Do irrational numbers exist?
Please note that this is not a question about whether numbers like Pi really exist in the real world, it is a question about whether irrational numbers, as they are defined, exist within the formal system of mathematics.
Perhaps this should go in Tilted Knowledge, because I'm pretty sure the idea I'm about to present is wrong, but I don't really understand why.
Irrational numbers are those numbers which cannot be expressed as a ratio of two integers.
Take the numbers between 0 and 1. Let us consider a rational number, such as 0.123. This can be expressed as 123/1000. Likewise:
0.1234=1234/10000
0.12345=12345/100000
0.123457= 123457/1000000
0.1234571= 1234571/10000000
0.1234571= 12345714/100000000
...
Since the set of integers is infinite and passes through every sequence of digits, there can be no sequence of digits which is not included in it, even those which are non-repeating and non terminating. Irrational numbers between 1 and 0 are expressed as a non-terminating, non-repeating sequence of digits after the decimal place. So why can we not extend the pattern above infinitely?
If this pattern cannot be continued infinitely, what is its limit?
If it can, then does that not mean that all the non-repeating, non-terminating decimals that make up irrational numbers can be expressed as a ratio of two integers and as such are simply infinitely long rational numbers and not truly irrational?
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Last edited by John Henry; 03-28-2005 at 02:32 AM..
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