Finger Multiplication

Tobias Dantzig makes mention of the lost art of finger multiplication in his book Number: The Language of Science, first published in 1930.  "Finger counting is a lost art among modern civilized people ... only a few hundred years ago finger counting was such a widespread custom in Western Europe that no manual of arithmetic was complete unless it gave full instructions in the method."  Using the method of finger multiplication, you can find the product of the numbers 6, 7, 8, and 9 with just your own ten fingers and a knowledge of the addition and multiplication tables up to five.

To use finger multiplication, hold out your hands palm down, label your thumbs 6, your index fingers 7, your middle fingers 8, your ring fingers 9, and your pinky fingers 10, and touch the tips of the two fingers that correspond with the two numbers you are multiplying together.  Then count all the fingers including and below the two fingers touching together and multiply that number by ten, and add to it the product of the left fingers above the two fingers touching together and the right fingers above the two fingers touching together. 

For example, let’s say you wanted to multiply 8 x 9.  You would hold your hands out palm down, and touch the tip of your left middle finger (which corresponds with 8) with the tip of your right ring finger (which corresponds with 9).  Including and below these two fingers there are a total of 7 fingers, and 7 x 10 = 70.  There are 2 fingers above the two fingers touching together on the left hand and 1 finger above the two fingers touching together on the right hand, and 2 x 1 = 2.  Adding these two numbers together gives 70 + 2 = 72.  So 8 x 9 = 72.

Or let’s say you wanted to multiply 6 x 7.  You would hold your hands out palm down, and touch the tip of your left thumb (which corresponds with 6) with the tip of your right index finger (which corresponds with 7).  Including and below these two fingers there are a total of 3 fingers, and 3 x 10 = 30.  There are 4 fingers above the two fingers touching together on the left hand and 3 fingers above the two fingers touching together on the right hand, and 4 x 3 = 12.  Adding these two numbers together gives 30 + 12 = 42.  So 6 x 7 = 42.

In general, if we want to multiply a x b using finger multiplication, then there are a – 5 fingers and b – 5 fingers including and below the two fingers touching together and 10 – a fingers and 10 – b fingers above the two fingers touching together.  The sum of all the fingers including and below the two fingers touching together and multiplying by ten can be represented as 10(a – 5 + b – 5), and the product of the left fingers above and the right fingers above can be represented as (10 – a)(10 – b).  Adding these together and simplifying would give 10(a – 5 + b – 5) + (10 – a)(10 – b) = (10a – 50 + 10b – 50) + (100 – 10a – 10b + ab) = ab, which is why finger multiplication works.

There is an easier method of finger multiplication, but it only works if one of the factors is 9.  Hold your hands out, count off the other factor you want to multiply by 9, and put that finger down.  Then the product will be the number of fingers to the left times ten plus the number of fingers to the right.

For example, let’s say you wanted to multiply 3 x 9.  You would hold your hands out and put down the third finger from the left (the left hand middle finger).  There are 2 fingers to the left and 7 fingers to the right, and 2 x 10 + 7 = 27.  So 3 x 9 = 27.

Or let’s say you wanted to multiply 8 x 9.  You would hold your hands out and put down the eighth finger from the left (the right hand middle finger).  There are 7 fingers to the left and 2 fingers to the right, and 7 x 10 + 2 = 72.  So 8 x 9 = 72.

In general, if we want to multiply a x 9 using finger multiplication for nines, then there are a – 1 fingers to the left of the finger that is down and 10 – a fingers to the right of the finger that is down.  The number of fingers to the left times ten plus the number of fingers to the right can be represented as 10(a – 1) + (10 – a).  Simplifying would give 10a – 10 + 10 – a = 9a, which is why this method of finger multiplication for nines also works.

As observed by Dantzig, finger multiplication is a dying art.  Few people know how to use finger multiplication for nines, and even fewer know how to use finger multiplication for the numbers six through nine.  It is likely that finger multiplication is no longer taught in schools anymore because for most people it takes just as much effort to memorize the multiplication table up to ten than to remember all the steps involved for finger multiplication.