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Alien Fingers and Bases

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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!
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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

of thing.

As an extra bonus, and at no extra cost to you, here is some particular

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/
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