Here we will look at the method Jakow Trachtenberg devised for doing basic multiplication, that is multiplying by the numbers zero to twelve, with out having to know the multiplication tables. For basic multiplication using the Trachtenberg system you don't do any multiplication at all. How cool is that! What you do instead is addition or some subtraction in a special way. Actually there is a different method of addition used depending on what number you are multiplying by.
Using these methods, it is possible to calculate the answer to a problem instead of having to have first learnt your times tables. In conventional math if you don't know the answer to "7 x 8" then your stuck, unless you start drawing seven rows of eight circles and then counting them up. Instead using these methods you could use either the method for multiplying by seven or the method for multiplying by eight to work out the answer of "56".
Before looking at the rules themselves we ought to explain the layout of the equations and some of the terms used a bit more.
For the Trachtenberg system you add a zero at the beginning of the multiplicand so that you don't stop too early in the calculation. In the most of the methods in the table above each digit of the multiplicand is used twice, once as the digit we are looking at then again as the neighbor of the next digit to the left. When you look at the pages for each multiplier how we use the digit and its neighbor will become clear. For basic multiplication how you layout the calculation does not make any difference. In either layout we simply write the answer below the multiplicand.
The Red Box
When we go through the calculation we will use a red box to highlight which digits we are referring to. The digit on the left in the box is always the digit we are looking at and the digit on the right in the box is the neighbor. Let me show you what I mean. Here we are looking at 5 and its neighbor is 6.
Adding 5 if Odd
Adding 5 to the number only if it is odd.
This means if the number, or digit, we are looking at is either : 1, 3, 5, 7 or 9 then it is odd. So we add 5 to any of these number.
If the number, or digit, we are looking at is either: 0, 2, 4, 6 or 8 then it is even and we do not add anything.
We will look at a few examples to make sure this is clear:
When we use the "half" we are referring to the integer part of the half value.
When the number is even then the "half" is the true half value.
We can represent even numbers by 2n such as:
6 = 2n where n = 3 as 2 x 3 = 6
8 = 2n where n = 4 as 2 x 4 = 8
We can represent odd numbers by 2n + 1 such as:
7 = 2n + 1 where n = 3 as 2 x 3 + 1 = 7
5 = 2n + 1 where n = 2 as 2 x 2 + 1 = 5
The number we are using as the "half" is the number represented by n in the examples above.
The table below will show you all the "half" values for all the digits from zero to nine.
When doing these multiplications we should add a leading zero to the multiplicand so that we do not stop too early in the calculation. A leading zero does nothing to change the number we are multiplying but gives us a placeholder.
In the examples above you will notice the leading zero, which is grey rather than black just to indicate it is added by us for reference in the calculation.
The table below list the multipliers from 0 to 12 and the rule(s) for each. For completeness, multiplying by 0, 1, 2 and 10 are included but there is nothing new.
The rules can look a little confusing when you first see them all together like this but each multiplier has a link to its own page where I will explain each rule in more detail.
The order of the multipliers is not in number order as you may expect but in an order that follows a pattern in the rules used for each multiplier. I believe that ordering them this way is helpful when remembering the rules.
As our brains like patterns I will list some of the patterns I have found in these rules:
- The higher numbers, 8 and above, use the neighbor if at all.
- The lower numbers, 7 down, only use "half" the neighbor if at all
- whenever "half" the neighbor is used then adding 5 if the number is odd is also used.
- The lower number using "half" the neighbor is always 5 less than the higher number using the neighbor.
I go into more details on these patterns when I look at how I remember the Trachtenberg Basic Multiplication System rules. For now when you are ready go through the multipliers in the order listed and practice each one before moving on to the next.
You can click on one of the multipliers above or click here to start with multiplying by eleven.
If you have any questions or comments please leave them below.