Is 1 = 0.999... ? Really?

“Number”

If memory serves, this is the first time you’ve suggested otherwise. But that’s sort of my point, (L) can’t be a real number for all the reasons I’ve pointed out, so if it’s not a reason then we agree there. I just think that’s a big problem for your argument, because you’re treating (L) as though it’s a real number.

I’m not ignoring it, I’m pointing out that it’s vague. (L) is clearly not a number like the ones we usually work with, and so I’m not making assumptions about what you mean by “number”. You haven’t clarified what happens when we subtract 1 from (L), what kind of number do we get? If it’s a real number, than that number plus 1 must be (L), in which case (L) is the sum of two real numbers. If it’s some number less than (L) but of the same type, then does (L-1 = L)? Is (L) hyperreal?

Failing to say explicitly what you mean lets you play calvinball around (L): no conclusion that could defeat your claim can be argued for, because you offer attributes of (L) only if and when they become necessary to undermine some argument.

Here is a great example: just confirm or deny the inequality!

I have yet to see a coherent definition of the “the largest number”, and this question makes me think whatever you have in mind has some problems. But I guess we’ll see when you tell me what you have in mind.

This here is treating (\infty) like a number, and it isn’t a number. I would say that that equation is meaningless. I’ve probably been sloppy in my language around this, and I apologize. When I say that two infinities are equal, I mean that every element of one set can be mapped to exactly one element of the other set. So, for example, (.\dot9) and (9.\dot9) have the same number of decimal places, in the sense that each decimal place in one corresponds to exactly one decimal place in the other.

Somewhat as an aside: in your system, (.\dot9) is ambiguous, since if I just say (.\dot9) we don’t know if it’s just the simple repeating decimal, or some different version of the repeating decimal obtained by multiplication by a power of 10 and subtracting the integer portion. This strikes me as a larger problem than you seem to acknowledge (how might 1 fewer zeros on the end of an integer’s decimal expansion change its value?).

(L - 1) is not a real number. It’s also not (L). Even if we say that (L) and (L - 1) are numbers of the same type (i.e. if we put them in the same category), it does not mean they are the same number, that they are equal.

I am not sure why you’re insisting on categorizing (L). What exactly do we gain by placing it in a category?

I don’t have to confirm or deny irrelevant claims. Not answering irrelevant questions isn’t evasion in the negative sense of the word. If it is an evasion, it is a positive kind of evasion (one that tries to push or pull the discussion in the right direction.)

If I recall correctly, your claim is that (0.999\dotso = 1). If this is true, it means that you think that an infinite number of non-zero terms (which is what (0.\dot9) is) can be equal to a number (such as (1).) In other words, it would imply that at least SOME infinite quantities are numbers (since (1) is a number, right?)

(0.\dot9) and (9.\dot9) have the same number of decimal places because you said so. It’s an arbitrary decision. Nothing is stopping you from picking a different position e.g. that (0.\dot9) has fewer decimal places than (9.\dot9).

And the result of dividing .9 recurring in half?

The question is are they significant decimal places? As example 125 = (1x100)+(2x10)+(5x1). further 3076 = (3x1000)+(0x100)+(7x10)+(6x1)

In the case of 0.9, 0.9 = (0x1)+(9/10) the number in decimal form is already telling you there are zero 1s. If there are zero 1s .9 recurring can’t be equal to 1 because there are zero 1s in the expression.

We gain the ability to have a conversation! If your argument depends on the properties of (L), then we need a full description of those properties (e.g. by placing (L) into an existing category) to evaluate your claims.

Do you not know the answer? You don’t have to do anything, but if you know the answer, it would be awfully kind of you to just say it. Or if the question doesn’t make sense I’d love to hear why. Or if the answer is completely toxic to understanding and also irrelevant, then make that case.

Your conclusion doesn’t follow. (0.\dot9) has infinite decimal places. It is not itself infinite, and infinity isn’t used as a number in defining that.

It follows from the definition of a repeating decimal. If you have some other definition, please specify (and really, you need to clarify what it means if you’re going to hold that (9+0.\dot9 = 9.\dot9 \neq 9.\dot9 = 10 * 0.\dot9))

The long division makes sense here, and doesn’t really inform the question.

(\frac{0.\dot9}{2}= .4\dot9 = .5)

Next up: Does 1 covid 19 infection = 0.999… covid 19 infection?

Interesting but you are not expressing the fraction .9 recurring. divided by 2 as a fraction.

In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero.

Your example is invalid as .9 recurring is not an integer. the representation .9 can be written as (0x1) + (9/10) Does it matter how many 9’s follow the expression as the expression itself clearly declares there are zero 1s.

My point was that the division works as expected: the long division algorithm produces (0.499999…)

So, I don’t think it answers the question either way because if (0.\dot9 = 1), then (0.4\dot9 = 0.5), and if it doesn’t it doesn’t.

There’s a lot of addition in there too, and I don’t think anyone’s denying that 0x1+5/10+5/10 =1

Yeah but you haven’t respected the notion of decimal formatting in your example either. You have added two fractions together from the same decimal position. 0.9 recurring still states there are zero 1s within it’s notation. Adding .5 + .5 is the math required to move the decimal position, .9 recurring leaves the decimal point where it is and leaves it as zero 1s.

Somehow, you forgot the fact that (0.\dot9) is an infinite sum, which makes it an infinite number of, non-zero terms. Are you going to tell me it’s a finite sum? a finite number? Is it finite or is it infinite?

Are you suddenly denying that (0.\dot9 = 0.9 + 0.09 + 0.009 + \dotso)? What’s at the right side of the expression if not an infinite number of things?

