Three finger tricks for multiplying

Many math sites teach the following method of using your fingers to remember the multiples of nine: to find the product of 9 times n, hold your hands out in front of you and fold down your n^th finger from the left to separate the tens and the ones. For example, to find 9×4, you would hold down your 4^th finger from the left as in the above photograph. The bent finger separates the tens and ones digits, so the configuration of 3 fingers (folded finger) 6 fingers gives the answer of 36.

While this method has enjoyed great popularity among students and teachers, there are two other lesser-known finger tricks for multiplying numbers.

The first is a way of multiplying {5,6,7,8,9, or 10} by {5,6,7,8,9, or 10}. To begin, subtract 5 from each of the two numbers and hold up the remaining number of digits. For example, to multiply 9 by 7 you would begin by holding up 9-5=4 and 7-5=2 fingers respectively.

Now for the fun part: To find the number of tens in the product, you add the raised fingers: in this case there are 4+2=6 tens. To find the number of ones in the product, you multiply the lowered fingers. In this case, the left hand has 1 lowered finger and the right hand has 3 lowered fingers, so there would be 1×3=3 ones in the product. Putting this together, we see that 9×7 has 6 tens and 3 ones, or 9×7=63! Amazing!

Warning: You may have to do a little bit of carrying. For example, to multiply 6×6 you’d hold up only 1 finger on each hand, leaving 4 lowered fingers on each. The method above gives 1+1=2 tens, and 4×4=16 ones so the product is 20+16=36.

But wait, there’s more! What if you want to find the product of {10, 11, 12, 13, 14, or 15} by {10, 11, 12, 13, 14, or 15}? You begin by subtracting 10 from each number and holding up the remaining number of digits.  For example, to multiply 14 by 12 you would hold up 14-10=4 and 12-10=2 fingers respectively:

With this trick, you only look at the raised fingers. To 100, add as many tens as the sum of the raised fingers (4+2=6) and as many ones as the product of the raised fingers (4×2=8). That is, 14×12=100+60+8=168. This method illustrates the algebraic fact

Warning: As before, you may have to do some carrying. For example, multiplying 13 by 14 would lead to holding up 3 and 4 fingers respectively. Using this method you’d start with 100, add 3+4=7 tens, and then add 3×4=12 ones; that is, 13×14=100+70+12=182. But even with the carrying, this method is so simple that it just makes my day.

These two latter methods are discussed in “Finger Reckoning,” History of Mathematics, Vol II, by D. E. Smith (©1925,1953, Dover Publishing, pp. 201-2). Smith explains that the first method is similar to other shortcuts used in the Middle Ages, while the final method is still being used in modern times. “Modern”, presumably, being 80 years ago.

Thanks to Emmett for agreeing to pose as the hand-model for this post!