In the Finger Multiplication post, we have learned how we can cleverly multiply numbers from 6 to 10 using  our fingers without memorizing the multiplication table. In this post, we will discuss the reason why this method works.  The proof can be written in a few lines, but since this article falls under the elementary school mathematics category, we will discuss it with more details.

Let’s examine the first example in the previous article as shown above. In the example, we wanted to multiply 8 by 7, so we connected the ring and the middle fingers. In this discussion, we will call the connected fingers and all the fingers below them down fingers, while we call all the fingers above them up fingers.

The Down Fingers

There are 3 down fingers on the left hand and 2 down fingers on the right (see figure). We multiplied each of them by 10, and added them up, which equaled 50. Using 8 , 7, and 5 (the total number of fingers on one hand), we can relate down fingers as shown in the following equation: 10(2) + 10(3) = 10(8-5) + 10(7-5). Can you see why?

The Up Fingers

Looking at the up fingers we have 2 at the left and 3 at the right and we  multiplied them to get 6. Note that we can also relate them to 10 (the total number of fingers) as well as 8 and 7, as shown in the following equation: 2 x 3 =  (10-8) x (10 – 7).

Adding them up and Generalizing

Using the equations above, we can express the product P of 8 and 7 using the following equation:

P =  [10(8-5) + 10(7-5)] + [(10-8) x (10 - 7)] = 56

As an exercise, verify if this method works in the second example in theprevious article.

Generalizing, what if we want to multiply the numbers m and n?

We now generalize by multiplying m and n.  The product P is given by the equation P = [10(m-5)+10(n-5) + (10-m)(10-n)].  Simplifying the equation we have P = 10m – 50 + 10n – 50 + (100 – 10m – 10n + mn) = mn. But mn is the product of m and n, which is what we want to show. That is the reason why the  method works.

Students often fear multiplication of numbers, which involve lot of 9’s in them. However the truth is exactly the opposite. The higher the number of 9’s in the question, the easier it is to calculate the correct answer.

There can be three cases with multiplication with series of 9’s

Here we are discussing Case 1 only and leaving the rest two cases for the readers to explore.

Question: Multiply 764 by 999

Question: Multiply 2345123 by 9999999

Subtract 1 from 2345123 to get 2345122 as left part of answer. Now subtract individual digits of 2345122 from 9 and write answer as 7654877. This becomes right part of answer. The final answer is 23451227654877

Use of Yavadunam Sutra for Finding square of a number near 100

Sutra Yavadunam function over a base value. The bases may be 10 , multiples of 10 or 100 , multiples of 100 or 1000 , multiples of 1000 etc. Here we limit ourselves to base 100 only.

The sutra says whatever be the difference of the number from the base add (if the number is more than the base) or subtract (if the number is less than the base) that much to the number and on the right hand side set the square of the difference. This gives us the final answer. Remember that the number of digits on right hand side is equal to the number of zeros in base.

Since 108 is 8 more than 100 therefore we have added 8 here as per the sutra and on the right hand side we set square of 8.

Since 98 is 2 less than 100 therefore we have subtracted 2 here as per the sutra and on the right hand side we set square of 2. See that we set 4 as 04 as the right hand side of the answer should have number of digits equal to the number of zeroes in the base.

I was surprised when I came across with an article on Polish hand magic, a strategy for multiplication using the fingers. I remember us doing it when we were in the first grade, but in a slightly different way.  Here in our country, we call it Finger Math or Finger Multiplication.

Finger multiplication is a strategy for multiplying numbers from 6-10.  It is used by pupils in the early grades who have not memorized the multiplication table yet.  The idea is to assign the numbers from 6 to 10 to each finger on both hands (see Figure 1): 6 to the pinkie, 7 to the ring finger, 8 to the middle finger, 9 to the index finger, and 10 to the thumb.

To multiply, we do the following (see Figure 2):

(1) Connect the fingers assigned to the numbers that we want to multiply. For example, in multiplying 8 and 7, we connect the middle finger and the ring finger.

(2) Next, we count by 10s the connected finger and all the fingers below them, then add them all up. In the figure, we have 50.

(3) Next, we count by 1s all the fingers above the connected fingers, and then multiply the number of fingers on the left hand to the number of fingers on the right hand.  In the second figure, we have 2 x 3 = 6.

(4) Add the total in 2 and the product in 3. We have 50 + 6 = 56.

In the second example (see Figure 3), we multiply 7  and 9. We do this by connecting the index finger and the ring finger. Again, we count by 10s the connected fingers below and all the fingers below it. This gives us 60. Next, we count the fingers above the connected fingers by 1 and multiply the total at the left by the total at the right which 3 x 1 = 3. We add up: 60 + 3 = 63.

Now, try multiplying the following if it works: (1) 8 and 6, (2) 9 and 6, and (3) 10 and 0.

Amazed?