Trachtenberg Speed Math

The Trachtenberg Speed System of Basic Mathematics

The Trachtenberg Speed System of Basic Mathematics is a system of mental mathematics which in part did not require the use of multiplication tables to be able to multiply. The method was created over seventy years ago. The main idea behind the Trachtenberg Speed System of Basic Mathematics is that there must be an easier way to do multiplication, division, squaring numbers and finding square roots, especially if you want to do it mentally.

When solving equations it is not enough to just find the answer you also should prove to yourself you found the right answer. The Trachtenberg System shows you the tools to not only get the answer faster and easier but also how you can check your results easily.

Jakow Trachtenberg spent years in a Nazi concentration camp and to escape the horrors he found refuge in his mind developing these methods. Some of the methods are not new and have been used for thousands of years. This is why there is some similarity between the Trachtenberg System and Vedic math for instance. However, Jackow felt that even these methods could be simplified further. Unlike Vedic math and other systems like Bill Handley's excellent Speed Math where the method you choose to calculate the answer depends on the numbers you are using, the Trachtenberg System scales up from single digit multiplication to multiplying with massive numbers with no change in the method.

The Trachtenberg Speed System of Basic Math can be taught to children once they can add and subtract. They do not need to have learned the multiplication tables before being able to multiply using this system. The basic multiplication method taught in this system is ideal for children and for adults who feel they are poor at multiplying. The rules are easy to learn and it does not take much practice to become proficient.

Once a child or adult is comfortable with the multiplication tables there is the direct method of multiplication. When multiplying a two digit number by a two digit number, and also when squaring two digit numbers, the Trachtenberg System makes use of binomial expansion to make the calculation easier and faster than the more traditional method of multiplication.

A two digit number, say 43 can be split into 40 + 3.
To generalize this for any two digit number we can replace the numbers with letters.
If we replace 40 with a and the 3 with b we get a + b.
Squaring this number we can represent as \left{(}a+b\right{)}^2
If we multiply this two digit number with another, which we can represent as c + d then it becomes \left{(}a + b\right{)} \times \left{(}c + d\right{)}

Binomial Expansion of \left{(}a + b\right{)}^2

    \[ \left{(}a + b\right{)}^2 = a^2 + 2ab + b^2. \]

Binomial Expansion of \left{(}a + b\right{)} \times \left{(}c + d\right{)}

    \[ \left{(}a + b\right{)} \times \left{(}c + d\right{)} = ac + ad + bc + bd \]

The methods for squaring two digit numbers and multiplying a two digit by two digit number make use of the expanded equations to create the rules you follow to do the calculation.

Don't be alarmed if you do not yet understand this very, very short explanation you do not need to know this to use the method but be assured what you are doing is solid math that always works and not a trick that may work in only some cases.

 

Once you start using the direct method of multiplication you find that the numbers you are having to mentally add up can get quite large and the totals also can run into several digits when multiplying by numbers with three or more digits. Jakow Trachtenberg also realized this and again he wondered if there was a way to make this easier. There was, and it is the masterpiece of the Trachtenberg System, it is the tens and unit method of multiplication. The tens and unit method is also referred to as the two finger method as you can use two fingers to help keep track of the calculation as you go along.

The calculation only uses only either the unit digit or the tens digit from a two digit result from multiplying two one digit numbers together. This greatly reduces the size of the numbers you are having to add up in your head.
Once learnt the two finger method will allow you to multiply any two numbers together and simply write the answer down as you mentally solve the equation.

This site is dedicated to teaching you about the Trachtenberg Speed System of Basic Mathematics. A very short introduction follows of the methods that will be covered. You will find more detailed explanations on all these methods throughout this site.
 

Easy Method of Basic Multiplication

The first part of the system involves a set of rules for multiplying any number by the numbers 0 through to 12. It was designed so that you did not need to know your multiplication tables so solving the problem involves arguably no multiplication at all!

For example, to multiply 427 by 12 you follow the rule "double the number and add the neighbor".
The steps you would follow are:

Prepare to start by putting a zero in front of the multiplicand then underline the multiplicand as the result will go below.

    \[ \text{Step 1: }\qquad \underline{\textcolor{Grey}{0}427} \times 12 \]

Starting on the right, double the 7. It has no neighbor to the right to add.

    \[ \text{Step 2: }\qquad 7 + 7 = 14\quad \text{(write 4 and carry 1)}\\ \]

    \[ \renewcommand\arraycolsep{1.5pt} \begin{array}{rl} &\\ \underline{\textcolor{Grey}{0}427}&\times 12 \\ 4& \end{array} \]

Moving left to the next column, double the 2 and add the 7, its neighbor, then add the 1 carried over.

