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.

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

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

THEN
(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

THE CALCULATION
((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

QED

© C. Dale Brittain 2016

The Abacus

Medieval people used Roman numerals.  The Arabs had used what we call Arabic numerals (big surprise), 1, 2, 3 etc., having probably gotten them ultimately from India.  Arabic numerals first became known in the West around the year 1000, and were pushed by the Italian scholar Fibonacci around the year 1200, but they were really not used widely until the late Middle Ages.

It is very difficult to do arithmetic with Roman numerals.  Try adding the following, and you'll see what I mean:

XLIV

+IX

The answer of course is LIII.  But you knew that.

Because arithmetic was (and is) very useful for a lot of things, medieval people used the abacus.  The most basic version is a series of wires with beads, set in a frame.  The bottom wire is for the "ones place," the next for the "tens place," and so.  (You learned about "tens place" in school, right?)  So the number LIII (53) would be represented by three beads pushed along on the bottom wire and five on the next wire.  The use of the abacus goes back at least to the ancient Greeks and probably Egypt and Babylon.

In the twelfth century in England, the royal treasury used what was essentially a checker board for the same purpose, putting stones on the squares rather than moving beads on a wire.  England's treasury is still known as the Exchequer.

Medieval people thus conceptualized numbers as we do, even if they wrote them as Roman numerals.  This is not at all strange.  Think about the following:

seventeen

+ eleven

We switch them into numbers in our heads to get the answer and don't even think about it.  Medieval people did the same thing.

Medieval arithmetic was complicated by not having a zero.  They of course knew about "nothingness." The number "ten" on an abacus was a 1 in the "tens place" and nothing in the "ones place."  But when counting backwards, minus-one was next to one, without a zero in between.  Thus, when AD-BC dating was invented in the fifth century AD, the year 1 BC came immediately before the year 1 AD.  Zero came from the Arabs and was first really explained as a number by Fibonacci.

© C. Dale Brittain 2016