I must know why

Yesterday I was reading Chapter 1 “Fingerprints” from Tobias Dantzig’s NUMBER: The Language of Science. Very fascinating stuff to me about primitive ways of counting –or not– and theories of how different numbering systems developed.

I got stuck on pg. 9 where Mr. Dantzig relates that

“to this day, the peasant of central France (Auvergne) uses a curious method for multiplying numbers above 5.”

He goes on to explain how they determine a number of fingers to bend down on each hand and then do some adding and multiplying of the up and down fingers and come up with the correct product. I tested it. It does work.  It also works with numbers below 5 except for the fact that it is (naturally) a little difficult to bend down a negative number of fingers.

But why does it work?

I tried to come up with the algebra formula that would explain it and I struggled with a portion of it. This was driving me crazy. I sought help from Jack. He and I ended up at the dining room table for at least an hour trying to figure this out. We determined at some point that it is simply because of the base 10 system, but that is still not enough for me. At one point I got James’ appropriately-named  Math-U-See blocks to determine if I could see what was going on.  That is, why do these numbers work this way?

After scrawling several pages of formulas and examining the problem from different angles, including using the Math-U-See blocks, we managed to make some sense of it and simplify the formulas some. But I’m still not satisfied.

(You can see now the truth of my opening statements on my About page.)

If anyone is interested in helping me solve a potentially unimportant puzzlement, here’s Mr. Dantzig”s  description of the little trick the French peasants do with their fingers:

If he wishes to multiply 9 x 8, he bends down 4 fingers on his left hand (4 being the excess of 9 over 5), and 3 fingers on his right hand (8 – 5 = 3). Then the number of the bent-down fingers gives him the tens of the result (4 + 3 = 7), while the product of the unbent fingers gives him the units (1 x 2 = 2).