Is 1 = 0.999... ? Really?

The question demonstrates a quirk of human perception and the limitations of the decimal system. It’s similar to an optical illusion but it’s a mental illusion. :smiley:

Emmmm…
… no. :sunglasses:

Well, that is what I was expecting for at least someone to say. That was Wiki’s take on it.

But happens to be wrong. :sunglasses:

If you say so then it must be no. :laughing:

I’ll give you a hint:

0.999… ≡ ∑[from 1 to ∞} of [9 * 10^-n]
That is:
0.9 +
0.09 +
0.009 +
0.0009 +
.
.
.

And for that reason, you know that 0.999… is NOT equal to 1.000…

Just think about what “…” actually means.

Why sum an infinite number of numbers when you only need to sum 9?

1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9=1

Represented in decimal form :

.111_ + .111_ … = .999_ and also = 1

That’s it in a nutshell. No infinite series required. KISS

Feels somehow unsatisfying?

Well, that’s life.

But is it really?
… not really.

So you say. :smiley:

What does the “…” really mean?

If there is no exact ‘1’ then there is no exact fraction of one.

therefore, no amount of integers even an infinite amount would ever add up to one, because there would be a numerical entropy caused by difference betwixt the whole and the inexact proportion.

The basic issue with this is that the “…” means that the string of decimal numbers never, never, never gets to an end point. And that means that it never gets to the point wherein it is actually representing the fraction that it was “trying” to get to, its “limit”.

1/3 does NOT actually = 0.333…

Base 10 decimal notation simply cannot be used to represent “1/3”. That is what the “…” is telling you.

No decimal number that ends with “…” ever truly represents its limit. And even though for all practical applications, it might as well be equal, it can be misleading to just ignore that missing final infinitesimal when combining more complex concerns or comparing numbers that are very close, yet infinitesimally different.

1 = 0.999… ← The way I read this is: you have 0.999 and then a little extra. ← That’s what the “…” means. It means 0.999 and more. Why can’t the “more” be 1 - 0.999?

Irrational or rational numbers with infinite places [ either decimal or non decimal ] are accepted just as much as real or whole or other rational numbers even
if they cannot be written in their entirety. The most famous irrational number of all is pi but no one has a problem with it having an infinite number of places
Every irrational number has to be abbreviated because that is the only way to express them. And if you think 0.333 … is not base ten then what is it ? Is there
a rule which says irrational or rational numbers with infinite places cannot be written in base ten ? No because 0.333 … actually uses the notation of base ten

I think that you’re missing the point, as do all of those claiming that it is merely an issue of notation. I would rather explain this to a mathematician, but…

The “…” notation, specifically means that there is no end to be obtained. That means that it never, ever gets up to being exactly 1.0.

And to prove the point:
The number “0.999…” is formed of the following infinite series:

90% of 1 +
90% of the remaining 10% +
90% of the remaining 1% +
90% of the remaining .1% +
90% of the remaining .01% +
90% of the remaining .001% +
.
.
.
Note that always and forever, only 90% of the remaining amount up to 1.0 is ever added to the sum. The very definition of the number “0.999…” forbids the inclusion of the very last infinitesimal amount that would allow it to get up to 1.0. It is always and forever only taking 90% and must always and forever leave 10% of whatever was left, thus never, never, never reaching 1.0.

It is forbidden to ever equal 1.0 by its very definition.

What ever formula was used to create the 0.999… does not resolve to 1.0, but always just an infinitesimal amount less.

0.9r leaves 0.1r, and both are [potential] infinities.

Surely that means 0.9r is infinitely less than 1.

its one of the reasons ‘why’ reality isn’t made of lego [why qm and relativity exists].

James, I think you’re confusing infinite sums and limits. An infinite sum is not waiting on anything, the ‘…’ doesn’t mean the summation is pending, it means that an infinite series can’t be written on a finite piece of paper (and it isn’t necessary since the form of the sum fully specifies the number).

Let me ask a related question: why isn’t it a problem that the number 1 is expressible as an infinite series 1 + 0/10 + 0/100 + 0/1000 etc.? Every integer can be written surrounded by zeroes extending infinitely in either direction, and every one of the zeros is necessary to fully specify the integer, e.g. ‘…0001.000…’. How are you making the distinction?