    \[ \text{Step 3: }\qquad 2 + 2 + 7 + 1 = 12 \quad\text{(write 2 and carry 1)}\\ \]

    \[ \renewcommand\arraycolsep{1.5pt} \begin{array}{rl} &\\ \underline{\textcolor{Grey}{0}427}&\times 12 \\ 24& \end{array} \]

Moving left to the next column, double the 4 then add its neighbor, 2, and add the carry.

    \[ \text{Step 4: }\qquad 4 + 4 + 2 + 1 = 11 \quad\text{(write 1 and carry 1)} \]

    \[ \renewcommand\arraycolsep{1.5pt} \begin{array}{rl} &\\ \underline{\textcolor{Grey}{0}427}&\times 12 \\ 124& \end{array} \]

Moving to the last column, doubling zero is still zero so we ignore this and just add the neighbor, the 4 then add the 1 carried over.

    \[ \text{Step 5: }\qquad 4 + 1 = 5 \quad\text{(write 5)} \]

The final result is:

    \[ \renewcommand\arraycolsep{1.5pt} \begin{array}{rl} &\\ \underline{\textcolor{Grey}{0}427}&\times 12 \\ 5124& \end{array} \]

The answer is 5124. You could do the whole thing in your head or at worst just write out each digit of the answer as you work it out. If you want a complete example of what is actually being done have a look at multiplying by twelve. This is a very simplified example for this introduction.

Multiplying by the numbers 0, 1, 2 and 10 are so simple that the rules are not new when multiplying by these numbers.

This was just the first chapter of the book the next chapters involved speed multiplication by the direct method and the "two finger" method. These methods involve a different approach to the actual multiplication, the "two finger" method in particular was designed to keep the calculations as simple as possible.

Direct Multiplication

The direct multiplication method is actually the same method as done in Vedic maths when doing the "crosswise and vertical" but is presented on one line.

We will have a look at a small example and will show the two ways the equations are generally done. The version on the left is how the equations are done in the Trachtenberg method, the second is how it is shown in Vedic Math.

The calculations are:

The first step is:
Direct Multiplication method Step 1

    \[ 3 \times 4 = 12 \qquad\text{so we write 2 and carry the 1} \]

The second step is:
Direct Multiplication method Step 2

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

which gives us 8 + 3 = 11
adding the carry from step 1 giving us 12.
We write 2 and carry the 1.

The third step is:
Direct Multiplication method Step 3

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

plus 1 from the carry gives us 3
and the answer is 322.
 
This method can handle larger numbers but it does mean that the calculations that need to be added together, especially in the middle steps can get rather difficult. This is where the "two finger" method Jakow developed allows you to multiply any two numbers together, no matter how large and be able to use simple multiplication.

"Two Finger" or Units and Tens Multiplication

The Units and Tens method breaks the multiplication down into a series of 1 digit multiplications and treats the results of multiplication as a two digit result:

    \begin{equation*} \begin{split} 2 \times 3 &= 06\quad\text{here the result is 6 so we put a zero in front to have two digits}\\ 8 \times 7 &= 56\\ 9 \times 9 &= 81 \end{split} \end{equation*}

As 81 is the highest possible result from 1 digit multiplication we know there will always be two digits, if we put zero in front of the single digit results. The two digits are a units digit and a tens digit.

When explaining the method we draw a line from the multiplier to the multiplicand that has a forked end to indicate that we will multiply the digit from the multiplier with two digits from the multiplicand.
Unit and Tens
For the line ending at the U we are only interested in the unit digit of the result of multiplying the digit from the multiplier with the digit in the multiplicand located under the line.
For the angled line ending at the T we are only interested in the tens digit of the result of multiplying the digit from the multiplier with the digit in the multiplicand located under the angled line.

Again this will be a very short example of the method, you can follow the link to read more on the two finger method. We will look at the same example we used above:

The first step is:
Two Finger method step 1

    \[ 3 \times 4 = 1\textcolor{red}{2} \]

we ignore the tens digit, the 1, and just use the units digit, the 2.

The second step is:
Two Finger method step 2

    \begin{equation*} \begin{split} 4 \times 2 &= 0\textcolor{red}{8}\quad\text{ we only use the units digit which is 8}\\ 4 \times 3 &= \textcolor{red}{1}2\quad\text{ we only use the tens digit which is 1}\\ 1 \times 3 &= 0\textcolor{red}{3}\quad\text{ we only use the units digit which is 3} \end{split} \end{equation*}

Adding these together we get

    \[ 8 + 1 + 3 = 12 \]

so we write 2 and carry the 1.

