Multiplying by Nine

Date: 01/10/97 at 03:28:45
From: Jeffrey Gleaves
Subject: Number nine

Hi,

Why is it that when you multiply nine times any number between one
and ten, the sum of the two digits added together equals 9? 

 9*1=9
 9*2=18 1+8=9
 9*3=27 2+7=9
 9*4=36 3+6=9

Please forgive my stupidity and thank you for your time.

Date: 01/10/97 at 14:54:24
From: Doctor Rob
Subject: Re: Number nine

No stupidity displayed here!

Basically, it is because 9 = 10 - 1, and 10 is the base of our
system of numerals.

   9*a = 10*a - a = 10*(a - 1) + (10 - a)   

   (by borrowing 1 from the 10's)

If 0 < a < 10, then the digits of 9*a are:  

     a-1 is the 10's digit,
and 
     10-a is the 1's digit.  

(a-1) + (10-a) = 9.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/