Where is the line (mathematically) between a math progression and infinity?

Factual Questions
61

As I have mentioned many times in previous such threads, 0.999… is not 1. They are of different types. One is an infinite series and the others is a number. The limit of the first is 1.

Note especially that there are operations you can perform on numbers that you cannot safely perform on infinite series. The two should not be confused.

It’s like equating “2” with “{2}”.

If you merely say “The limit of 0.999… is 1.” then problems like the OP has goes away.

62

You’re thinking about it backward.

There are infinite series that sum to pi, or to 1/3, or to e, or to the square root of 10. There are also infinite series that sum to 1, or 2, or any other number. The difference is that pi and e, unlike 1, can’t be expressed except as the sum of an infinite series or equivalent, and 1/3 can’t be expressed in decimal notation other than as the sum of an infinite series.

You can stop at any point in the expansion of Pi or 1/3 and that will be a finite number, it just won’t be one that is exactly pi (or 1/3), but you can get arbitrarily close.

63

Whether a repeating decimal like 0.999… or 0.333… represents a number itself or only an infinite series is a question of notation. And I think most people, including most mathematicians, use it to represent the number itself. So if you insist otherwise, you’re using nonstandard notation.

64

We’ve gone over this many, many times before, on this board and many others across the Interwebs. In brief:

  1. 0.999… is a real number, and it’s equal to 1. It isn’t a sequence (except a constant one), and it doesn’t “tend” to anything; it’s just an ordinary real number. (One can construct R as a set of sequences in Q, but that’s a different matter.)

  2. Your intuition about real numbers, infinity, sequences, etc. is wrong. That’s fine; that’s part of the reason math exists. If you want to resolve the matter, you’ll have to go beyond intuition and look up the formal definition of real numbers.

  3. This is a solved problem. The difficulty some people have in accepting the solution is that they have some fuzzy notion of what a real number is that doesn’t have any mathematical basis. This is why we have definitions.

65

It’s also worth noting that

.3 does not equal 1/3
.33 does not equal 1/3
.333 does not equal 1/3
.3333 does not equal 1/3

but a decimal point with an infinite number of 3s after it DOES equal 1/3.

It’s not necessary to invoke pi or “point nine repeating” in order to see this phenomenon.