The third step is:
Two Finger method step 3

    \begin{equation*} \begin{split} 4 \times 0 &= 0\quad\text{ this is zero so we ignore it}\\ 4 \times 2 &= \textcolor{red}{0}8\quad\text{ we only want the tens digit which is zero so we ignore it.}\\ 1 \times 2 &= 0\textcolor{red}{2}\quad\text{ we only want the units digit which is 2.}\\ 1 \times 3 &= \textcolor{red}{0}3\quad\text{we only want the tens digit which is zero so we ignore it.}\\ \end{split} \end{equation*}

We add up the 2 plus the 1 carried over and we have 3, we write down 3 and we have our answer of 322.

This example does not do the method justice as it comes into its own when the digits are larger, like 7, 8 and 9. What I did want you to see is that the two methods are similar, the pattern followed is the same. If you first learn the direct method then the two finger method is easier to follow although you do not have to learn the direct method and can simply jump straight into the two finger method.
Once the two finger method is mastered it becomes very quick doing the calculations and they are very easy to do mentally.
 

Checking Results with Digit Roots

Jakow also covered two methods of checking results, which although have been known for hundreds of years, have fallen out of favor in recent times with the advent of the pocket calculators. The methods are casting out nines and casting out elevens.

Very quickly, these methods involve finding a digit root, which is basically the remainder if you divided the number by nine or eleven, depending on which method your using.

For nines remainder the digit root is found by adding all of the digits of the number together and if that sum has more than one digit then adding its digits together until only one digit remains.

For elevens remainder, there are several ways you can calculate the digit root, one way is to start on the right hand digit and add every odd column digit. You then add all the even column digits then subtract this total from the first.

    \[ \textcolor{red}{2}\textcolor{blue}{5}\textcolor{red}{7}\textcolor{blue}{6}\textcolor{red}{4} \]

Step 1 : add the odd column numbers, those in red

    \[ 4 + 7 + 2 = 13 \]

Step 2 : add the even column numbers, those in blue

    \[ 6 + 5 = 11 \]

Step 3 : subtract the second total from the first.

    \[ 13 - 11 = 2 \]

So the elevens remainder of 25764 is 2
There are some other considerations, like what to do if the second total is larger than the first. I won't cover that here you can read more about elevens remainder or casting out elevens here.

In addition, adding the digit root of the numbers added should be the same as the digit root of the answer. This method also works for subtraction, multiplication and division although for subtraction it is better to check it as an addition and for division it is better to check it as a multiplication.

An example using nines remainder in addition:

    \[ 25 \times 13 = 325 \]

The digit sum of 25 is:

    \[ 2 + 5 = 7 \]

The digit sum of 13 is:

    \[ 1 + 3 = 4 \]

The digit sum of 325 is:

    \begin{equation*} \begin{split} 3 + 2 + 5 &= 10\\ 1 + 0 &= \textcolor{red}{1} \end{split} \end{equation*}

To check we multiply the digit sums of the factors:

    \[ 7 \times 4 = 28 \]

taking 28 to a digit root is:

    \begin{equation*} \begin{split} 2 + 8 &= 10\\ 1 + 0 &= \textcolor{red}{1} \end{split} \end{equation*}

So both digit roots are one so our result should be correct.
You can read more about nines remainder or casting out nines here.
 

Speed Addition

In the book a method of speed addition is presented in which you add up the numbers in columns, the order you do each column is not important as each column is separate from the others.

What makes it faster is the one rule of this method of addition, that you do not count past eleven, as soon as you go past eleven while adding up you simply subtract eleven from the total, make a mark next to the figure that caused the total to reach or exceed eleven, and continue on using the reduced total.

Lets have a look at a simple example:

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ &&2&&&3&&&5&&&6& \\ &&3&&&6&&&8&&&2& \\ &&&&&4&&&5&&&7& \\ &&5&&&4&&&2&&&9& \\ \cline{3-13} \text{running total:}&&&&&&&&&&&& \\ \text{marks:}&&&&&&&&&&&& \\ \cline{3-13} \text{total:}&&&&&&&&&&&& \\ \end{array} \]

When adding up a column when your total is greater than 11 you subtract 11 from the total and make a mark next to the number that caused the total to go over 11.
At the bottom of the column write down the total, which will be a maximum of 10, this is part of your running total.

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ &&2&&&3&&&5&&&6& \\ &&3&&&6&&&8&\textcolor{blue}{\text{/}}&&2& \\ &&&&&4&\textcolor{blue}{\text{/}}&&5&&&7&\textcolor{blue}{\text{/}} \\ &&5&&&4&&&2&&&9&\textcolor{blue}{\text{/}} \\ \cline{3-13} \text{running total:}&&10&&&6&&&9&&&2& \\ \text{marks:}&&&&&&&&&&&& \\ \cline{3-13} \text{total:}&&&&&&&&&&&& \\ \end{array} \]

Below the running total you write down the number of marks in each column.

