Here are 20 skills to improve your Trachtenberg Basic Multiplication.
Like most things the more familiar you are with something the easier it seems to be and so to help improve your Trachtenberg Basic Multiplication skills you need to become more familiar with the techniques involved. Much of the Trachtenberg Basic Multiplication is meant to be done mentally and although none of the calculations you need to do are difficult the more familiar they are to you the faster you will be able to do them.
The Trachtenberg Basic Multiplication arguably does not involve you doing any multiplication. Yes, you do double numbers but as well as being considered multiplying by two, doubling can also be considered as just adding a number to itself. The rest of the technique involves adding, subtracting and “halving” which are very simple things to do.
I am not going to repeat the rules you need for the Trachtenberg Basic Multiplication as I assume you have already read through the rules and read how I remember those rules. Instead what we are going to do here is look at the individual techniques that are involved and try to help you get more comfortable with them so that when you practice the method it will all seem very familiar.
I have broken down all the rules involved for all the multipliers from zero to twelve and come up with 20 skills to improve your Trachtenberg Basic Multiplication, that is twenty distinct mental calculation techniques you need to know to multiply using the Trachtenberg Basic Multiplication method. By becoming familiar with these techniques you will become faster when doing the calculations and we show you how to think more efficiently as well.
When you are doing calculations, and probably just about everything else for that matter, there is a part that you are consciously doing and a larger part you are doing subconsciously. To get more efficient at doing the calculations we need to get more of the work being done subconsciously rather than consciously.
An example would be seeing the number 2 and knowing that you need to double it, instead of thinking to yourself “2 plus 2 is 4” you need to just think “4” and let your subconscious do the work of adding the two plus two. We will go into this further when we look at the mental calculations techniques. Not all the techniques involve a single calculation, some involve up to four separate calculations that need to be done. However, each technique is a complete calculation that needs to be done on a single digit using the Trachtenberg Basic Multiplication. With a little practice you will be able to see the digit, know the technique required and just have the answer “appear” in your conscious thoughts.
As you go through each technique try covering up the answers on the page and see how fast you can come up with the answer. A few minutes practice with these techniques will improve your skills because you will do more and more of the work subconsciously and only consciously think the answer.
In many of the lists below where the number or digit we are looking at is in a larger number then the digit we are looking at is in red. The neighbor, if any, is always the next digit on the right of our number.
20 Skills To Improve Your Trachtenberg Basic Multiplication
01. Multiplying By Zero
Zero conquers all, so any number multiplied by zero the answer will always be zero. As soon as you see “multiplied by zero” the other number is irrelevant the answer will be zero.
02. Reduce By One
Reduce by one, or to put it another way “subtract one”. For this we simply look at the digit we are interested in and subtract one. I did say most of these techniques are very simple.
Yes, there will be times where you may get a negative number in your calculation but this is usually offset by a carry from the previous digit that will bring your total back to at least zero.
03. Reduce By Two
Reduce by two, or “subtract two”. For this we simply look at the digit we are interested in and subtract two.
04. Double The Number
Doubling the number or adding it to itself, whichever way you approach it the result is what is important.
05. "Half" The Number
This “half” is only a true half if the number is even, such as four then half is two, however, for the odd numbers we are only interested in the whole number and we ignore any decimal value. For example half of 5 is 2.5 but we are only interested in the 2 and we ignore the point five.
06. “Half” And Reduce By One
We have two separate calculations here, first working out the whole number “half” of the digit we are looking at, as we just did above, then subtracting one from that.
With a little practice you can get this down from two thoughts to one and go straight to the answer.
07. “Half” And Reduce By Two
Same as the previous technique except we are subtracting two from the “half” of the digit we are looking at.
To start with you will calculate the answer in two steps, first the “half” then subtracting two.
With a little practice you should try to just think of just the final answer in one step.
08. Subtract From Ten
Here we are using complements of ten so the number you think of when you look at a digit is the number you need add to that digit to equal ten.
The red figures are the figures we are looking at while the black figures help us to show the figure in different positions in the larger number. This applies to all the techniques from here.
You may have noticed that when subtracting from ten we only ever looked at the right hand digit, this is because you will only ever subtract from ten for the right most digit in the number.
