Is 1 = 0.999... ? Really?

9 (recurring) over 10 (recurring)

[attachment=0]fraction.jpg[/attachment]

Interesting, although it doesn’t move us forward; (1\dot0 - \dot9 = 0), so that fraction equals (1 = 1/1 = 0.\dot9)

Let me go back to this:

So we have ‘1’, the symbol, which points to 1, the concept. We can say of the concept that there is no decimal part, it’s an integer. By convention, we treat the symbol ‘1’ as pointing to that concept, and we assume that where ‘1’ doesn’t explicitly specify the decimal expansion, it implies it. But as you say, if there is any decimal part that is not 0, we agree that it would not be the concept 1. By convention, the symbol ‘1’ implies that there is no decimal part.

We can point to the same concept in a number of different ways: (\frac{2}{2}, x^0, 0!, -e^{i\pi}) – those are all the same concept.

So the claim is that (0.\dot9) is another symbol for that concept. To show that it doesn’t, you need to show that they are conceptually distinct. Similar to what I say above, (1 - 0.\dot9 = 0), and there’s no other value we can coherently put on the right side of the equality. Magnus’ approach, if I understand it correctly, seems to be to posit (0.\dot01), but that too is equal to the concept (0), because by its construction there’s no difference between them.

Related: do you agree that (\infty+1 = \infty) (or rather, that the cardinality of the set of integers is the same as the cardinality of the union between the set of integers and the set {1.234})?

Wow. the site is slow today, painfully.

This is taking me no where. I can’t even copy and paste what you have written. I disagree with the language. 1 as an integer does not require the concept of division or exponents, or … Your examples include them as part of the concept of the integer 1.

I’m no Georg Cantor. The notion of infinity has messed with better heads then mine.

You are a more informed mathematician then I.

The proof involving let x=.9 (recurring) introduces a rounding error in the equation. The presence of the function of the infinitely recurring property is lost in the process and therefore the equation isn’t balanced. At best the result is an approximation. That’s my best guess. Rather like rounding up .6 (recurring) to .67 or truncating π to a specific decimal place. That’s what my gut tells me, but i can’t prove it. An infinity long line and a finite length 180 degree arc share the same number of points along their paths because a finite length path can be divided infinitely. All infinities are not created equally. Thank you Georg Cantor. Perhaps this “proof” ignores this detail, but I’m not the mathematician to prove it.

Using quote from the normal view should show the LaTeX markup. The quote buttons in the “Topic Review” below the reply box are kind of misleading, they only tack on the poster name, they don’t actually quote from the post they’re attached to (you can see this by selecting arbitrary text anywhere on the page and then clicking the quote button: it will format the selected text as a quote by the author of the post that the button is attached to).

What does “=” mean? If (\frac{2}{2} = 1), what do we mean by “=” there? I wouldn’t say that (\frac{2}{2}) is “part of” the concept of the integer 1, but it is equivalent to the integer 1, and it’s certainly an important part of the concept that it is the multiplicative identity, of which (\frac{2}{2} = 1) is an example.

I think it messed with Cantor’s head as well. He seems to have somewhat agreed with Magnus, in that he thought there was a greatest number, though he also seems to have recognized that the concept leads to contradiction. His first quote on that page is downright mystical, though, basically saying that “absolute infinity” is tantamount to god.

Sorry, I got use to the select, copy, paste. The quote button always grabs the whole post, but it does work better to see the code that makes it all pretty.

I guess equals means more to me then you, huh? And to add insult to injury, the extra meaning is likely invalid. I get side tracked easily, had to come up with some grip of this LaTeX thingy.

Anyway, 1, the integer is a counting unit. It requires no ‘function’ to be performed on it to be it. In the case of 2 divided by 2 an element has been added to the units as counting digits that requires a division function to be performed. So when you concluded “-- those are all the same concept.” I thought there was more then just one concept in play. So yes, the value resulting from the division function results in 1 which is equal to 1. Yet there is something ‘more’ taking place on the left (in your example) of the equals sign then on the right. Not arguing what you mean just how you said it; as if the two, as a package, didn’t make the transit. Likely the confusion was on my end.

Anyway, I am more likely to suspect that some quality is not being taken into account, or there is yet more at play then meets these eyes. But if the question comes up in a quiz I’ll be better prepared to answer counter intuitively.

If on a math test I came upon the question x equals two minus one, or x plus one equals 2, solve for x. If I answered .9 (recurring) would my answer be considered correct? Or if presented the question; two plus two equals x, solve for x, and my answer was 3.9 (recurring)?

I deeply regret that I did not think of testing this when I was in school. There are infinite answers to any “solve for x” problem, and I now realized the many missed opportunities to be a precocious wise ass.

The repeating decimal is an interesting case here. By comparison, if the question is (2-1=x), then (x=2^0) is true, but I suspect the examiner would say that it’s not in its most “reduced” form, and that seems legitimate since there’s an unresolved operation baked in. But I don’t think that objection works as well for (0.\dot0): it’s not really any less “reduced” than (1), at least not as obviously as an answer that contains an operation.

Ultimately, “is it correct” and “would it be considered correct” are different questions. It is correct, but I suspect many teachers wouldn’t accepted it. Lots of teachers punish wiseassery, even when it obeys the letter of the law, and appealing to mathematical trivia to provide a technically correct but counterintuitive and less-clear answer would be read as being a wise ass.

I haven’t read much at the intersection of math and philosophy of language, but my impression is that popular conceptions give it more meaning than it needs. I think it’s possible to build math as a purely formal language, with no real-world analogues to the transformations and relationships necessary to specify how its objects are interconnected. Define rules about how we can manipulate symbols, and then use the rules to show that two symbols satisfy the relationship “=”.