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ &&2&&&3&&&5&&&6& \\ &&3&&&6&&&8&\textcolor{blue}{\text{/}}&&2& \\ &&&&&4&\textcolor{blue}{\text{/}}&&5&&&7&\textcolor{blue}{\text{/}} \\ &&5&&&4&&&2&&&9&\textcolor{blue}{\text{/}} \\ \cline{3-13} \text{running total:}&&10&&&6&&&9&&&2& \\ \text{marks:}&&&&&1&&&1&&&2& \\ \cline{3-13} \text{total:}&&&&&&&&&&&& \\ \end{array} \]

To get the total, add up the running total and the marks starting from the right column and working left.

    \[ \text{column 1: }\quad \textcolor{red}{2} + \textcolor{red}{2} = \textcolor{blue}{4} \\ \]

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ \cline{3-13} \text{running total:}&&10&&&6&&&9&&&\textcolor{red}{2}& \\ \text{marks:}&&&&&1&&&1&&&\textcolor{red}{2}& \\ \cline{3-13} \text{total:}&&&&&&&&&&&\textcolor{blue}{4}& \\ \end{array} \]

Moving left to the second column, we add up in an L shape. To the subtotal, add the number of marks in this column and also add the number of marks in the column to the right which gives us the L shape.

    \[ \text{column 2: }\quad \textcolor{red}{9} + \textcolor{red}{1} + \textcolor{red}{2} = \textcolor{blue}{12}\quad \text{ write 2 and put a dot to indicate the carry.}\\ \]

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ \cline{3-13} \text{running total:}&&10&&&6&&&\textcolor{red}{9}&&&2& \\ \text{marks:}&&&&&1&&&\textcolor{red}{1}&&&\textcolor{red}{2}& \\ \cline{3-13} \text{total:}&&&&&&&\textcolor{blue}{^1}&\textcolor{blue}{2}&&&4& \\ \end{array} \]

Moving left to the third column, add the number of marks in this column and also add the number of marks in the column to the right to the subtotal.

    \[ \text{column 3: }\quad \textcolor{red}{6} + \textcolor{red}{1} + \textcolor{red}{1} + \textcolor{red}{1} = \textcolor{blue}{9}\quad \text{ the third 1 is from the carry in column 2}\\ \]

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ \cline{3-13} \text{running total:}&&10&&&\textcolor{red}{6}&&&9&&&2& \\ \text{marks:}&&&&&\textcolor{red}{1}&&&\textcolor{red}{1}&&&2& \\ \cline{3-13} \text{total:}&&&&&\textcolor{blue}{9}&&\textcolor{red}{^1}&2&&&4& \\ \end{array} \]

Moving left to the fourth column we add up in an L shape.

    \[ \text{column 4: }\quad \textcolor{red}{10} + \textcolor{red}{1} = \textcolor{blue}{11}\quad \text{ just write 11 on this last column no need to carry}\\ \]

    \[ \renewcommand\arraycolsep{0.5pt}\def\arraystretch{1.1} \begin{array}{rrrlrrlrrlrrl} &&&&&&&&&&&& \\ &\textcolor{white}{00}&\textcolor{white}{00}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}&\textcolor{white}{0}& \\ \cline{3-13} \text{running total:}&&\textcolor{red}{10}&&&6&&&9&&&2& \\ \text{marks:}&&&&&\textcolor{red}{1}&&&1&&&2& \\ \cline{3-13} \text{total:}&&\textcolor{blue}{11}&&&9&&^1&2&&&4& \\ \end{array} \]

There we have the total of 11924 as our answer.
 

Long Division

Jakow Trachtenberg also came up with a completely different way of doing long division that does not involve doing any division at all.

A Fully worked example looks like this:

long division

I won't go into a full explanation here but you can go to Fast Long Division and to find out more.

 

Squaring Numbers

Jakow took advantage of a math technique known as binomial expansion to come up with a method to easily find the squares of any two or three digit numbers as well as a specific method for two digit numbers ending in 5 as well as a specific method for two digit numbers where the tens digit is 5. Vedic Math uses the same technique.

Examples of the specific methods are:

Squaring a two digit number ending in 5
Any two digit number ending in 5, when squared, the last two digits of the answer are always 25.

