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 .
To generalize this for any two digit number we can replace the numbers with letters.
If we replace with and the with we get .
Squaring this number we can represent as
If we multiply this two digit number with another, which we can represent as then it becomes
Binomial Expansion of
Binomial Expansion of
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.
Starting on the right, double the 7. It has no neighbor to the right to add.
Moving left to the next column, double the 2 and add the 7, its neighbor, then add the 1 carried over.
Moving left to the next column, double the 4 then add its neighbor, 2, and add the carry.
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.
The final result is:
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.
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:
The second step is:
which gives us
adding the carry from step 1 giving us .
We write 2 and carry the 1.
The third step is:
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:
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.
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:
we ignore the tens digit, the 1, and just use the units digit, the 2.
The second step is:
Adding these together we get
so we write 2 and carry the 1.
The third step is:
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.
Step 1 : add the odd column numbers, those in red
Step 2 : add the even column numbers, those in blue
Step 3 : subtract the second total from the first.
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:
The digit sum of 25 is:
The digit sum of 13 is:
The digit sum of 325 is:
To check we multiply the digit sums of the factors:
taking 28 to a digit root is:
So both digit roots are one so our result should be correct.
You can read more about nines remainder or casting out nines here.
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:
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.
Below the running total you write down the number of marks in each column.
To get the total, add up the running total and the marks starting from the right column and working left.
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.
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.
Moving left to the fourth column we add up in an L shape.
There we have the total of as our answer.
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:
I won't go into a full explanation here but you can go to Fast Long Division and to find out more.
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.
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.
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.
To get the last two digits of the answer we square the 6
To get the first two digits of the answer we add the unit digit to 25:
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.
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.
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.