Friday, July 29, 2016

Multiplication on your fingers

There were no calculators in the Middle Ages.  So if one needed to make a calculation, one might do it either in one's head or on one's fingers.  For adding or subtracting, an abacus came in very useful, as I discussed in an earlier post, but it's hard to do multiplication on an abacus.

Medieval people did however work out a way to do multiplication on their fingers.  Here's how it worked.

First, they memorized the "times table" (as school children often call it) up through 5 X 10.

Then they pondered the nature of the "ten's place" and "one's place."  In the number 46, for example, the 4 is in the "ten's place" (four tens), and the 6 is in the "one's place" (six ones).  This is extremely easy to visualize on an abacus.  On an abacus one adds, for example, 27 and 25.  Add the numbers (beads) in the ten's place (on the second wire) and get 4.  Add the numbers in the one's place (first or bottom wire) and get 12.  The 2 goes in the one's place in the answer, and the 1, which is in the ten's place, is added to the 4 already there, by flicking a bead over.  The answer is 52.

Okay, all those following along at home, remember how they explained numbers back in middle school, get out your abacus if necessary, and let's keep going.

For multiplication, they had a finger-calculating method to be used when both numbers were between 5 and 10 (if one number was smaller, you had to just have memorized the answer).  One hand represents each number.  On each hand, put up the number of fingers by which the number is greater than 5.

Example, suppose you are multiplying 7 X 7.  On each hand, you put up two fingers, because 7 is two greater than 5.  Now add the upright fingers together.  This is the ten's place.  So the ten's place in your answer will be 4.  Now look at your hands again, at the tucked-down fingers.  There are 3 of them on each hand.  Multiply them together, 3 X 3.  The answer is 9.  This goes in the one's place.  So the answer is 49.

Or multiply 6 X 8.  One hand has one finger sticking up, the other three.  Add them together.  You get 4 for the ten's place.  And how many fingers are tucked down?  Four on one hand, two on the other.  Multiply them to get 8 for the one's place.  Answer, 48.

Or multiply 6 X 10.  One finger sticks up on one hand, five on the other (because 10 is 5 more than 5).  Add them.  The ten's place is 6.  The tucked down fingers are four and none.  None times four is none.  So just a 6 for the ten's place and nothing for the one's place, giving 60.

Or multiply 6 X 7.  One finger from one hand plus two fingers from the other hand, added together, gives 3.  But multiplying four tucked-down fingers from one hand times three tucked-down fingers from the other hand gives 12.  So the 2 (of 12) goes in the one's place, the 1 gets added to the ten's place.  Answer, 42.  The meaning of life.

Practice.  Fool your friends.

My brother the engineer worked out the mathematical formula that explains this:

a = First number (left hand)
b = Second number (right hand)
5 ≤ a ≤ 10
5 ≤ b ≤ 10

(a - 5) = Number of fingers UP on left hand
(b - 5) = Number of fingers UP on right hand
(5 - (a - 5)) = Number of fingers DOWN on left hand
(5 - (b - 5)) = Number of fingers DOWN on right hand

((Number of fingers UP on left hand) + (Number of fingers UP on right hand) x 10) + ((Number of fingers DOWN on left hand) x (Number of fingers DOWN on right hand)) =
(((a - 5) + (b - 5)) x 10) + ((5 - (a - 5) x (5 - (b - 5)) =
((a + b - 10) x 10) + ((10 - a) x (10 - b)) =
(10a + 10b - 100) + (100 - 10b - 10a + (a x b)) =
a x b


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