The Trachtenberg Two Finger Multiplication method is done in a similar method to the Trachtenberg Direct Multiplication method with one difference. Instead of each digit of the multiplier connecting with one digit of the multiplicand as in the Direct Multiplication, in the Two Finger Multiplication method, each digit of the multiplier connects with two digits of the multiplicand to create a pair product.

Pair Product


A pair product is a number obtained by multiplying a pair of digits by a separate (multiplier) digit in a special way: we use the multiplier digit to multiply each digit of the pair separately, and then we add together the units digit of the product of the left-hand digit of the pair and the tens digit of the product of the right-hand digit of the pair. Another name for the Two Finger Multiplication is the Unit-Tens Multiplication method.

Trachtenberg Two Finger Multiplication Pair-Product

Taking the pair of digits 3 and 2, we multiply both 3 and the 2 by 8.

    \begin{equation*}   \begin{split}    3\times 8 &= 2\textcolor{red}{4}\quad\text{\textcolor{red}{U}}\\    2\times 8 &= \textcolor{blue}{1}6\quad\text{\textcolor{blue}{T}}   \end{split} \end{equation*}

We are only interested in the units digit of the result of the first equation and the tens digit of the result of the second equation. We now add these two digits together.

    \[    \textcolor{red}{4} + \textcolor{blue}{1} = 5 \]

The result of this pair product is 5.

Two Digit Result for Single Digit Multiplication


The Two Finger Multiplication relies on each single digit multiplication having a two digit result. In the few cases where the result is actually a single digit result we make it a two digit result by putting a leading zero, which does not affect the value of the result.

    \begin{equation*}   \begin{split}    0\times 1 &= 00\\    1\times 2 &= 02\\    2\times 3 &= 06\\    3\times 4 &= 12\\    5\times 6 &= 30\\    7\times 8 &= 56\\    9\times 9 &= 81   \end{split} \end{equation*}

Leading Zeroes


When using the Two Finger 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. 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.

 

One Digit Multipliers


We will first look at one digit multipliers where each product pair one digit of the final answer.

Step One: We setup the equation on one line and we put a single leading zero on the multiplicand as we are using a single digit multiplier. We put the U of the UT over the position where we will write the next, or in this case the first, figure of the answer.

trachtenberg-two-finger-multiplication-1-digit-multiplier-1

The U is above the 6 in this case so we want the unit value from 6 times 7.

    \[   6 \times 7 = 4\textcolor{red}{2} \]

The T is not above a digit so has nothing to do in this step. We write the 2 as the first figure of the result.

Step Two: We move the UT one digit to the left. We now have both a U and a T equation.

Trachtenberg Two Finger Multiplication - 1 Digit Multiplier - 2

    \begin{equation*}   \begin{split}    8 \times 7 = 5\textcolor{red}{6}\\    6 \times 7 = \textcolor{blue}{4}2   \end{split} \end{equation*}

The U value is 6 and the T value is 4, we add these together to get the pair product.

    \[  \textcolor{red}{6} + \textcolor{blue}{4} = 10 \]

We write zero and carry the 1.

Step Three: We move the UT one digit to the left and do the UT equations.

Trachtenberg Two Finger Multiplication - 1 Digit Multiplier - 3

    \begin{equation*}   \begin{split}    4 \times 7 = 2\textcolor{red}{8}\\    8 \times 7 = \textcolor{blue}{5}6   \end{split} \end{equation*}

The U value is 8 and the T value is 5, we add these together then add the carried 1.

    \begin{equation*}   \begin{split}    \textcolor{red}{8} + \textcolor{blue}{5} = 13\\    13 + 1 = 14   \end{split} \end{equation*}

We write 4 and carry the 1.

Step Four: We move the UT left over the next digit of the multiplicand.

Trachtenberg Two Finger Multiplication - 1 Digit Multiplier - 4

    \begin{equation*}   \begin{split}    3 \times 7 = 2\textcolor{red}{1}\\    4 \times 7 = \textcolor{blue}{2}8   \end{split} \end{equation*}

The U value is 1 and the T value is 2, we add these together then add the carried 1 to get the product pair.

    \begin{equation*}   \begin{split}    \textcolor{red}{1} + \textcolor{blue}{2} = 3\\    3 + 1 = 4   \end{split} \end{equation*}

We write 4.

