Decimal System

Date: 08/12/2003 at 17:00:24
From: Benjamin
Subject: Decimal system

How universal is math, and in what ways is it merely an artificial 
language? 

I realized the other day that while math is universal in function, the 
decimal system is just an artificial form of expression. For example 
suppose there were instead a "heximal system," 1 2 3 4 5 6 10. 
The decimal system is so engrained in us from such an early age, that 
this idea of an alternate base number seems absurd at a first glance. 

In any case, my question is why the decimal system? Why not 12 as the 
base number since it is divisible by more, and I would imagine would 
be more docile in math functions?

Thanks,
Benjamin

Date: 08/12/2003 at 17:57:03
From: Doctor Warren
Subject: Re: Decimal system

Hi Benjamin,

Congratulations on your very astute observation. You are absolutely
correct that numbers exist independently of the numeral system used to
represent them.  The number fifteen is an abstract concept that exists
independently of its decimal representation (the numeral one followed
by the numeral five).

You can dream up any base you'd like, and represent your numbers in
that base.  For example, the base commonly called "octal" is composed
of the symbols (numerals):

   0 1 2 3 4 5 6 7

(there are no such symbols as 8 or 9 in octal.)  There are also bases
with MORE than ten symbols, such as "hexadecimal."  The symbols used
in hexadecimal are, by convention:

   0 1 2 3 4 5 6 7 8 9 A B C D E F

The symbols A through F mean the numbers "ten" through "fifteen." 
Another common base is called "binary," and has only two symbols:

   0 1

Binary is used by all digital machines, like your computer. A digital
switch is either on or off, and you can represent numbers in binary by
strings of ones and zeros, just as you can represent numbers in
decimal by strings of decimal numerals.

The standard place system we use in decimal numbers is commonly used
with other bases, too. For example the symbols "123" mean the
following things in decimal, octal, and hexadecimal:

       decimal: 1        2        3
                1*10^2 + 2*10^1 + 3*10^0

       octal:   1        2        3
                1*8^2  + 2*8^1  + 3*8^0

   hexadecimal: 1        2        3
                1*16^2 + 1*16^1 + 1*16^0

In other words, the symbols "123" means the number "one hundred 
twenty-three" in decimal, but they mean the number "eighty-three" in 
octal, and they mean "two hundred ninety-one" in hexadecimal. (Try 
adding 1*64 + 2*8 + 3*1 and 1*256 + 1*16 + 3*1.)

You're right, the decimal system is very much ingrained in our notion
of numbers. When I say the number "eighty-three," I don't necessarily
mean the numeral 8 followed by the numeral 3; what I mean is the
number that follows eighty-two and precedes eighty-four. There is no
way for us to name a number, however, except in terms of numerals in
some specific base, like decimal.

You can do all your arithmetic the same way in different bases, too. 
For example, if you want to add two numbers, consider the following
"rule:"

   decimal:     9 + 1 = 10
   octal:       7 + 1 = 10
   hexadecimal: F + 1 = 10

If you want to try your hand at doing math in different bases, the
Windows calculator (Start Menu->Programs->Accessories->Calculator) can
do math in decimal, octal, hexadecimal, and binary in its "scientific
mode" (View menu->Scientific).

As for your second question: why base 10?  The simple answer is that
we have ten fingers. Some other cultures have used number systems with 
more "symmetries."  The Babylonians used a base-60 system, largely 
because of the number of ways you can divide 60. Here's a great page 
on the history of mathematics in different cultures:

http://www-gap.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html 

Take a look at the numerals of the ancient Babylonians.

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