Finger Multiplication for the 9s, Explained

Date: 04/22/2002 at 10:29:39
From: Sarah Richards
Subject: 9 Times Table Finger Trick

I long ago learned the nine times table 'finger trick' that you 
feature in the Dr. Math archives, but am puzzled as to exactly WHY it 
works.

Thanks a lot.
Sarah

Date: 04/22/2002 at 12:17:46
From: Doctor Peterson
Subject: Re: 9 Times Table Finger Trick

Hi, Sarah.

You are referring to this page, I believe:

   Finger Multiplication for the 9s
   http://mathforum.org/library/drmath/view/59085.html 

Suppose you multiply a single digit number N by 9. This is the same 
as multiplying it by 10-1:

    9N = (10-1)N = 10N - N

This looks almost like the expanded form for a two-digit number, where 
"AB" means 10A + B. Unfortunately, we have a negative "second digit," 
-N. So we borrow one from the tens:

       = 10(N-1) + (10-N)

Since N-1 and 10-N are both single digits, this tells us that 9N has 
tens digit N-1 (the number of fingers to the left of the Nth finger), 
and ones digit 10-N (the number of fingers the to right of the Nth 
finger). Note also that the sum of the digits is N-1 + 10-N = 9, so 
that it uses up all your fingers except the one you bent down.

Does that help?

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