Why Are All Repeating Decimals Classified Rational?

Date: 11/12/97 at 13:50:25
From: Beh Auge
Subject: Irrational numbers

I teach an advanced 7th grade math class and we are discussing 
rational vs irrational numbers. The text states that .6 repeating is 
considered a rational number because it can be expressed a/b by using 
2/3. But if you try to express .6 repeating over ten, or 100, or  
1000, etc., it is impossible because you never know exactly what power 
of ten will be in your denominator. Please explain to us how .6 
repeating can be a rational number when expressing it over 10's, 
100's, etc. Why are all repeating decimals classified rational?

Thanks.    Beth Auge

Date: 11/12/97 at 17:45:17
From: Doctor Ceeks
Subject: Re: Irrational numbers

Hi,

You are correct that 2/3 cannot be expressed as a ratio of two 
integers p/q where q is a power of ten.

However, it is not necessary to do so to have a rational number.

A rational number is any number that is expressible as the ratio of 
two integers. Since .6 repeating is 2/3, it is rational. The 
definition of a rational number makes no condition on the prime 
factors of the integers involved.

To understand why repeating decimals are all rational, you have to
understand geometric series, which you can find discussed in the
Math Forum archives.

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

Date: 11/12/97 at 17:54:10
From: Doctor Tom
Subject: Re: Irrational numbers

Hi Beth,

It's just an accident of nature that we work in base 10 - we happen 
to have 10 fingers. Since the decimal system uses 10, when we write 
decimal forms for certain fractions, they come out even, (like 1/2, 
1/5, ...), and for others, they repeat: 1/3 = .33333...  If humans had 
8 fingers and we used a base 8 system, the same problems would exist, 
except that a different set of fractions would "come out even," and a 
different set would have a repeating "octal" (not decimal) expansion.

The best way, perhaps, to answer your question, is to say that 1/3 is 
not equal to .3333. It's also not equal to .33333333. In fact it's not 
equal to .3333(a million of these 3s), although this is VERY close to 
1/3. It is only equal to the infinite series of 3s following the 
decimal point.

Here's something that might convince advanced 7th graders. Suppose
somebody asks you what is the value of .3333.... (the infinite
decimal expansion)?

Well, let's call the unknown number x, and write out what x must be:

x = 3/10 + 3/100 + 3/1000 + 3/10000 + 3/100000 + ...   (equation 1)

Multiply x by 10:

10x = 3 + 3/10 + 3/100 + 3/1000 + 3/10000 + ...        (equation 2)

Now, subtract equation 1 from equation 2:

10x = 3 + 3/10 + 3/100 + 3/1000 + 3/10000 + ...
-x =     -3/10 - 3/100 - 3/1000 - 3/10000 + ...
-----------------------------------------------
9x = 3

so x = 3/9 = 1/3.

The same trick can be used to convert any repeating decimal to a 
fraction. For example, what is .123123123123... ?

x = 123/1000 + 123/1000000 + 123/1000000000 + ...
1000x = 123 + 123/1000 + 123/1000000 + 123/1000000000 + ...

Subtract them, and you get:

999x = 123

so x = 123/999 = 41/333.

If you try this with a bunch of examples, you'll see that if you just 
put the repeating part over an equal number of 9s, you get the answer, 
so:

   .27272727... = 27/99 = 3/11
   .5124512451245124 = 5124/9999 = 1708/3333

et cetera.

To do the reverse, in other words, to convince yourself that any 
fraction repeats, consider what happens in long division. Eventually, 
the remainders have to repeat, so the numbers in the quotient will 
begin to repeat at the same time.

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