To square 35
The first one or two digits of the answer are found by multiplying the first digit of the number to be squared by the next larger digit.

    \[ 35^2 \]

We know the answer will end in 25.
To find the initial digits of the answer we take the 3 and multiply it by 4, the next larger digit.

    \[ 3 \times 4 = 12 \]

So the answer is 1225

Squaring a two digit number starting with 5
When squaring a two digit number starting with 5 the last two digits are always the units digit squared.
To get the first two digits of the answer we add the units digit to 25.

    \[ 56^2 \]

To get the last two digits of the answer we square the 6

    \[ 6 \times 6 = 36 \]

To get the first two digits of the answer we add the unit digit to 25:

    \[ 25 + 6 = 31 \]

So the answer is 3136
Why add to 25? Because 5 squared is 25.
Follow these links to find out more about squaring two digit numbers or squaring three digit numbers. I suggest reading about squaring two digit numbers first.
 

Square Roots

Jakow Trachtenberg covered finding the square roots of 3 to 8 digit numbers but the method used can be used for even larger numbers. To see an example of how to find the square root of three or four digit numbers here.

For me the method for square roots was the hardest method in the book to get used to but none of the methods I have seen for finding a square root of a large number are easy.

 
The final chapter of the book included some of the algebraic proofs for the Trachtenberg System. Most people would not be interested in the algebraic proofs but they are there to show that the methods do work and there is real math behind the methods.

 

My Thoughts

This book contains gold and I wish I had been taught these methods as a kid. I will be teaching them to my son once he is old enough.

The methods Jakow Trachtenberg distilled from his years of trying to simplify common basic mathematics are wonderful and imaginative. Although he did not invent some of the methods he was able to take that
knowledge and distill it down further than it had been done before and come up with the two finger method and the basic multiplication rules.

I can understand why some would scoff at his achievements, doing multiplication without actually using multiplication, crazy! To those who have already spent the time to memorize the multiplication tables yes it would seem crazy but what about those who have not yet learnt the multiplication tables or those who have trouble learning them. Being given another way to be able to find the answer when you are struggling with multiplication tables is far better than letting them lose confidence in themselves and their math ability.

I have looked into Vedic math as well as reading some of Bill Handley's books. The Vedic math is made to seem almost mystical by the mainly Indian teachers. The methods used by Bill Handley are very similar to the Vedic math. If you have any interest in finding yet another way to do basic mathematics I would recommend Bill Handley's books rather than the Vedic Math for yourself or your children.

Why would I be recommending another method of learning math when I have a whole site here dedicated to the Trachtenberg System? Well the answer is that not everyone learns the same way and sure there are plenty of people for whom the method taught in school is enough. What about the rest? Maybe something here on this site will click for them or maybe it is Bill's methods that will do it.

Why not learn several ways to do the same math problem? Confidence in math gives confidence in other areas as well. Don't just rely on a calculator to do all your calculating and let your brain rot, put your mind to work and you will never regret it. The maths you learn here is the type of math you can use everyday.

Spend some time on this site an have a look around, join up for free and download worksheets to practice. Watch the videos and if you have any suggestions or questions contact me and will do what I can to help you.

 

 
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Comments

    • Hi Elliedan,
      Thank you for taking the time to stop by. I’m glad you found it interesting. If you love math you should really enjoy the different approach to math that Trachtenberg developed and is shown on this site.

      Reply

    • Hi Andre,
      Thank you for the compliment on the site.
      Yes I think the Trachtenberg System is great and gives you a different way to look at basic mathematics which is helpful and even fun.

      Reply

  3. Great site!
    I liked the system enough to buy it off Amazon as a book, but there doesn’t appear to be any subtraction in the book.

    Reply

    • Thank you and your right subtraction is not included in the Trachtenberg System.
      A method I like for simplifying subtraction involves rounding the number to be subtracted then adjusting for the amount rounded.
      For example: 124 – 68
      1) round number to be subtracted: 68 + 2 = 70
      2) subtract rounded number: 124 – 70 = 54
      3) adjust for rounding: 54 + 2 = 56
      This method is great when mentally subtracting numbers.

      Reply

  4. I Love math even when I’m not a genius or not even good at it. But one thing I love is seeing some one that is good at it, it make me feel so happy.
    I would love to see you guys solving problems and I hope I don’t bother any one by doing so.
    I just admire intelligent people and want to see problems I don’t realized are so small one.
    Thank you guys.

    Reply

    • I am not great at math as I relied on calculators too much. That is what started me looking for a way to improve my math and make myself think. When I found the Trachtenberg System it was great to find something that worked generally not just in specific cases. It is also great to see alternate ways problems can be solved. Those math geniuses don’t solve problems the way they are taught in school.
      Glad you like the site and be sure to try some of the worksheets, they have the answers at the back.

