is Set of Real Numbers Between 0 and 1 really uncountably Infinite?

math set infinite set-theory countable
Ayub Khan picture Ayub Khan · Oct 8, 2013 · Viewed 18.2k times · Source

Cantor's Set of Countable infinite and Uncountable infinite Infinites

You may know and you may have proved that Set of Real Numbers Between 0 and 1 are Uncountably Infinite. Mean we Can not Map Every number of that set on a different Natural Number.

I got a Technique by which I would be able to Map all Real numbers between 0 and 1 on a different Natural Number. Technique is Simple Replace the Decimal Point with 1 and Map the Original on that Number Such that Map 0.0003 on 10003 and 0.03 on 103

By using this Technique we Would be able to Map all Real Numbers Between 0 and 1 on Natural Numbers. And All of those Natural Numbers will be starting with 1 so we will be having other Numbers as well on which No Number will be mapped like 2 or 211 or 79 So This Means Set of Natural Numbers is Grater then Real Numbers Between 0 and 1. So Set of Real Numbers Between 0 and 1 is Countably Infinite.

What's Ur Opinion ?

Answer

Joni picture Joni · Oct 8, 2013

The set of real numbers between 0 and 1 is uncountably infinite, as shown by Cantor's diagonal argument which you are familiar with.

What may be surprising to you is that the set of rational numbers between 0 and 1 is countably infinite. That is, there is a 1-to-1 correspondence between the integers and all fractions and numbers with a finite decimal expansion. You can find the proof here.

Related questions

How can I check for NaN values?

float('nan') results in Nan (not a number). But how do I check for it? Should be very easy, but I cannot find it.

Read More
Math.random() explanation

This is a pretty simple Java (though probably applicable to all programming) question: Math.random() returns a number between zero and one. If I want to return an integer between zero and hundred, I would do: (int) Math.floor(Math.…

Read More
Calculate distance between two latitude-longitude points? (Haversine formula)

How do I calculate the distance between two points specified by latitude and longitude? For clarification, I'd like the distance in kilometers; the points use the WGS84 system and I'd like to understand the relative accuracies of the approaches available.

Read More