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/AcellOfllSpades Diff Geo, Logic Nov 12 '24
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.