Multiplication: A parlor finger trick

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Click numbers on the left and on the right to set up an example:

This Flash applet teaches you how to compute times tables from 6 to 10, on your fingers. The trick was first recorded around 15th century, when merchants collected techniques called finger reckoning to calculate their profits. Back then, calculating devices were bulky. But you can do finger reckoning under the table or in your pocket, away from prying eyes of competitors. Now we do finger reckoning for fun.

Finger Reckoning Trick

Instructions in words

Number your fingers, starting from 6 on thumbs. You may want to write numbers on your nails.

  • thumbs 6
  • index 7
  • middle 8
  • ring 9
  • pinkies 10

Bring together the fingers with the numbers you are multiplying, for example, index (7) and middle (8) for 7*8. The fingers that touch, and any fingers under, count as tens, so you get 50 in this case. Multiply the other fingers, left and right, in this case 3*2=6. So you get 56 as the total, which is the answer to 7*8.

Why does it always work? Here is a formal algebraic proof…

Suppose you multiply numbers A and B on your fingers. The trick tells you to count this many tens on either hand: (A-5) + (B-5) as your tens. And your ones are (10-A)*(10-B).

The total is:
(A-5+B-5)*10+(10-A)*(10-B)=
10A+10B-100+100-10A-10B+AB=
AB

But I bet this isn’t how did the merchants of old came up with the system! Was it something about their counting boards or abacuses that suggested the finger trick?

Do you know a plausible explanation about the origins of this trick?

 

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