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Rational Numbers are Countable

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					       Rational Numbers are Countable
Rational Numbers are Countable

Rational numbers are the numbers which can be expressed in form of p/q
where p & q are integers such that q ≠ 0.

If we are given a pair of rational numbers say 2/7 and 4/7. If we look at glance
on these two rational numbers, we say that there exist only one rational number
i.e. ¾ between them.

But if the question is find 5 rational numbers b/w the given two rational
numbers. Now , we multiply numerator & denominator by 4, we get (2x4)/(7x4)
and (4x4)/(7x4) Or 8/28 and 16/28.

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Now, we have 9/28, 10/28, 11/28, 12/28, 13/28, 14/28 and 15/28 as the rational
numbers lying between the two rational numbers 8/28 and 16/28. We can say
that they are rational numbers lying between 2/7 and 4/7.

Similarly, if we multiply and divide this pair of rational numbers by 6, we get
(2x6) / (7x6) and (4x6) / (7x6) =12/42 and 24/42 Here we find that there 11
rational numbers between the given two rational numbers.

Thus we conclude that rational numbers are not countable as there can be any
number of rational numbers between the given two rational numbers.

Though it is to be remembered that some of them are equivalent but they exist
in the different forms.

A set is countable if you can count its elements. Of course if the set is finite,
you can easily count its elements.

If the set is infinite, being countable means that you are able to put the
elements of the set in order just like natural numbers are in order.

Yet in other words, it means you are able to put the elements of the set into a
'standing line' where each one has a 'waiting number', but the 'line' can still
continue to infinity.
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In mathematical terms, a set is countable either if it is finite, or it is infinite and
you can find a one-to-one correspondence between the elements of the set and
the set of natural numbers.

Well the infinite case is the same as giving the elements of the set a waiting
number in an infinite line...

And here is how you can order rational numbers (fractions in other words) into
such a 'waiting line'.

It's just positive fractions, but after you have these ordered, you could just slip
each negative fraction after the corresponding positive one in the line, and put
zero leading the crowd. I like this proof because it is so simple and intuitive yet
convincing.
   Thank You




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