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Problem with numbers

I read somewhere that

"Q is countable but R is not" (reference-R means real numbers and Q means rational numbers)

On thinking i found that this statement doesn't make any sense.Or does it??? Can you explain this statement???

(I WOULD BE PLEASED IF EXPLANATION IS LUCID WITH SOME CRUNCH OF ADVANCED MAtHEMATICS)

Note by Somesh Rout
3 years, 11 months ago

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\(Q\) and \(R\) are both infinite sets. But \(Q\), the set of rational numbers, is countably infinite. On the other hand, \(R\), the set of real numbers is an example of an uncountable infinity. So far, this shouldn't make much sense, so bear with me.

Even though rational numbers are found everywhere on the number line, they leave a lot of gaps. Think about this: if you had an infinitely thin pin and started to stab a piece of paper with it, you'd make a lot of holes but you'd never be able to destroy the paper. The rational numbers are like those pin pricks on the number line. Using a countable infinity like the set of rational numbers, you can’t construct any kind of continuous set.

However, with real numbers, you can make a continuous set (the number line). This is called an uncountable infinity.

This post here goes into much more detail. Check it out and you should have all your questions answered.

The point to be noted here is that the word 'countable' is not being used in the usual sense. In the usual sense, the elements of an infinite set can't be counted (unless you're given infinite time). Hope this helps! Mursalin Habib · 3 years, 11 months ago

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@Mursalin Habib I knew the countably and uncountable infinite stuff,but your explanation with the pin is amazing.just loved it. Aahitagni Mukherjee · 3 years, 11 months ago

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@Aahitagni Mukherjee You're welcome. But the credit goes to the post I mentioned earlier. Mursalin Habib · 3 years, 11 months ago

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@Mursalin Habib thanks for the great explanation Somesh Rout · 3 years, 11 months ago

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@Somesh Rout You're welcome. Mursalin Habib · 3 years, 11 months ago

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