Step Five: We move the UT left over the last figure, the leading zero in front of the multiplicand.

Trachtenberg Two Finger Multiplication - 1 Digit Multiplier - 5

    \begin{equation*}   \begin{split}    0 \times 7 = 0\textcolor{red}{0}\\    3 \times 7 = \textcolor{blue}{2}1   \end{split} \end{equation*}

The U value is 0 and the T value is 2.

    \[    \textcolor{red}{0} + \textcolor{blue}{2} = 2 \]

We write 2 and we have the answer 24402.

Trachtenberg Two Finger Multiplication - 1 Digit Multiplier - 6

You may have noticed that each digit of the multiplicand is used twice, once under the U and again under the T thus using both digits of their product with the multiplier.

 

Two Digit Multipliers


We will now go through an example using a two-digit multiplier. This time, we will have two UT pair products to sum for each digit of the answer.

Step One: We set up the equation on one line and we put two leading zeroes on the multiplicand as we are using a two-digit multiplier. We put the U of the UT from the unit figure of the multiplier over the position where we will write the next, or in this case the first, figure of the answer.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 01

    \[    2 \times 7 = 1\textcolor{red}{4} \]

We write 4.

Step Two: We move the UT linked to the 7 of the multiplier left over the next digit of the multiplicand. We also introduce the UT linked to the 8 of the multiplier and put the U of the UT above the right-hand figure of the multiplicand.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 02

We now have three equations to find the U and T values.

    \begin{equation*}   \begin{split}    7 \times 7 &= 4\textcolor{red}{9}\\    2 \times 7 &= \textcolor{blue}{1}4\\    2 \times 8 &= 1\textcolor{red}{6}   \end{split} \end{equation*}

    \[    \textcolor{red}{9} + \textcolor{blue}{1} + \textcolor{red}{6} = 16 \]

We write 6 and carry the 1.

Step Three: We move each UT one place to the left.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 03

Now we have four equations to find the U and T values.

    \begin{equation*}   \begin{split}    9 \times 7 &= 6\textcolor{red}{3}\\    7 \times 7 &= \textcolor{blue}{4}9\\    7 \times 8 &= 5\textcolor{red}{6}\\    2 \times 8 &= \textcolor{blue}{1}6   \end{split} \end{equation*}

    \[    \textcolor{red}{3} + \textcolor{blue}{4} + \textcolor{red}{6} + \textcolor{blue}{1} = 14 \]

We then add the 1 from the carry.

    \[   14 + 1 = 15 \]

We write 5 and carry the 1.

Step Four: We move each UT one place to the left.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 04

We find the U and T values, add them together then add the 1 from the carry.

    \begin{equation*}   \begin{split}    6 \times 7 &= 4\textcolor{red}{2}\\    9 \times 7 &= \textcolor{blue}{6}3\\    9 \times 8 &= 7\textcolor{red}{2}\\    7 \times 8 &= \textcolor{blue}{5}6   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{2} + \textcolor{blue}{6} + \textcolor{red}{2} + \textcolor{blue}{5} = 15\\    &15 + 1 = 16   \end{split} \end{equation*}

We write 6 and carry the 1.

Step Five: We move each UT one place to the left.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 05

We find and add the U and T values, then add the 1 from the carry.

    \begin{equation*}   \begin{split}    0 \times 7 &= 0\textcolor{red}{0}\\    6 \times 7 &= \textcolor{blue}{4}2\\    6 \times 8 &= 4\textcolor{red}{8}\\    9 \times 8 &= \textcolor{blue}{7}2   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{4} + \textcolor{red}{8} + \textcolor{blue}{7} = 19\\    &19 + 1 = 20   \end{split} \end{equation*}

We write 0 and carry the 2.

