That’s like saying the number 1.2345 “gets up” to being exactly 1.2345, like it goes throw a process of development to become 1.2345. It starts at 1, and then all of a sudden 2 emerges on the right of the decimal, then 3 shows up, then 4, and finally 5. Only then is it completely 1.2345.
0.999… is meant to denote the immediate presence of the quantity so denoted.
You might say it would take an eternity for the 9s to reach 1.
What 0.999… = 1 is saying is that at eternity, with an infinity number of 9s, you get 1.
Since (non-zero) decimals which repeat endlessly cannot be written out completely, one can say that the ‘…’ notation indicates a symbol for a number rather than an actual real number. Therefore, 0.111… is a symbol representing the number which is exactly equal to 1/9. Then also, 0.999… is a symbol representing 9 * 1/9 or the actual number 1.
“At Eternity” is like “at infinity” - there is no “at infinity” to be reached. The “…” is saying “this quantity cannot be represented”.
No. It is a symbol that is expressing that 1/9 cannot be expressed in base 10 decimal form. The “…” means “endless”, which means that there is no end to be obtained such as to finally satisfy the formula “1/9”.
Again, this begs the question. If .999… = 1, then a zero followed by infinite decimal expansion of nines is equal to a one followed by an infinite decimal expansion of zeros.
So when you say that an infinitely repeating 9 is always slightly less than 1, it’s just as true (or rather, just as false) to say that an infinite string of 0s is always slightly more than 1.
This may be a semantic point, but to ensure we’re on the same page I’ll address it: the word “superset” means a set that includes a set within it. The hyperreals include the reals, the reals do not include the hyperreals. The real numbers are a “subset” of the hyperreals. This follows the standard use of the prefixes ‘super-’, meaning greater or larger or above, and ‘sub-’, meaning lesser or smaller or below. In Arminius’ diagram, the real numbers are shown as a subset of the complex numbers.
The point being, the real numbers don’t include the hyperreals, the hyperreals do include the reals. (Arminius, I’m not sure how complex numbers relate to the hyperreals.) Wiki makes clear: “The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R…” An “extension”, i.e. they are the larger set. Wolfram says the same thing: “Hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers.” That is, there are numbers included in the hyperreals that are not included in the reals, thus the set of hyperreals is larger than the set of reals, and the set of hyperreals contains all the real numbers. So the hyperreals are a superset of the reals.
It doesn’t “beg the question”. It answers the question.
Do you agree that for the two numbers to be equal, there must be absolutely zero difference between them?
You know that no matter how many times you add 2+2, you will always and forever get 4 and there will forever be absolutely zero difference between the first time you did it and the last time. You know that because there is no opportunity for change concerning that operation. If nothing changes concerning the operation, you will always and forever get the same result.
That series of operations has no opportunity to become different no matter how long it is carried out. That is why it is “endless” or “infinite”. The result of every operation is that there is required to still be a small percentage between the accumulated result of the operation and 1.0. As long as the operation adds only 90%, there will always be a 10% not added into the accumulated sum. Thus the difference between the sum and 1.0 can never become zero. QED
Not at all. Where did you get that thought? The infinitely growing string of 0s is always equal to 1.0 from the beginning eternally. It is never above 1.0 such that it must settle down to become 1.0. But the string of 9s begins certainly below 1.0 and grows closer to 1.0, yet is never allowed be ever become 1.0.
Those are very different issues.
I well know what “super” vs “sub” means. And Arminius’ diagram didn’t show the hyperreals. The hyperreals are the extended portion of the real number line past first order infinity such as to be called the “nonstandard” set of reals.
The “nonstandard reals” plus the “standard reals” makes up the superset called “reals”.
The number 2 is not a hyperreal, nonstandard, number. It is NOT included in the set of hyperreals. The number 2 is a standard real number, outside the set of the nonstandard hyperreals. There is no “standard hyperreal”.
If allowed to proceed to absolute zero (the limit of the hyperreals), it might be arguable that the number string reaches that absolute zero difference between itself and 1.0. But if confined to the standard reals, the string cannot get even down below a single order infinitesimal difference (the rule being that you either cannot divide an infinitesimal or that if you divide an infinitesimal, you get the same infinitesimal remaining). So in the standard reals, there is always a first order infinitesimal difference between the two numbers.
