trachtenberg Direct Multiplication-Three Digit Multiplier-7

So our answer is 56,088

Four Digit Multiplier


When we have four digit multipliers, we will use up to four number pairs similar to what we did with the three digit multipliers.

We will look at 3241 times 1234 and we write out the equation on one line. As we have a four digit multiplier we put four leading zeros on the multiplicand. The answer will be written directly beneath the multiplicand.

Trachtenberg Direct Multiplication-Four Digit Multiplier-1

As we have done for all the earlier examples we start with one number pair from the units digit of the multiplier and multiplicand and in each step, the outer pair of the units of the multiplier moves one digit to the left in the multiplicand until we reach the final zero. As we progress more inner pairs will be available until we are using all the multiplier digits in number pairs.

First step: We multiply the outer number pair from the unit digit of the multiplier and the multiplicand and get a result of 4.

Trachtenberg Direct Multiplication-Four Digit Multiplier-2

    \[   1 \times 4 = 4 \]

Second Step: We now have two pairs available.

Trachtenberg Direct Multiplication-Four Digit Multiplier-3

    \begin{equation*}   \begin{split}    4\times 4 &= 16\\    1\times 3 &= 3   \end{split} \end{equation*}

We add the results together.

    \[    16 + 3 = 19\\ \]

We write the 9 and carry the 1.

Third Step: We now have three pairs available.

Trachtenberg Direct Multiplication-Four Digit Multiplier-4

    \begin{equation*}   \begin{split}    2\times 4 &= 8\\    4\times 3 &= 12\\    1\times 2 &= 2   \end{split} \end{equation*}

We add the results together and then add the carried 1.

    \begin{equation*}   \begin{split}    8 + 12 &= 20\\    20 + 2 &= 22\\    22 + 1 &= 23   \end{split} \end{equation*}

We write the 3 and carry the 2.

Fourth Step: We now have the maximum of four pairs available.

Trachtenberg Direct Multiplication-Four Digit Multiplier-5

    \begin{equation*}   \begin{split}    3\times 4 &= 12\\    2\times 3 &= 6\\    4\times 2 &= 8\\    1\times 1 &= 1   \end{split} \end{equation*}

We add the results together and then add the carried 2.

    \begin{equation*}   \begin{split}    12 + 6 &= 18\\    18 + 8 &= 26\\    26 + 1 &= 27\\    27 + 2 &= 29   \end{split} \end{equation*}

We write the 9 and carry the 2.

Fifth Step: We multiply each of the next four pairs.

Trachtenberg Direct Multiplication-Four Digit Multiplier-6

    \begin{equation*}   \begin{split}    0\times 4 &= 0\\    3\times 3 &= 9\\    2\times 2 &= 4\\    4\times 1 &= 4   \end{split} \end{equation*}

We add the results together and then add the carried 2.

    \begin{equation*}   \begin{split}    0 + 9 &= 9\\    9 + 4 &= 13\\    13 + 4 &= 17\\    17 + 2 &= 19   \end{split} \end{equation*}

We write the 9 and carry the 1.

Sixth Step: We multiply each of the next four pairs.

Trachtenberg Direct Multiplication-Four Digit Multiplier-7

    \begin{equation*}   \begin{split}    0\times 4 &= 0\\    0\times 3 &= 0\\    3\times 2 &= 6\\    2\times 1 &= 2   \end{split} \end{equation*}

We add the results together and then add the carried 1.

    \begin{equation*}   \begin{split}    0 + 0 &= 0\\    0 + 6 &= 6\\    6 + 2 &= 8\\    8 + 1 &= 9   \end{split} \end{equation*}

We write the 9.

Final Step: We multiply each of the next four pairs.

Trachtenberg Direct Multiplication-Four Digit Multiplier-8

    \begin{equation*}   \begin{split}    0\times 4 &= 0\\    0\times 3 &= 0\\    0\times 2 &= 0\\    3\times 1 &= 3   \end{split} \end{equation*}

We add the results together.

    \begin{equation*}   \begin{split}    0 + 0 &= 0\\    0 + 0 &= 0\\    0 + 3 &= 3   \end{split} \end{equation*}

We write the 3.

We do not need to go to the last zero because we do not have any carry from this step.

Trachtenberg Direct Multiplication - Four Digit Multiplier - 9

So our answer is 3,999,394

Multipliers Of Any Length


I hope it was becoming obvious, that no matter how many digits there are in the multiplier, using the direct multiplication we are following the same pattern. The only difference is how many number pairs we use as this is determined by the number of digits in the multiplier, assuming the multiplier is always smaller than the multiplicand.

This pattern will also work when the multiplier and multiplicand are different sizes. The multiplicand can be any length that is the same or larger than the multiplier.

Multiplying Numbers Ending with Zeros


When using direct multiplication on numbers that end in one or more zeros we can simplify the multiplication by simply removing the trailing zeros from the numbers then adding them back to the answer.

We can state this rule as:

Collect all zeroes at the end of the multiplicand and at the end of the multiplier and put them at the end of the answer. Then go ahead and multiply without paying any further attention to them.

Look at the following equation.

    \[        234000 \times 3200 \]

There is a total of 5 trailing zeroes, three on the multiplicand and two on the multiplier. We can rearrange the equation as follows.

    \[     234 \times 32 \times 100,000  \]

Now ignoring the 100,000 we can setup the equation as so:

Trachtenberg Direct Multiplication - Trailing zeros-1

Following our rule above before we start we can write the five zeroes at the end of our answer before we even begin the equation.

Trachtenberg Direct Multiplication - Trailing zeros-2

The final solution would look like this:

Trachtenberg Direct Multiplication - Trailing zeros-3

The direct multiplication is a shortcut method compared to the traditional multiplication method as we only write down the answer. However, when multiplying numbers that include larger gidits like 7, 8 and 9 and the multiplier is several digits or more then the numbers added together become quite large. These larger numbers can make the direct multiplication more difficult. Trachtenberg also noticed this and so he tried to refine the method further and he came up with the unit-tens method of multiplication which greatly reduces the size of the numbers to be added together after each multiplication on a number pair.