But that’s beyond me, and I’m only about 80% sure it’s true.

Funny, in my previous post I was going to ask if it would be correct or I’d get sent to the office for being a wise ass, or maybe both. I was being precocious asking it.

It’s the decimal. 0. means something different then .0, following or preceding yet, on each own they mean the same thing. And within the infinity between zero and one there is an infinite opportunity for counting. Save for the first from no thing to some thing. Save between counting of something and counting nothing. How can no thing be counted? Absence? the decimal on it’s own, merely a divider, a quantifier of a whole or a part. That is how much difference exists between .9 recurring and one, the smallest thing remaining countable is as infinite as .9 recurring.

This might be a pedantic over-emphasis of your specific wording, but the real numbers aren’t countable. There’s no next real number. For any two distinct real numbers, there are infinitely many real numbers between them. So the term “the smallest thing remaining countable” is undefined, there can be no smallest real number distinct from zero (“smallest” in absolute value, I assume that is your intent as well).

If it isn’t zero; the decimal point on it’s own, it is counting by default. Any unit recognized as not zero is a form of counting.

It seems we are getting hung up on this notion of size. Zero is not the “smallest” in absolute value, it represents a value as non existent regardless of size, sequence and series. Any unit recognized as existing can not be zero. This is perhaps more an existential question then one of math. If there were not things to count we would have no reason to create a symbol representing the absence of the countable things. We wouldn’t have a zero as a symbol for no value if we didn’t have things we place values upon.

Zero represents nothing to count. Zero is a symbol given to not counting. As soon as any other symbol shows up counting has taken place. You’ve recognized a thing and things as distinguishable from the absence of the thing and between things. It is sort of binary, zero or one. You’ve presented me with a range of an infinite set of numbers between 0 and 1 which are real numbers. Given this information, there are three values involved. A) No thing represented by zero (0) B) “a” thing represented by (1) where X is not equal to 1 and 1 is not equal to 0 And as a given C) there is an infinite subset of “somethings” between them N is not (0) AND n is not (1). I can count there is at least one thing between 0 and 1, N. All real numbers can be counted just not in an infinite sequence that is dependent on an infinite number of digits. There is/are N of them between 0 and 1. They can be counted in sequence as 0, N, 1. Where N can not be less then or equal to zero or equal to or more then 1.

I consider the extents in a range as not included in the subset of the range defined by them. Key word being “between”.

Sure, but between (1) and (0), there are a bunch of other numbers; we can point to any of them (0.123, for example) and prove that (1) and (0) are distinct numbers. There are no numbers between (1) and (0.\dot9), which isn’t possible if they are distinct real numbers.

the decimal point. It is not, to the best of my understanding, a number in and of itself. But is does reside between 0.9 and 1.0 in our numbering system.

Count to 1.0 by .9s recurring? What is used to move the decimal point and assign the next most significant digit? The application of the convention of counting natural numbers. So we have all these different descriptions of groups of numbers; natural, whole, fractions, integers, rational, irrational, prime, positive, negative, real and complex and imaginary.

Not done.

Darn, composed for too long and my service provider killed the connection.

I will agree that within a hypothetical convention where infinity exists 1 = 0.9 recurring. I would also argue that using the natural numbers as a convention the numbers 0 and recurring decimals 9’s don’t exist to have equivalence.

It is a matter of convention and not all conventions are created equal. Yet it is a postulate that .9 recurring using an infinite convention is equal to 1 limited by another convention of the infinite. Different sets that are not inclusive.

We “let” it be equal in an infinite set as an axiom, while it doesn’t exist within the convention of natural numbers as other example of an infinite set. We have already determined all infinite sets aren’t created equal such that a .9 recurring in a set does not require it be equal to a 1 from every other case of an infinite set. 1 can not be equal to .9 recurring in a set that does not include .9 recurring as a member of it’s set.

It is presented as a one or the other case while there is an “and” option in place of “either/or”. Just what exactly are we “letting” the unequal non-equivalent sets include axiomatically?

OK. this ‘seems’ to “sum” up the argument as reasoning thus far. (don’t know, could be other members of an infinite set, axiomatically). I don’t know maybe the first draft was a member from a set that was a more accurate representation. What are we going to “let” axiomatically?

phyllo nailed it and didn’t.

Differing sets of convention.

What exactly is vague about the statement “A number larger than every other number”?

Closure

If you add two real numbers, you will get another real number. What makes you think that (L) is a real number and that what applies to real numbers also applies to it?

Of course, you’re aware of the possibility that (L) is not a real number. Earlier you said “So, (L) is some special type of number, it doesn’t follow normal rules of arithmetic. So it’s possible that (L-1 = L), we can’t really say because we don’t really have a definition of what (L) is.” But we do have the definition of (L). You’re merely choosing to ignore this for some reason.

Associativity

Earlier, you said “Right, so (L + (1-1) \neq (L+1)-1). That means (L) isn’t a real number, because addition and multiplication aren’t associative on it.” I never confirmed or denied this inequality since it doesn’t strike me as particularly necessary. What makes you think that (L + (1 - 1) \neq (L + 1) - 1) can’t be the case?

I’m now going to return to this:

What about (\sqrt{-1})? Is it undefined? Is there a real number the square of which is equal to (-1)? If the answer is “no”, does that mean you reject complex numbers? If you do not reject complex numbers, why do you reject pseudo-numbers such as numbers larger than the largest number? And why do we even have to speak of such numbers?