      Reply

  5. I was watching the movie Gifted, such a coincidence that I watched while preparing my kids next year curriculum. I have one child is doing addition and subtraction ready for more and one that was working on memorizing multiplication but from this discovery I think I’m switching my methods.

    Would switching the child learning multiplication have issues learning this instead?

    Reply

    • The Basic Multiplication is different to learning the multiplication tables. It may be more interesting to learn more than one way to multiply. Eventually, the multiplication tables are needed but this method can help make multiplication interesting and a bit more fun. It also gives a way to work out the answer if the multiplication tables have not been fully memorized.

      Reply

        • The basic multiplication is the best place to start, you can go to the Multiplication section of this site to find more on this. There is also a link at the footer to a free copy of the original book that has been made available on the web.

          Reply

    • Thank you. It is a fun technique, and I am glad you like it. I also like the fact you can see how it works and use the method without having to understand why it works. If you do want to look at why it works, you can’t avoid some algebra, but it is just as interesting.

      Reply

  8. although i dont love maths but with the view of this i love because this movie “GIFTED” really make me like maths. i know it wont me easier for me to start loving maths. by the way trachtenberg method is best of all.

    Reply

    • Most people seem to either love or hate math, my feelings towards math have gone up and down over the years.
      I’m glad you like the Trachtenberg method, I think it is great too, it renewed my interest in math again.

      Reply

  9. Squaring a two digit number starting with 5. so you can only square if the number is 50 to 59 column and what if the number is between 70 to 79 is there a different way you will solve that too

    Reply

    • Yes, there is a general method for squaring any two digit numbers. There are two slightly quicker methods for the special cases where the numbers either start with five or end in five.
      The general method is explained below the two special case methods.

      Reply

  10. Hi! I just knew about this method about a month ago from ‘Gifted’ movie and I found this site is really helping, thank you! I’m planning to teach my daughter about this system in the future. How old do you think for her to be ready to be taught about this Trachtenberg system? Thank you.

    Reply

    • It is more math skills rather than age that will decide when a child is ready. Once your child can count to thirty, subtract single digit numbers from ten and can add any numbers up to a total of thirty they would be able to start learning the Basic Multiplication method.

      Reply

  11. Hi I really like your site. It has great example and I’m literally shocked. Can’t wait to master this method then teach my son. So much our children can achieve if this method can be tought in schools. Thanks a lot keep it up

    Reply

    • Learning the Trachtenberg method is an excellent way to help build confidence in math and confidence in maths helps with self-confidence in other parts of children’s lives.

      Reply

  13. we all perform math from when we where small children, like when a child counts how many friends will get a popsicle from the refrigerator. Basic stuff subtraction/addition.

    a child may start school and discover multiplication is fun, I was excited To learn more methods and try to create my own.

    when a mathematician performs multiplication for the first time. What happens after this? do they even need practice? Do They instantly have this ability to remember large sequences of numbers?

    the biggest struggle I had was remembering a set of numbers and not loosing them inside my head.
    326
    x 24
    —-
    It does help to see the problem on the board but I challenged myself to complete the task completely in my head with remembering the least amount of numbers.

    =
    144
    48
    72
    —–
    7824

    I would only have to remember 3 sets of Small numbers. I was getting to bigger numbers 123×456…

    this was something I did every day in math class, challange my self go as far as possible.

    I look forward to teaching my kids this method you share. Great article!

    Reply

    • Thank you, Michael. You have to exercise your brain to keep it fit, doing math all in your head as you were, every day is an excellent way to get it in shape. The trick is not to try to remember too much too fast and get yourself frustrated. Starting small and working your way up as you did is ideal. A little bit every day goes a long way.

      Reply

  14. I was watching the movie Gifted today and that’s where i heard about this system. I was always interested in trying some new and interesting stuff and this is really cool. Its just sad that so many young people now a days have no interest in getting knowledge like this…

    Reply

    • Several people mentioned the movie Gifted references the Trachtenberg Method, so I had to watch it. Only a quick reference but I am not complaining. 🙂
      The Trachtenberg Method is a cool method, and yes it would be nice if more people could learn it.

      Reply

  15. After watching a nice movie Gifted yesterday, i realy wanted to know about this Trachtenberg method n it is realy a fabolous method for math lovers ……..

    Reply

  16. I am so glad I found this article. I will definitely teach this to my grandchildren!!! Thanks for sharing this knowledge!

