Alien Fingers and Bases

Date: 12/07/98 at 17:54:09
From: Cristin Rabik
Subject: What base are they counting in?

Using the most powerful telescope ever made, scientists observe a class 
of young aliens on a planet millions of light years away. On the 
blackboard, their teacher has written these equations:

   13 + 15 = 31
   10 * 10 = 100
     6 * 3 = 24

How many fingers do they have? (We count in base ten. What do they 
count in?)

I am really lost. I don't even understand what 'base 10' means. And 
I don't know how to use different bases. I would really appreciate 
some help with understanding this. Thanks!

Date: 12/07/98 at 18:50:27
From: Doctor Mike
Subject: Re: What base are they counting in?

Hi Cristin, 

I think I can help you out. First of all, we have a faq on number 
bases, at:

   http://mathforum.org/dr.math/faq/faq.bases.html   

This will lead to quite a bit of background information about this kind 
of thing. 
  
As an extra bonus, and at no extra cost to you, here is some particular 
information about your aliens. 

The number that we call ten is special in our number representation 
system because the digits count powers of ten. For instance, 6345 means 
6 times 10 cubed, plus 3 times 10 squared, plus 4 times 10, plus 5. If 
the powers of a number smaller than ten, say B, are used, then we call 
it base B. Now you are all set for the first clue.
   
- In base B, whatever B means, 13 is B+3, 15 is B+5, and 31 is 3*B+1.  
So what choice of B will make it true that: 
   
   B+3 + B+5 = 3*B+1
  
You need to use simple algebra to solve this. For a hint in checking 
your answer, I'll just say that it is interesting that you sent in this 
question on December 7th.   
   
- The fact is that 10 * 10 = 100 is true in *every* base representation 
system.
  
To check out the third fact, verify that 6 * 3 = 2*B + 4 for your B 
value.
   
There are two more things I should mention. The fact that the digit 6 
is used in the statement of the problem guarantees that the correct 
base must be larger than 6. If the base is *greater* than the number we 
call ten, some new symbols have to be used as digits. The hexadecimal 
(base 16) system is often used in connection with digital computers. In 
this system, the "numbers" less than "hexadecimal 10" are the 
following:
   
   1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
    
After F comes hex 10, and the hexadecimal "teen" numbers, namely:
  
   11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F
   
And then comes hexadecimal 20. But, I digress. I guarantee to you that 
the base B you want is less than what we usually call ten.   
   
Have fun. I hope this helps.
   
- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/