Last Step: We move each UT one place to the left.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 06

We find and add the U and T values, then add the 2 from the carry.

    \begin{equation*}   \begin{split}    0 \times 7 &= 0\textcolor{red}{0}\\    0 \times 7 &= \textcolor{blue}{0}0\\    0 \times 8 &= 0\textcolor{red}{0}\\    6 \times 8 &= \textcolor{blue}{4}8   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{0}0 + \textcolor{red}{0} + \textcolor{blue}{4} = 4\\    &4 + 2 = 6   \end{split} \end{equation*}

We write 6.

Trachtenberg Unit-Tens Multiplication - 2 Digit - 07

We have our answer, 606564.

Notice that although we are still just doing single digit multiplication by just using the U and T values the number we need to add or carry in each step is much smaller than it would have been if we used the Direct Multiplication method.

 

Four Digit Multipliers


We will now go through an example using a four-digit multiplier. This time, we will have up to four UT pair products to sum for each digit of the answer.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 01

Step One: We set up the equation on one line and we put four leading zeroes on the multiplicand as we are using a four-digit multiplier. We put the U of the UT from the unit figure of the multiplier over the unit figure of the multiplicand.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 02

    \[    2 \times 1 = 0\textcolor{red}{2} \]

We write 2.

Step Two: We move the UT linked to the unit of the multiplier left over the next digit of the multiplicand. We also introduce the UT linked to the tens digit of the multiplier and put the U of the UT above the unit figure of the multiplicand.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 03

We now have three equations to find the U and T values.

    \begin{equation*}   \begin{split}    8 \times 1 &= 0\textcolor{red}{8}\\    2 \times 1 &= \textcolor{blue}{0}2\\    2 \times 5 &= 1\textcolor{red}{0}   \end{split} \end{equation*}

    \[    \textcolor{red}{8} + \textcolor{blue}{0} + \textcolor{red}{0} = 8 \]

We write 8.

Step Three: We move each UT one place to the left. We also add a third UT linked to the hundreds digit of the multiplier and put the U of the UT above the unit figure of the multiplicand.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 04

Now we have five equations to find the U and T values.

    \begin{equation*}   \begin{split}    7 \times 1 &= 0\textcolor{red}{7}\\    8 \times 1 &= \textcolor{blue}{0}8\\    8 \times 5 &= 4\textcolor{red}{0}\\    2 \times 5 &= \textcolor{blue}{1}0\\    2 \times 6 &= 1\textcolor{red}{2}   \end{split} \end{equation*}

    \[    \textcolor{red}{7} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{1} + \textcolor{red}{2}  = 10 \]

We write 0 and carry the 1.

Step Four: We move each UT one place to the left. We also add a fourth UT linked to the thousands digit of the multiplier and put the U of the UT above the unit figure of the multiplicand.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 05

We find the U and T values, add them together then add the 1 from the carry.

    \begin{equation*}   \begin{split}    4 \times 1 &= 0\textcolor{red}{4}\\    7 \times 1 &= \textcolor{blue}{0}7\\    7 \times 5 &= 3\textcolor{red}{5}\\    8 \times 5 &= \textcolor{blue}{4}0\\    8 \times 6 &= 4\textcolor{red}{8}\\    2 \times 6 &= \textcolor{blue}{1}2\\    2 \times 3 &= 0\textcolor{red}{6}   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{4} + \textcolor{blue}{0} + \textcolor{red}{5} + \textcolor{blue}{4} + \textcolor{red}{8} + \textcolor{blue}{1} + \textcolor{red}{6} = 28\\    &28 + 1 = 29   \end{split} \end{equation*}

We write 9 and carry the 2.

Step Five: We move each UT one place to the left.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 06

We find and add the U and T values, then add the 2 from the carry.

    \begin{equation*}   \begin{split}    0 \times 1 &= 0\textcolor{red}{0}\\    9 \times 1 &= \textcolor{blue}{0}9\\    9 \times 5 &= 4\textcolor{red}{5}\\    4 \times 5 &= \textcolor{blue}{2}0\\    4 \times 6 &= 2\textcolor{red}{4}\\    7 \times 6 &= \textcolor{blue}{4}2\\    7 \times 3 &= 2\textcolor{red}{1}\\    8 \times 3 &= \textcolor{blue}{2}4   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{5} + \textcolor{blue}{2} + \textcolor{red}{4} + \textcolor{blue}{4} + \textcolor{red}{1} + \textcolor{blue}{2} = 18\\    &18 + 2 = 20   \end{split} \end{equation*}

We write 0 and carry the 2.

