We also have a look at the error checking included in this addition using the digital root to check and double-check our results. We will also have a look at what to do if our tests do find a problem. The rule for addition is: Never count higher than eleven What this means is as we are adding up the numbers down a column, when the running total becomes eleven or higher, we subtract eleven from the running total, make a mark next to the digit that we just added, then continue with the reduced running total. When we finish the column, we write down this running total below the column we just added up. The mark can be anything you like, but usually, a small stroke, check mark or tick is best.
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We write 2 and carry the 1. The third step is: plus 1 from the carry gives us 3 and the answer is 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. 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 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 is 2 There are some other considerations, like what to do if the second total is larger than the first. 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 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. 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: 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 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.
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. 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 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 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 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 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 So the answer is Why add to 25? Because 5 squared is 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. 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.
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? Why not learn several ways to do the same math problem? Confidence in math gives confidence in other areas as well.
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.
To keep himself sane whilst living in an extremely brutal and harsh environment, Trachtenberg immersed his mind in a world of mathematics and calculations. As concentration camps do not provide books, paper, pen or pencils nearly all of his calculations had to be performed mentally. This forced Trachtenberg to develop methods and shortcuts for performing calculations mentally. Trachtenberg developed his discoveries into a complete system of mathematics. After the second world war, Trachtenberg started teaching his system of mathematics. He started teaching the more backward children to prove that anyone could learn his system. In he founded the Mathematical Institute in Zurich, where both children and adults were taught the system.
Trachtenberg Speed Maths - Pdf
Continue with the same method to obtain the remaining digits. The calculations for finding the fourth digit from the example above are illustrated at right. The arrow from the nine will always point to the digit of the multiplicand directly above the digit of the answer you wish to find, with the other arrows each pointing one digit to the right. The vertical arrow points to the product where we will get the Units digit, and the sloping arrow points to the product where we will get the Tens digits of the Product Pair. If an arrow points to a space with no digit there is no calculation for that arrow. As you solve for each digit you will move each of the arrows over the multiplicand one digit to the left until all of the arrows point to prefixed zeros.
The Trachtenberg Speed System of Basic Mathematics