It’s pretty clear that infinites and finites overlap. An infinite number of zeroes is a zero. One raised to infinity is one. And so on. If you can calculate the result of (\infty \times 1) what’s the reason you can’t calculate the result of (\infty - \infty)?

If you say that (9.\dot9) and (0.\dot9) have the same number of (9)'s following the dot, then it follows that (9.\dot9 - 0.\dot9 = 9). That’s doing arithmetic with infinites.

If you say that (9.\dot9) and (9.\dot9) have the same number of (9)'s after the dot, then it follows that (9.\dot9 - 9.\dot9 = 0).

If you say that (\infty) and (\infty) refer to the same number, then it follows that (\infty - \infty = 0).

There’s ABSOLUTELY NOTHING meaningless about (\infty - \infty = \infty). That’s just you not willing to play the game. You don’t win the game by not playing the game.

How about you place (i) in one of the categories that existed before people invented the category of complex numbers? How useful is that? Very unuseful, isn’t it? What’s the point of placing something where it doesn’t belong?

How about placing rationals in the category of integers?

And why is any of this relevant? I don’t even remeber why I mentioned (L). That’s how relevant it is. It served some very limited purpose, one I can’t even remember, precisely because it’s so limited.

The question; Is 1 = 0.999…?

Another way of asking the same question might sound something like; is an infinite number of smaller and smaller “parts” equal to 1 finite “whole” ?

If I had 1 volume of space and I divided it, equally in half, an infinite number of times, the result would be an infinite number of parts. I guess if I could cut a volume of space an infinite number of times, it would still have come from the 1 original volume.

@Carleas

Your argument rests on this: infinity minus infinity is meaningless but (0.\dot9 - 0.\dot9) is equal to zero. Not sure how to say it but don’t you realize that infinity is equivalent to a sum of ones? Like so: (\infty = 1 + 1 + 1 + \cdots). And that (0.\dot9) is also equivalent to a sum of numbers? Like so: (0.\dot9 = 0.9 + 0.09 + 0.009 + \dotso). Now, you take (0.9 + 0.09 + 0.009 + \cdots) and you subtract from that (0.9 + 0.09 + 0.009 + \cdots) AND YOU HAVE NO PROBLEM WITH THAT. You go like “Hm, okay, take first numbers from both sums, subtract them, we get – hehe – zero! Now, let us take second number, what do we get? oh yeah baby, we get yet another zero! and so on and so forth!” But you take (1 + 1 + 1 + \cdots) and subtract from that (1 + 1 + 1 + \cdots) and you tell me it’s MEANINGLESS? As in meaning-fucking-less no less. Caps lock doesn’t have big enough letters to communicate the bewilderment I am currently suffering through :slight_smile: What is going on in here?

Infinity ((\infty)) times infinitesimal ((\frac{1}{\infty})) equals (1). No doubt about that. But (0.\dot9) is a different beast – one that never adds up to (1).

That’s false Magnus. Infinity times infinitesimal still equals infinity.

Sure is fascinating to think about.

You need to level up.

It’s not either true or false per se but either true or false in relation to the starting premises. If you start with the premise that (\infty + \infty = \infty), then yes, (\infty \times \frac{1}{\infty} = \infty \div \infty = \infty). But if you start with the premise that (\infty + \infty = 2 \times \infty) a different conclusion follows.

The important thing is that, whatever premises you pick, the mentioned Wikipedia proof is invalid.

I want very badly to dismiss this argument, but I am forced to admit it’s stronger than I hoped. The best response I have, which is not much response at all, is that it is question-begging: if (.\dot9 = 1), then it isn’t true that there are zero 1s when there are 9s in every decimal place.

I think we need to be careful with our language. “An infinite number” is not the same as “the number ‘infinity’”. The former treats infiniteness as a property of multiple numbers, the latter treats infinity as a number.

This might seem pedantic, but I think it matters. If there’s more than one infinity, if it can be a property of a number but not a number itself, then the equations using (\infty) as a number are at best ambiguous, so our answers will be uninsightful.

It isn’t. Both (9.\dot9) and (0.\dot9) are finite numbers.

I think those answers seem intuitive, but I think they’re playing fast and loose with notation that has a formal definition that makes those equations meaningless in a rigorous sense. Using the symbols that way is a sort of expressive math, an uncertain poetry built of the symbols of rigorous expression.

I’m willing to talk about what (\infty - \infty) might mean, I just don’t think it’s well-defined (“meaningless” might be too strong). (\infty) can refer to multiple quantities that are provably distinct. I’m not sure what the subtraction symbol means here, given that I don’t think (\infty) is a number. My most natural reading is set subtraction.

So let’s play, but I don’t want to play calvinball.

But we agree that “complex numbers” is not a category, right? Are you defining a new category with (L)?

It is the diminishing value of each place that is filled. I don’t know how to notate it but it’s the .1 in an infinite number of decimal places where it comes up short to clear the decimal point and increment the digit on the “whole” side of the decimal. In it’s form, .9 recurring; it is presented as a fraction and I don’t think infinity can change that. It may simply be the largest fraction possible. And with little more then a place holding decimal point between them they could be equal. A fraction turns into a whole through infinite recursion. Fraction or integer?

And what happens when you multiply .49 x 2, would the result not be the same in an infinite number of decimal places? Try the operations reciprocal.

1 x 2 = 2 and 2 / 2 =1. If .9 recurring is equal to 1 then the math should be that simple.

When you can invoke Murphy’s law in math? You’re having fun.