Earlier quoted context omitted.
Why is 0.000... bigger than 0?
It isn't.
You are correct. This was a rhetorical question to get traderjane to question whether "bigger than 0" really applies here.
0.999...= 1
371–380 of 647 posts
Re: 0.999...= 1
#372Earlier quoted context omitted.
Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?
Can you explain what you mean with "real" in that sentence? Because in the context of maths, a real number is "a number in ℝ", which this absolutely qualifies for. Whereas in plain English the term doesn't really have a clear definition. You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long,…
Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is meaningless with respect to the real numbers.
Re: 0.999...= 1
#373Earlier quoted context omitted.
> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thi…
Ask for a number between .9 repeated and 1
"There isn't one, 1 is the very next number right after 0.999... Checkmate atheists." (In all seriousness I don't think it's a very convincing argument for someone who doesn't buy the proofs -- it requires you to believe and have internalized the idea that there are an infinite number of reals between any two distinct reals, and therefore that any pair of reals with nothing between are the same number. Those seem like bigger logical leaps to me than the simple proofs for someone who hasn't thought about this stuff.)
Re: 0.999...= 1
#374Earlier quoted context omitted.
Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?
You would define it as a digit sequence, using ω + 1 as the indexing set. I would consider it to be most naturally an element of the hyperreal numbers, although it is also contained in smaller extensions of the real numbers.
Sure, you could define some sort of number system that has an infinite point digit. I doubt that’s what the original poster had in mind though.
Re: 0.999...= 1
#375Earlier quoted context omitted.
I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…
> it could just as easily be reframed as 0.000...0001 = 0 But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.
My argument is that "noun" is a purely human abstraction, and that phenomena that act noun-like in nature are at best snapshots of iterative processes. Within the bounds of the noun abstraction, sure, I'll cede that point.
But if one eschews that abstraction and looks at it purely as a process (I want to render 1/3 in decimal notation, then multiply that decimal notation by 3), there is always that niggling 0.000...1 remainder at every snapshot. The "never stopping" bit is what smuggles verbiness into the "0.999..." noun, while simultaneously pretending it's a static value.
Re: 0.999...= 1
#376Earlier quoted context omitted.
> 0.00...1 And what does this mean? I will remind you that for an integer d between 0 and 9, 0.ddd… means the limit of \sum_{i=1}^N d/10^i as N tends to infinity.
0.000...1 = 1/∞
And what does the right hand side of that mean? Division is commonly defined for a real numerator and a real, non-zero denominator. You are using the common symbol, but with ∞ in the place of the denominator. Since ∞ is not a real number, you must be using a non-standard definition of division, and have to define what you mean.
Re: 0.999...= 1
#377Earlier quoted context omitted.
There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.
> There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1.
I think people intuitively see that infinitely zero equals zero.
Re: 0.999...= 1
#378Earlier quoted context omitted.
> An infinite number of zeroes. . .and then a one. . .wait, you can't do that. why not? why can't an infinitely small number exist?
It can, and infinitesimals are a part of so-called nonstandard analysis, but you cannot write an infinitesimal using decimal notation. "0.999…1" is simply meaningless, a contradiction. If you have a "…" it means there's no place where you could put a "last" digit. If "0.999…1" doesn't feel impossible enough, then what would "0.999…9" mean?
People in this thread seem to think that “9 repeating” is not an infinity of nines but is instead “write or think of nines until you get bored and then write something else”
Re: 0.999...= 1
#379Earlier quoted context omitted.
great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in…
I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 dig…
Interesting. I find it funny to imagine what if you did have pictorial representations. They'd have more and more complex strokes and knots, and you would also need an infinitely large paper or infinitely precise pen in order to not end up repeating a number sooner or later as you enumerate them.
Come to think of it, this is one way of thinking about the relationship between symbols and geometry.
Re: 0.999...= 1
#380Earlier quoted context omitted.
You're repeating the same wrong thing you said earlier. It's 0.999... and not 0.999...0 In the same way, it's 0.000... and not 0.000...1.
0.999... = 0.999...9 0.999...9 + 0.000...1 = 1 0.999...0 + 0.000..1 = 0.999..1 0.000...1 = 1/∞ 0.999...9 = 1 - 1/∞ 0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞ If x/∞ = 0, then 0.999...x = 1. If x/∞ ≠ 0, then 0.999...x ≠ 1.
Ok, now you're saying that infinite decimals have final digits.