The deeper truth is that even if including the hyperreals, the very definition of “0.999…” still forbids any remaining difference to ever be totally consumed into the accumulated sum. Thus by definition, there must always be a difference between the two numbers.
The “…” simply means, “you can’t get to the limit from here”.
But there are no infinitesimals in the standard reals. They are part of the hyperreal extension of the set. In the standard reals, two numbers with a “first order infinitesimal difference” have no difference, they are the same number.
You keep saying things like “…forbids any remaining difference to ever be totally consumed into the accumulated sum…”, which sounds a lot like you’re talking about a process. All the summing done for .999…, there’s no “ever” to get to, it’s not something that’s happened, it’s a static value.
Like above where you treat the summing as getting 90% closer, one step at a time. But the infinite series .9 + .09 + .009 + … is static, it’s not a limit, it’s not approaching, it’s value is a single, standard real number. It isn’t being “carried out” at all. All the infinite steps are complete, and they sum to a value, and that value is .999… = 1
Reading more about this question, there do seem to be number systems (like the hyperreals) where it is at least ambiguous as to whether .999… = 1, mostly because .999… doesn’t seem to fully specify a number, or because there are different conventions by which it specifies a number. But in the standard reals, there is no such ambiguity. .333… is the decimal expansion of 1/3, and 3 times either equals .999… = 1
The term “infinitesimal” merely means “immeasurably small”. So in that sense, an “infinitesimal” isn’t actually a number at all, just like “infinite” isn’t. But the real number line includes any number of the form “[1+1+1+…+1]” and “1 / [1+1+1+…1]”, whether it is actually measurable or not, eg 10^(-10^100,000,000) = real number, although never to be measurable = “infinitesimal” (they can’t even measure down to 10^-31).
Yes, but it is a “static value” that is defined to NOT HAVE an END, hence “never getting to” thus “not being able to represent” the approaching limit involved.
The “…” is indicating that the 999… process cannot ever satisfy the ratio and therefore does not satisfy the ratio even in the infinitely distant stage.
Because it cannot ever end by becoming identical to the limit, it cannot ever have a resolve, especially not that limit.
I agree with the mathatician, but at some point the philosopher predicates.and validates.his point, that.at the criitical.point of the ‘thinness’ of the gap between the absolute and the relative’space’ between them, the difference merely.will cease to be differential. Here Being and existence will only become a non-functional idea ,indiscernible to men discernible only to god. IF IT WAS NOT SO, EXISTENCE AND CREATION COULD NEVER BE CONCEIVED.
The definition requires that at no time, shall more than 90% of that difference be taken. Thus that difference can never be allowed to become no longer “differential”.
I’m guessing that you have no supporting argument for that assertion.
Yes. But if , more than 90% Was allowed, the differential would become.impractical, for most purposes.
That would indicate preferentially in placing the mathematical pre-rerequisite
Without that, the argument may be sustained, purely on onto-logocal basis.
The problem of this thread is exactly analogous to issues arising from fractional results in the integer number system.
In the integer system, you can do a division like 3/2 but the result is not an integer. You can represent the result with a symbol like 1…
When it it comes time to convert the symbol to a valid number, you only have a choice between two numbers - either 1 or 2 depending on whether you use rounding up or truncation. There are no other reasonable numbers on the number line. If you round up, then the symbol 1… is equal to 2 within the integer number system.
You can do the same thing with an infinite sum :
9/10 + 9/100 + 9/1000 + …
That sum is going to equal 1 within the integer system, because it can’t equal anything else.
Except that it is knowingly “not equal”, but merely an accepted approximation, a “pseudo-equality”. It is also like random number generation in that the numbers are not actually random, merely as close as the system allows.
So “0.999…” is as close as the decimal system allows you to get to 1.0 without actually being 1.0.
IOW, within the framework and limitations of the Real number system, 0.999… is equal to 1. But it is possible to conceive and use an alternate number system where 0.999… is not equal to 1.
