Here we will have a look at how to multiply double digit numbers. First using a method dubbed the Direct Method by Jakow Trachtenberg and the second using his “two finger” method. Both these methods will work for any two digit number combinations.
If you are interested in looking at multiplying by numbers up to twelve, have a look at basic multiplication.
The Direct Method
The Direct Method is rarely taught in schools but has been known for centuries. In school, you are usually taught to write the result of multiplying each digit of the multiplier on a separate line then adding up the total.
In the Direct method, you don’t need to write down the sub-totals. Instead, you only write the answer.
To achieve this, you are doing a pair of calculations in each step. Pairs that equate to nothing are ignored.
These pairs are called the outer and inner pairs. The outer pair always joins the unit digit of the multiplier with the digit in the multiplicand we are currently looking at. The inner pair always joins the tens digit of the multiplier with the digit to the right of the digit we are working on in the multiplicand.
This method is essentially the same as used in Vedic Mathematics when they use the “vertical and crosswise” sutra when multiplying two digit numbers. The equation style is the only real difference. In Vedic Mathematics the equation is written on two lines, as seen below. For the direct method the equation is on one line with the answer under the multiplicand.
I will show the method with the equation in both styles for the first example so that you can see how it works in each style.
You can watch the video about direct multiplication using two digit multipliers or you can continue reading the following examples.
Let’s look at an example,
For the Direct method, we put leading zeros on the multiplicand. The number of leading zeros is always the same as the number of digits in the multiplier, so in multiplying 2 digit numbers, we always add 2 leading zeros.
The next thing is we multiply the two unit digits together
This step involves multiplying the tens digit of one number with the units digit of the other.
When writing the equation on a single line, if we draw curved connecting lines between the multiplied digits, we get an outer pair and an inner pair. When writing the equation on two lines, we get a cross when we draw straight connecting lines between the multiplied digits.
Adding the results of these two equations we get 14, so we write 4 and carry the 1.
In this step, we multiply the tens digits of each number.
3 plus 1 from the carry in step 2 gives us 4, so we write 4, and we have our answer of 448
Note: When writing the equation on a single line, the outer pair in this step connects to a zero so the result of this pair is zero and can be ignored.
In this example the mental calculations we need to do are relatively simple and since we are doing fewer steps than the traditional method of multiplication it is faster. However, there is a drawback to this approach, especially when the digits involved are larger.
Lets look at another example,
We multiply the two unit digits together
So we write the 2 and carry the 7.
This is where it gets tough, especially if you are trying to mentally do the calculation,
Adding 81 + 56 gives us 137 then we add the carry of 7 from step 1 to give us 144.
So we write the 4 and carry the 14.
We have 63 to which we add the carry of 14 to give us 77. We write down 7 and carry the 7.
Following the original method and the reason for the leading zeros, we have an extra step because of the carry.
So we have zero plus the carry of 7 which is 7. We write down 7 which gives us our answer of 7742.
This step may seem redundant, and we could have just written down the carry in the last step, but as you are learning the method, it is better to follow the whole equation through until you are familiar enough with the method to take the little shortcuts.
As you can see when the numbers include the digits 7, 8 and 9, the math becomes more complicated, especially if you are trying to do it mentally.
Jakow also realized this, and he tasked himself to find a simpler way to accomplish this. Enter the “two finger” method as he called it, which simplifies the calculations you need to do.
Before going on to the two finger method we need to get some more background information for single digit multiplication
Tens and Units
When multiplying two single digit numbers together the result can only be a one or two digit number. The highest single digit number is 9 and .
If we put a zero in front of any single digit result, we can treat all the results of multiplying two single digit numbers as two-digit results, a units digit and a tens digit.
We will use this concept as we go through the calculation using the “two finger” method. The tens will be represented by "T" and the units by "U"
Before we look at the two finger method, let’s have a look at 98 x 79 again but we will write down each step of the multiplication in a different row as we go along. We will place the results in the right columns based on whether they are units, tens, hundreds or thousands.
As you can see each step in the multiplication can be seen as multiplying two single digit numbers.
The Two Finger Method
In this method, we will be only taking the units digit of the result for the vertical connector and only the tens unit for the sloping connector.
So in the above we have two calculations:
When mentally doing the calculations we know , and we only want the units digit so when looking at the we try just to think "1". For the we just try to think "7" as we only require the tens digit.The more you practice this the faster and easier it will become.
Now we have the two results, 1 and 7; we now add them together to give us the "pair-product," which in this case is 8.
A way to remember which digits to combine for the pair-product is to imagine or write the first few times, the results of the first multiplications just below the numbers in the multiplicand. Of the four digits in the answers, you always take the two center digits.
Taking the unit digit from the left-hand number, 81, and the tens digit from the right-hand number, 72, we get the pair-product of 8.
Except for the first and last step, each digit of the multiplier will product a pair-product, but since we are adding, we can just add the values together as we go along. We do not need to calculate the first pair-product then calculate the second one and then add them together.
To see this in action lets re-look at our earlier example,
We ignore the 7, the tens digit, and just use the 2, the units digit. We write down 2.
For the outer pair
For the inner pair
Adding the pair-products we we get:
So we write the 4 and carry the 1.
For the outer pair
For the inner pair
Adding the pair-products and the carry we get:
So we write the 7 and carry the 1.
For the outer pair
For the inner pair
Adding the 6 and the carry, we get 7.
So we write the 7 giving us our answer of 7742.
As you can see the actual calculations are the same as you would do in the direct method but since you are only taking one digit and adding those one digit numbers, the adding of the results is a lot simpler.
Now if we look again at the example where we wrote each result of the multiplication on separate lines, and compare that with the numbers we just used in the four steps of this example.
You will notice that the numbers we used in each step to create the pair-products correspond to the numbers in each column, although not necessarily in the same order in a column. Step 1 corresponds to the right-hand column, step 2 with the next column to the left, and so on. In both cases, we have the same carry and so we end up with the same result.
When doing these calculations, you would not be writing everything down as I have done here to explain the method. Instead, you would do all the workings mentally and only write down the final answer. With a little practice, the calculation of the pair-products becomes faster and easier.
I hope that you have found the methods of multiplying double digit numbers displayed here both interesting and useful.