Whether a time concern or not in the case of 01.000…, there would be “0” left remaining at every point along the summation out infinity. If there is zero remaining at any point along such a summation, the sum is already achieved.

I am aware that the sum isn’t a time concern, merely a descriptive notation.

It is the description that is saying that there is never a point wherein zero remaining will ever occur, and and does not exist (ie. “endless” = “infinite”).

Now you consider that if you stick to only the first order infinity (as opposed to infinity^n), the lowest number would be one infinitesimal:

[1.000…:00] - [0.000…:01] = [0.999…:99]
and
[0.999…:99] + [0.000…:01] = [1.000…:00]

And also
[1.000…:00] + [0.000…:01] = [1.000…:01]

Else, you can’t add or subtract an infinitesimal amount from any number without claiming that you still have the same number, which of course means that you didn’t really add or subtract anything, which in turn means that there is no such as thing an infinitesimal amount.

And if your ontology is going to imply that there is not such a thing as an infinitesimal amount, what is the lowest number?

1.0 is equally well continues as 1.00 and 1.09, so each zero really does further specify the value of the number. Just as you point out that each subsequent value added in the infinite sum of 9/10^n only get you closer to 1, so does each subsequent zero added to 1.0000…

This makes sense to me: there are exactly two decimal expansions of each number integer: one that ‘approaches’ from above, and one that ‘approaches’ from below. The limit of each is 1 as the values summed go to infinity, and the value of each infinite sum is exactly 1.

Any two real numbers have an infinite number of other numbers between them, so assuming by ‘lowest’ you mean a number closest to another number, the answer is that all real numbers have exactly the same number of other numbers between them.

There is such a set as the hyperreals, which includes infinitesimals. I’ll admit that I don’t understand the math of hyperreals, so I’m not sure the infinitesimals there refer to what you’re referring to, but those infinitesimals don’t exist on the real number line (as you seem to be using them): they are the reciprocals of numbers that are larger than any number expressible as a sum of integers.

I keep forgetting and give you credit for some graciousness in communication.

“Zero remaining” does not mean merely a single “0” in the line of zeros.
Σ 9/10^n never, ever, ever leaves zero remaining and can never have done so because it is defined to be only 90% of whatever is still there.

No. One was not “approaching from above”, one was already there and merely re-noting the fact of it.

That wouldn’t be true even if related to the issue.

The issue is that if you can add/subtract an infinitesimal amount to a number and still have the exact same number, then you didn’t actually add/subtract anything at all. And that would mean that 1 infinitesimal = nothing = 0. That in turn means that any number n times an infinitesimal is still absolute zero, which in turn denies Newton’s calculus (among other concerns).

The inference is that all infinite series that have a limit of 1.0 are equal = “1.0”. Yet it is provable that some numbers that have a limit at 1.0 are smaller than others. How could that be if infinitesimals don’t exist?

You don’t want to understand the hyperreals because they are on my side (or I am on their side, whichever). But one thing you should know about the hyperreals, is that they are a subset of the reals.

But look, if we don’t know subsequent decimal places, neither 1.00 nor .99 are equal to 1. .99 is at most 1/100th less than 1, and 1.00 is at most .9999…/100ths more than 1.

When you talk about 1.000…, you’re implicitly assuming that infinitely repeating line of zeros. If you don’t do that, you have the same problem as you have with .999… You need every single repetition of zero and 9 for either to equal the integer 1.

I think this is a good point, but I don’t think it’s decisive. d/dx isn’t really being multiplied by infinity in calculus, because infinity isn’t a number.

Calculus is based on limits, and as you point out, limits can be defined even in situations where the value is undefined. So I don’t think the fact that an infinitesimal is actually equal to zero implies that calculus based on the limit as the infinitesimal goes to zero is necessarily broken.

I’m not sure that they are on your side, because you’re treating things that only exist in hyperreals as though they’re on the real number line. Hyperreals are a superset of the reals. If you’re just looking at the set of real numbers, there are no infinitesimals, so they aren’t on the real number line.