Is a Ratio Rational or Irrational?

Date: 8/2/96 at 8:27:20
From: RAMNATH.S
Subject: Is a Ratio Rational or Irrational?

Hello!

Let me introduce myself. I am Ramnath. I am doing my Masters in Math 
through correspondence here in Bangalore, India.

Q) Is there any way by which we could ascertain
   the irrationality of a ratio (e.g. 4/1, 22/7)
   without actually resorting to division.
   If no, then, 22/7 leads to an infinite sequence
   of non-repeating digits.
   What is the guarantee that the division is endless? 
   As there is nothing that can prevent us from supposing
   that the next digit will not divide the no perfectly?

Could you clarify that please ?

Date: 8/2/96 at 12:19:9
From: Doctor Anthony
Subject: Re: Is a Ratio Rational or Irrational?

The numbers you quote as irrational are in fact rational.  Any RATIO 
of two integers is by definition 'rational' - that is where the name 
comes from.  

You have doubtless been told that pi is irrational (and indeed it is), 
but pi is NOT equal to 22/7.  That is just one fairly crude 
approximation, used for convenience, if you are only working to two or 
three places of decimals. If you divide 7 into 22 you will soon get a 
repeating decimal, as you will get with ANY division you might like to 
try. 

Irrational numbers are often worked out to many places of decimals 
using infinite series, e.g sqrt(2), pi, or e.  There are innumerable 
series which can generate the approximations, but they will always be 
approximations.  For example the series for e is: 
1 + 1 + 1/2! + 1/3! + 1/4! + .... to infinity.

If a decimal starts to recur, that is evidence enough that the number 
is rational, and it is a simple matter to find what the two numbers 
are forming the ratio.

Example: what ratio is represented by 5.1672672672... ?

 This could be written 5 + (1/10) + 0.0672672....

Let x = .0672672  10x = 0.672672....
               10000x = 672.672672....

Subtracting     9990x = 672  and so x = 672/9990 = 112/1665

The number is therefore  5 + (1/10) + (112/1665) which is rational.

   = 17207/3330


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

Date: 8/5/96 at 19:47:30
From: RAMNATH.S
Subject: Re: rational

Sir,
    
I was simply thrilled by your reply! It was very exhaustive and put 
to rest about any doubts whatsoever about rational/irrational numbers.

Thanks once again,

With regards,

Ramnath
(Bangalore, India)