Step Six: We move each UT one place to the left.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 07

We find and add the U and T values, then add the 2 from the carry.

    \begin{equation*}   \begin{split}    0 \times 1 &= 0\textcolor{red}{0}\\    0 \times 1 &= \textcolor{blue}{0}0\\    0 \times 5 &= 0\textcolor{red}{0}\\    9 \times 5 &= \textcolor{blue}{4}5\\    9 \times 6 &= 5\textcolor{red}{4}\\    4 \times 6 &= \textcolor{blue}{2}4\\    4 \times 3 &= 1\textcolor{red}{2}\\    7 \times 3 &= \textcolor{blue}{2}1   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{4} + \textcolor{red}{4} + \textcolor{blue}{2} + \textcolor{red}{2} + \textcolor{blue}{2} = 14\\    &14 + 2 = 16   \end{split} \end{equation*}

We write 6 and carry the 1.

Step Seven: We move each UT one place to the left.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 08

We find and add the U and T values, then add the 2 from the carry.

    \begin{equation*}   \begin{split}    0 \times 1 &= 0\textcolor{red}{0}\\    0 \times 1 &= \textcolor{blue}{0}0\\    0 \times 5 &= 0\textcolor{red}{0}\\    9 \times 5 &= \textcolor{blue}{4}5\\    9 \times 6 &= 5\textcolor{red}{4}\\    4 \times 6 &= \textcolor{blue}{2}4\\    4 \times 3 &= 1\textcolor{red}{2}\\    7 \times 3 &= \textcolor{blue}{2}1   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{4} + \textcolor{red}{4} + \textcolor{blue}{2} + \textcolor{red}{2} + \textcolor{blue}{2} = 14\\    &14 + 2 = 16   \end{split} \end{equation*}

We write 7 and carry the 1.

Step Eight: We move each UT one place to the left.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 09

We find and add the U and T values, then add the 1 from the carry.

    \begin{equation*}   \begin{split}    0 \times 1 &= 0\textcolor{red}{0}\\    0 \times 1 &= \textcolor{blue}{0}0\\    0 \times 5 &= 0\textcolor{red}{0}\\    0 \times 5 &= \textcolor{blue}{0}0\\    0 \times 6 &= 0\textcolor{red}{0}\\    9 \times 6 &= \textcolor{blue}{5}4\\    9 \times 3 &= 2\textcolor{red}{7}\\    4 \times 3 &= \textcolor{blue}{1}2   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{5} + \textcolor{red}{7} + \textcolor{blue}{1} = 13\\    &13 + 1 = 14   \end{split} \end{equation*}

We write 4 and carry the 1.

Last Step: We move each UT one place to the left.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 10

We find and add the U and T values, then add the 1 from the carry.

    \begin{equation*}   \begin{split}    0 \times 1 &= 0\textcolor{red}{0}\\    0 \times 1 &= \textcolor{blue}{0}0\\    0 \times 5 &= 0\textcolor{red}{0}\\    0 \times 5 &= \textcolor{blue}{0}0\\    0 \times 6 &= 0\textcolor{red}{0}\\    0 \times 6 &= \textcolor{blue}{0}0\\    0 \times 3 &= 0\textcolor{red}{0}\\    9 \times 3 &= \textcolor{blue}{2}7   \end{split} \end{equation*}

    \begin{equation*}   \begin{split}    &\textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{0} + \textcolor{red}{0} + \textcolor{blue}{2} = 2\\    &2 + 1 = 3   \end{split} \end{equation*}

We write 3 and we have our answer 346049082.

Trachtenberg Two Finger Multiplication - 4 Digit Multiplier - 11
 

This technique will work with numbers of any size, the number of pair products used is the same as the number of digits in the multiplier. This method is a little different to how you are used to multiplying so will require some practice to get comfortable with it.