    Reply

  19. Hello sir i found this article really helpful for calculations.but in how much time would i be able to master it and do calculations real quick in my mind.or how much time should i give this method daily

    Reply

    • When learning something new a little bit of practice every day is better than trying to learn it all at once. Since you do not need any tools you can practice anytime you have a few moments to spare. Start with two digit numbers then when you are comfortable with that, try three digit numbers, then later four digit numbers. Try to keep it fun. Exercising your brain is just as important as exercising your body.

      Reply

  20. I happened upon this web page after seeing the movie “Gifted”. I noticed what appear to be errors in the steps for the calculations in the Speed Addition section. Specifically, the terms “right” and “left” appear to be reversed, starting at the point where the marks have been counted:

    “To get the total you add up the running total and the marks starting from the left column and working right.”

    Using the example shown, the 2’s are in the rightmost column, not the leftmost, and every reference to the terms “right” and “left” is incorrect (i.e. reversed) for the remainder of that section.

    I thought you’d want to correct this as soon as possible so as not to confuse anyone using this site to learn some of these techniques.

    Reply

  21. Dear Sir,

    After watching the movie Gifted, I was intrigued and was very interested to know about this method. So I searched the internet and found this article. This article is very interesting and helpful. I am not the genius type but I love Math very much and this article makes me love Math more. I’d like to introduce these methods to my nieces and nephews. And if I were teaching, I’ll be teaching these methods. I hope these methods be shared to other students too.

    Thank you for sharing this. 🙂

    Reply

      • My daughter will turn 5 in Feb 2018. She can count up to 100. I am now starting to teach her addition. Do you have any suggestion on how to teach her addition and later on introduce her to this method? Thank you.

        Reply

        • Hi Dila,
          The Trachtenberg method for addition was designed for a long list of numbers like a bank or accountant would have had to deal with. The idea behind the Trachtenberg Method is trying to teach algorithmic thinking which is a way of getting to a solution through the clear definition of the steps needed.
          Taking this approach to mental addition for a child I have used the following to teach my 5 yr old addition.
          First, some simple rules, One, when adding numbers, changing the order gives the same result. Second, a larger number can be split into two smaller numbers when adding.
          For example, adding 5 + 8, you can split the 8 into 5 and 3 making the problem 5 + 5 + 3.
          When adding two double-digit numbers, add the tens digits together then add the units digits then add the two together.
          For example, 35 + 43, add 30 + 40 = 70, then 5 + 3 = 8, finally 70 + 8 = 78.
          If the child is wrong, I never give the answer or say it is wrong, instead, I will ask questions that will lead to the correct answer.
          For example, if a wrong answer was given for 9 + 7, I would ask something like “What plus 1 equals 7?”. If the answer is correctly given, I would ask “What is 9 + 1?”, if given 10 as the answer, finally ask “What is 10 + 6?”
          My son like to do stuff on his own and he is happier to realize himself that his answer was wrong. Also, I found introducing subtraction the same way works great too. Once your daughter is comfortable with the addition and subtraction then you can start to teach multiplication.
          I have just begun to teach my 5-year-old son the basic multiplication method and have written a simple app for his iPad to help him learn and practice the method.

          Reply

          • Do you have that app on the store? I found apps for the android platform, but none for ipad.

          • Hi Kim,
            No, it is not in the App store. It is in development and is usable on my iPad. However, getting it ready for the App store will take some work but I plan to get it there when I can.

        • I think you should first teach her how to multiply two digit numbers by one digit numbers using the method after she learns addition.

          For example you could teach her how to multiply numbers by 1 then give her exercises on multiplying by 1.
          The next day you could teach her how to multiply by two (also using the trachtenberg system of addition NOT of multiplication, it is important she gets used to the system)
          Then multiplying by 3 etc… all the way to multiplying by twelve.

          I would suggest that you teach her the different methods verbally with a frequency of at least 1 method per day and follow it up with exercises for that particular method. At the end of each exercise I recommend you add a section on pure addition to exercise her mind.

          It is extremely important that you mark her answers and go through her mistakes with her as well as teaching her in the system not in multiplication. An example of what I mean is when teaching her to multiply by 3 you should say: “Subtract the rightmost digit from 10. Subtract the remaining digits from 9. Double the result. Add half of the neighbor to the right, plus 5 if the digit is odd. For the leading zero, subtract 2 from half of the neighbour.” and not “add the number to itself 3 times to multiply by 3 e.g.) 7+7+7.” You should stick with this way of teaching so that she can later easily solve for big numbers using the “2-finger method” or the “direct method.”

          The rules and examples for the different multiplications from 2 to 12 (not including 10) can be found on https://en.wikipedia.org/wiki/Trachtenberg_system#Multiplying_by_2 (copy and paste it if need be.)

          After she has learned the basics of multiplying by addition and subtraction, you can go on to teach her the technique for multiplying large numbers, which can be found online or on this site even.

