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### Is a Ratio Rational or Irrational?

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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 ?
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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/
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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)
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Associated Topics:
High School Analysis

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