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An easy proof that rational numbers are countable.
An easy proof that rational numbers are countable.
It's very counter intuitive that one could be able to "count" the set of rational numbers (those numbers that are quotient of two integers, say a/b), because such is an infinite set, but here's the proof:

[Image: rationals-countable.gif]

Quote: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" is allowed to continue to infinity.

In mathematical terms, a set is countable either if it s 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. Notice, the infinite case is the same as giving the elements of the set a waiting number in an infinite line Smile.

And without delay Peter went quickly out of the synagogue (assembly) and went unto the house of Marcellus, where Simon lodged: and much people followed him...And Peter turned unto the people that followed him and said: Ye shall now see a great and marvellous wonder. And Peter seeing a great dog bound with a strong chain, went to him and loosed him, and when he was loosed the dog received a man's voice and said unto Peter: What dost thou bid me to do, thou servant of the unspeakable and living God? Peter said unto him: Go in and say unto Simon in the midst of his company: Peter saith unto thee, Come forth abroad, for thy sake am I come to Rome, thou wicked one and deceiver of simple souls. And immediately the dog ran and entered in, and rushed into the midst of them that were with Simon, and lifted up his forefeet and in a loud voice said: Thou Simon, Peter the servant of Christ who standeth at the door saith unto thee: Come forth abroad, for thy sake am I come to Rome, thou most wicked one and deceiver of simple souls. And when Simon heard it, and beheld the incredible sight, he lost the words wherewith he was deceiving them that stood by, and all of them were amazed. (The Acts of Peter, 9)

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An easy proof that rational numbers are countable. - by Jehanne - February 22, 2018 at 3:57 am

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