          I really hope this helps and wish the best for your daughter. Inshaalah she will become a genius.

          If you would like to contact me please use this: hayder.g@icloud.com (not the alias I put in the Email section to avoid junk mail)
          I would be delighted if you choose to do so.

          Thank you/

          Reply

  22. A cool method to try! I am poor in arithmetic and looking forward to learn how to do quick math. Thank you so much for this and I’m pretty sure there will be some improvements in my arithmetic skills. Would love to share this to my friends too!

    Debra

    Reply

  24. Hello Tony,
    thanks for this great info.

    At what moment do you suggest this method should be teached to childeren?
    It makes me wonder how much more of techniques there are available that we don’t know of, for math and other subjects. Any suggestions?

    thanks,

    Reply

    • Sorry, already got the reply in the posts,
      you have posted:

      “It is more math skills rather than age that will decide when a child is ready. Once your child can count to thirty, subtract single digit numbers from ten and can add any numbers up to a total of thirty they would be able to start learning the Basic Multiplication method.”

      Reply

  25. This is awesome, I got to know about this from the movie ‘Gifted’ and I am glad I decided to check it out.

    I don’t understand how you arrived at your answer in the addition problems

    Reply

  27. A friend and I discovered this book in grade school in about 1966. We used the techniques at the chalkboard in class, to ‘smoke’ the other students!

    Reply

    • Doing the speed, or two-finger, multiplication on 497 x 4 you should get the following:

      For the units digit of the answer, you get the U value of 7 x 4 which is 8.
      For the tens digit of the answer, you add the U value of 9 x 4 and the T value of 7 x 4, which is 6 + 2 = 8.
      For the hundreds digit of the answer, you add the U value of 4 x 4 and the T value of 9 x 4, which is 6 + 3 = 9.
      For the thousands digit of the answer, you use the T value of 4 x 4, which is 1.
      Which gives the correct answer of 1988.

      Reply

  31. This was a very useful website. I am currently learning about volume and surface areas but I looked at this website again when I remembered about the movie ‘Gifted’ that I watched last year. I ever searched about the Trachtenberg System, but I decided to focus on my school topics.
    Now I got the chance and I learnt a really helpful way of calculating mental math.
    So far, this was a very good website and you should share it to others.😉

    Reply

  32. hello I found your sight from the movie Gifted thank you for putting this sight assessable I have 4 homeschooled children i am wondering your suggestions on were to start with children to implant this type of method ages 8 to 12

    Reply

    • For children that have already memorized the multiplication tables, it can be fun and informative to look at the basic multiplication method. I personally find it easier to use the basic multiplication method when multiplying by 12, 11 or 5 in particular.
      After reviewing the basic multiplication method, look at the direct multiplication method, where the answer is calculated in one pass even for multi-digit multipliers.
      Learning more than one method gives someone to more tools in their math toolbox and hopefully makes math a bit more interesting to learn.

      Reply

        • The Basic Multiplication is for multiplying any number by zero to twelve, without needing to know the multiplication tables. The direct multiplication can be used to multiply any two numbers, although larger numbers have bigger numbers you must sum and carry. Then there is the two-finger multiplication that can be used to multiply any two numbers.

          Reply

  33. I signed up. How long should it take for my email to arrive?

    Looking forward to be able to use worksheets for my grandchildren.

    Love what you’re doing on this site. Already downloaded the pdf and printed it. Can’t wait to show off lol.

    Reply

    • Hi Cherie, it should not take long, it is always a good idea to check your spam or junk mail folder as well.
      Thank you for the encouragement that the work here has been appreciated.
      If you have any questions let me know.

      Reply

  34. Hi mister,
    Im glad to say it is so helpfull to me sir.
    Im 17 year old and im so low in math but I love math very much.
    Im thanking you for this site, sir.

    Reply

  36. Trachteinberg was a genius..i think if this could have been taught in our schools before..i think no one could have seen maths as a tough subject..i love this article.

    Reply

  37. I have just started to homeschool my kids. One is 5 and the other is 9. the 9 year old knows some basic addition and subtraction, has been introduced to multiplication and division, but her confidence in math from the very beginning has been poor. The schools just push them through the lessons so fast that the kids don’t pick most of it up and then are soon left behind. I think this method is amazing, but it looks like they need to learn their basic addition and subtraction before they can start this. Correct? Or can I just get them started and they will pick it up throughout the lessons?

    Reply

    • I would try starting as you will see where they have trouble then you can work on the weaker areas. You can work through the multipliers in the basic multiplication such that you gradually introduce new concepts in the method.

      Reply

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