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 â‰ 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. Âž 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. Know More About Is a whole number a Rational Number
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. Learn More About Is Negative Number a Rational Number
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.