Uses of Bases Other Than Base 10

Date: 09/28/2004 at 12:38:10
From: Julie
Subject: different math bases & when/why they're used

Why are there different math bases and what would they be used for?

I know that in the US we go by base 10, because of our 10 fingers, but
I don't know who would use different bases and why that would be
important for us to know.

Date: 09/28/2004 at 14:22:35
From: Doctor Douglas
Subject: Re: different math bases & when/why they're used

Hi Julie.

Thanks for writing to the Math Forum.  

It's true that for most people, the decimal (base-10) system is
familiar, easy to use, and works well for most everyday purposes.  It
is a reasonable compromise between the number of different digits (in
the sense of "symbol", {0,1,2,3,4,5,6,7,8,9}, and in the compactness
of notation.  And it is natural for us because we humans have ten
digits (in the sense of "fingers").

But there are other base systems that can and have been used.  The
Mayans and Babylonians used base-20 and base-60 systems.  These might
have been more convenient for certain mathematical operations.  For
example, we now use 360 degrees in a circle.  Why 360 and not 100 or
1000?  Certainly the number 360 is conveniently divisible by many 
small numbers, such as 2,3,4,5,6,12,15, etc., and 100 and 1000 are 
less convenient in this regard.  Similarly, the English system of 
measurement units uses 12 inches in a foot, and 36 inches in a yard.  
If all we ever did was in base-10, then those of us in the United 
States probably would have converted to the metric system a long time 
ago.  Even if we did convert to metric for most things, it's likely
that the measurement of time would still retain the concepts of day 
and year, which aren't related to each other by a factor of 10, but by 
astronomical quantities (the rotation of the earth and the revolution 
of the earth about the sun) that don't come in a convenient ratio of 
10.  

The now widespread use of cheap computing power and storage has made 
the compactness issue less important, and so they use the binary 
number system, based on the two digits {0,1}, and its closely related 
variants of octal (eight digits:  {0,1,2,3,4,5,6,7}) and hexadecimal 
(sixteen digits:  {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}).  You and I might 
write out the number two thousand and four as 2004, and here is how a 
computer might represent them:

   2004 (decimal) = 7D4           (hexadecimal)
                  = 3724          (octal)
                  = 11111010100   (binary).

For a computer, the last form leads to the simplest circuitry 
necessary to manipulate the number, while the hexadecimal form is 
compact to write in text (so you might see this in compression
algorithms, or compact text representations of programs), and of 
course the decimal form is the form most familiar to us.

There are also fractional bases, of which the natural base is the most 
important:

  e = 2.71828..., the Base of Natural Logarithms
    http://mathforum.org/dr.math/faq/faq.e.html. 

You might also come across a hybrid base, where different digits use 
different bases.  You might think that this is weird, but there is one 
very familiar example--a digital clock:

   ##:## AM
   wx yz

Here the digits wx represent the hours and the digits yz represent the 
minutes.  You can see that the digit z simply counts through the 
numbers zero through nine, so that it is a base-10 digit.  But the 
digit y only takes values from zero through five, so that it is a 
base-6 digit.  When taken together as a single unit, the combination 
yz represents a number from zero to fifty-nine (base-60), which 
naturally enough means that there are 60 minutes in each hour.  
Similarly, the digit x is a base-10 digit by itself, but the 
combination wx is a base-12 object.  And don't forget the "A" in AM or 
PM--that too is a digit (base-2) where the two symbols are not 
numerals, but letters:  A and P.  So depending on how you look at it, 
there are two or three number bases being used simultaneously in the 
simple process of telling time--yet it doesn't seem that unfamiliar to 
us.

There is another place where I've seen hybrid bases is the recording 
of innings in baseball games:  for example, a starting pitcher might 
throw 5 innings and get 2 more batters out before leaving the game.  
An inning is three outs per side, so the old way of saying this is 
that the pitcher went "five and two-thirds" innings, but now many stat 
boxes report this as "5.2 innings".  The number that comes after 5.2 
is 6.0, then 6.1, etc.  Note that the two digits in this quantity 
represent different things:  one represents the number of complete 
innings and the digit after the decimal point represents the number of 
additional outs.  And 3 outs (not 10) make a complete inning for one 
team.

For much more information about number bases (how to represent numbers 
in them, and convert between them), please check out our archives, 
where there is lots of additional information.

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