09. Use The Neighbor
This is the first of only two techniques that do not actually use the number or digit we are looking at. The neighbor in this case is always the next digit to the right of the digit you are looking at. This one does not need any calculation at all as you ignore the number you are looking at and simply use the value of the neighbor, if any.
As you can see there may not always be a neighbor or the neighbor may be zero, in both cases the answer is always 0.
10. Add The Neighbor
This time you need to add the neighbor, the next digit on the right from the number you are looking at. Try just looking at the two digits and just think of the answer of adding those two digits together.
In the case where there is no neighbor, or the neighbor is zero, in both cases you add 0, or more simply you just use the number.
11. Subtract From Ten and Double
You are looking for the complement of ten for the number you’re looking at then doubling the complement value. Again you initially will need to think of this in two steps but try to practice doing it in one.
As mentioned earlier you will only ever subtract from ten for the right most digit in the number.
12. Subtract From Nine and Add The Neighbor
You are looking for the complement of nine for the number you’re looking at then adding the neighbor to the complement value.
You may have noticed that the number you subtract from 9 was never the rightmost digit of the number but can be any of the other digits. We saw in the previous technique you only subtract from ten on the right-most digit of a number.
13. Double And Add The Neighbor
Double the number or add it to itself, then add the neighbor, the next digit to the right.
When there is no neighbor or when the neighbor is zero you can stop as soon as you have doubled the number as you will only add zero to what you already have.
14. Subtract From Nine, Double And Add The Neighbor
This is the first technique we come across that has a maximum of three parts to it; subtracting from nine, doubling that result and then adding the neighbor, if there is one.
As we are subtracting from nine in the first step this means we can do this on any digit except on the right-most digit of the multiplicand.
When we get a zero from the subtraction we can leave out doubling it and go straight to adding the neighbor.
15. Subtract From Ten, And Add Five If The Number Is Odd
We are looking for the tens complement of the number we are looking at and we add five only if the number in red is odd.
Remember, we only subtract from ten on the right-most digit. This is a one step technique for even numbers and two step only for odd numbers.
16. Subtract From Ten, Double, And Add Five If The Number Is Odd
This technique is similar to the previous except we double the result after subtracting from ten and before adding five if the digit is odd.
This is a two step technique for even numbers and three step only for odd numbers.
As we are subtracting from 10 the digit we are looking at is always the right-most digit of the number.
17. Double And Add Five If The Number Is Odd, Add “Half” The Neighbor
Here we double the number we are looking at, then only if the number is odd we add five to the result of the double. Finally, if there is a neighbor, we add "half" the neighbor. As usual the "half" we refer to is the whole number ignoring any decimal places.
As you can see above where the number is even and there is no neighbor, or the neighbor is zero, this technique is reduced to a single step.
So you don't forget to do it, it is better to add the five to odd numbers before adding "half" of any neighbor.
18. Subtract From Nine, Double And Add Five If The Number Is Odd, And Add “Half” The Neighbor
Well this technique is the most complex of all these twenty techniques as it has at most four possible steps. The order the steps are listed in the title is probably the best order in which to do the steps.
Subtract the number we are looking at from 9, then double the result. If the number is odd we add 5 to the result and finally we add "half" the neighbor.
Note that since we are subtracting from 9 the digit we are looking at is never the right-most digit of the number so there is always a neighbor.
19. Add Five To The Number Only If It Is Odd; Add “Half” The Neighbor
20. Use “Half” The Neighbor and Add Five If The Number Is Odd
This is the second of the two techniques that does not use the number or digit we are looking at.
Another way to put this technique is; if the number is even start with zero and if the number is odd start with 5 then simply add "half" the neighbor, if there is one. Which is how we will approach this technique.
If your thinking to yourself "this stuff is too easy" then you are right and that is the beauty of the method. Jakow Trachtenberg spent years finding an easier way to mentally calculate numbers and these are the techniques he came up with for basic multiplication.
When multiplying by numbers such as 10, 11 or 12 you only use one of the techniques above to multiply any number. You use the same technique on every digit in the multiplicand.
For numbers such as 8 or 9 you will combine three of the techniques together to multiply any number.
When combining the techniques you only use one technique on a digit at a time. You will use one technique on the right-most digit of the multiplicand, then a second technique on all the other digits and finally you will use a third technique on the left-most digit of the multiplicand.
For the free members there are plenty of Basic Multiplication worksheets you can download and use to help you practice.