Quote:
Originally Posted by John Henry
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
...
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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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What you have designed above is a limit---not a ratio of two integers, but a rational expression that approaches a real number. An infinitely long integer then is not an integer proper---it's infinity---and hence not in the set of integers.
Since a rational number is composed of two integers, and what you have above are actually two infinitely long strings of digits composing an irrational number, then we can see that what you have built is not a rational number. In actuality, it's a sequence of rational numbers that get ever and ever larger (but never equals) the irrational number, as per the definition of limit. In other words, irrational numbers are not types of rational numbers or vice versa, because talking about infinite integers doesn't make much sense.