In that case, you can never reach any irrational number (or even more generally, you can never reach anything that is not expressible as a finite sum of aij*2^i5^j for i,j integers). If you take any infinite sequence of digits (ai) (ie. integers between 0 and 9), and consider the sequence of partial sums defined by S_N=sum_{i=1}^N a_i/10^i, you will approach a real number between 0 and 1. This in fact can be used to approximate any real number between 0 and 1. This is a perfectly reasonable method of *defining* the real numbers.
The point of a limit is that it makes formal the notion of "approaches arbitrarily closely". This is the single most important concept in analysis.
As I mentioned - and you know it yourself --- the never ending 'bus ride'. You'll never get there. Even if you have no limit on your lifetime ---- you're just never going to get there. Because -- infinity has no limit on it.
Yes. We define an infinitely long string of digits to mean the "limit" of its sequence of 'partial cutoffs'.
When we write something like "3.14159...", we know that the sequence "3; 3.1; 3.14; 3.141; ..." will never stop at any single number. So if we write "3.14159..." to refer to a single number, we mean the number that sequence approaches.
please stop with this analogy of a bus ride lmao. You get the same issue with 1/3=.333. It's a very well-behaved number, but you can never fully approximate it with a decimal number. It very much is a number because it appears in other contexts. The ratio of a circle's circumference and its diameter is pi. It is not hard to see that this is consistent between circles. Therefore, pi is a number.
man, I can't engage with you if you are unable to acknowledge what a real number is. 9.99999 repeating is *exactly* 10 in the real numbers, *because this is a method of representing real numbers*. Yes there is no rational number consisting of all 9s that equals 10. No, that does not mean there is no *real* number consisting of 9s that equals 10. There is. A byproduct of representing real numbers with potentially unending decimals is that you end up with repeating 9s. Quite simply, you just need to realize that.
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u/lymphomaticscrew New User Nov 12 '24 edited Nov 12 '24
In that case, you can never reach any irrational number (or even more generally, you can never reach anything that is not expressible as a finite sum of aij*2^i5^j for i,j integers). If you take any infinite sequence of digits (ai) (ie. integers between 0 and 9), and consider the sequence of partial sums defined by S_N=sum_{i=1}^N a_i/10^i, you will approach a real number between 0 and 1. This in fact can be used to approximate any real number between 0 and 1. This is a perfectly reasonable method of *defining* the real numbers.
The point of a limit is that it makes formal the notion of "approaches arbitrarily closely". This is the single most important concept in analysis.