Non-terminating Decimal Representations of Fractions

Date: 11/10/97 at 19:07:48
From: MRS.SWIZ
Subject: Non-terminating decimal representations of fractions

Dear Dr. Math,

Why is it that you can take a perfectly finite, limited quantity like 
one-third (you take a pie, divide it into three simple, equal pieces, 
and you get three and only three pieces - no more - no less) but when 
you turn it into a decimal you get .333... on into infinity? 

It seems intuitively that something is inherently wrong with the 
decimal representation of that quantity. I am an Education grad 
student with a BA in Fine Arts and no clue (obviously) about math. 

Thanks. Diane

Date: 11/10/97 at 20:08:07
From: Doctor Tom
Subject: Re: Non-terminating decimal representations of fractions

Hi Diane,

Unfortunately, the short answer is, "That's the way it is."

Since we happen to have ten fingers and thus use base 10 (decimal)
arithmetic, we're stuck with it. I'm not sure if you know anything
about other number bases, but the choice of 10 is totally arbitrary.

Base 2 and base 16 are heavily used in computers, but base ten
(decimal) is so heavily entrenched that the only other example I can 
think of where another base is commonly used is in old-style counting:  
dozen = 12, gross = 144 = 12*12, great gross = 1768 = 12*12*12.  
It's a sort of a start of a base 12 system.

In case you don't know about base systems, think about counting in 
decimal. Everything is regular in the units digit until you get to 
nine, but the next step "overflows," and drives the 9 to zero, but 
pumps up the next digit over (which may overflow itself -  
99 => 100 -- and so on).  In base 5, for example, counting would
go like this:

0,1,2,3,4,10,11,12,13,14,20,21,22,23,24,30,31,32,33,34,40,41
42,43,44,100,101,102,103,104,110,...

Base 3 is like this:

0,1,2,10,11,12,20,21,22,100,101,102,110,111,112,120,121,122,
200,201,202,210,211,212,220,221,222,1000,1001,1002,...

Stare at these until you get the pattern. It's just like the decimal 
system, but the overflow occurs earlier.

In decimal, if you look at a number like 123.45, the "1" represents
the number of 10*10s, the "2", the number of 10s, the "3" the number 
of 1s, the "4", the number of 1/10s, and the 5, the number of 
1/(10*10)s.

In base 5, the number 12.34 represents 1 of 5, 2 of 1, 3 of 1/5 and 4 
of 1/(5*5) = 1/25, so to convert 12.34 (in base 5) to base ten, you 
get: 1*5 + 2*1 + 3*(1/5) + 4*(1/25) = 7 19/25 = 7.76

In base 3, the fraction 1/3 has a great expression:  0.1, but then you 
have trouble with other fractions.  For example, 1/2 (one-half) is 
expressed by .111111... (base 3).

No matter what base you choose, some fractions will have a clean
expression in a "decimal-like" form, and others won't. I put "decimal-
like" in quotes, since "decimal" refers to 10, and if you're working 
in base 3, it's technically called a ternary system, the fraction is 
not a "decimal fraction" but a "ternary fraction," and it's not a 
"decimal point" but a "ternary point."

It's an interesting mathematical question to find the relation between 
the base and the numbers that come out "nicely." It has to do with 
what numbers divide evenly into the base. For 10, only 1, 2, 5, and 10 
do. For base 3, only 1 and 3 do. Base ten will thus tend to have more 
numbers that work out with clean decimal expansions.

But if only we used base 360, we'd be in heaven! Look what goes evenly 
into it:

1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180, and 
360.

Of course it would be a giant pain to have 360 different characters
to memorize instead of our current set of 10: "0", "1", ..., "9" :^)

I hope this helps.

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

Date: 11/10/97 at 22:08:05
From: MRS.SWIZ
Subject: Re: Non-terminating decimal representations of fractions

Dear Dr. Math,

Please disregard my question to you about non-terminating decimals. 
I have found the answer and it is so simple that I feel somewhat 
foolish. Decimals are based on tens and three will never divide ten 
evenly! Never mind. Sorry.

Diane