Algebraic Explanation for Wrist Multiplication

Date: 09/10/2004 at 22:01:14
From: Heidi
Subject: Why does wrist multiplication work?

I am learning wrist multiplication and am curious as to why and how 
it works for the numbers 11-15.  It is easier and quicker to use this 
application but I don't understand how it works. 

For example, 12 x 13.  Your wrists count as 50 each, each finger is
10.  Hold up 2 fingers on one hand and 3 fingers on the other.  Touch
your wrists together and count this as 100.  You should have 5 fingers
showing and this counts as 50 (10 per finger).  Then multiply 2
fingers by 3 fingers and count this as 6.  Add 100 + 50 + 6 and it
equals 156!!! Why and how??!!

Date: 09/11/2004 at 06:40:43
From: Doctor Luis
Subject: Re: Why does wrist multiplication work?

Hi Heidi,

If you think about it, you can write the numbers 11,12,13,14,15
as 10 + x, where x is between 1 and 5 (the number of fingers in
a typical hand).

Then, to multiply two such numbers, you get:

  (10 + x)(10 + y) = 10*10 + x*10 + 10*y + x*y
                   = 100 + 10(x+y) + x*y

So, the first 100 can be accounted for with the two wrists (50 each).
The x+y weighed by 10 is the value you gave to the total number of 
fingers, which do add up according to our algebraic formula above.  
Of course, the product of x and y is just the product of the actual 
number of fingers we're holding up in each hand, which was 2*3 = 6 in 
your example.  This is a nifty mnemonic way of multiplying two numbers 
that I hadn't seen until now.

Please note that the formula itself holds no matter what x and y are,
since it's an algebraic identity.  The reason we restrict them to the 
values 1 thru 5 is simply that we find it difficult to hold up six
fingers in a single hand, much less the 28 extra fingers that would be
required to multiply 13 x 42 using this method.  The sixty-fingered
creatures that live in Alpha Proxima Centauri can probably have
better luck with that last multiplication.

I hope this helped!

Let us know if you have any more questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/11/2004 at 23:18:27
From: Heidi
Subject: Thank you (Why does wrist multiplication work?)

Thank you so much for your help.  The formula really helps to 
understand the concept.