Standards for peer review have thankfully evolved in the last two millennia.
You never replied to my previous post, so I guess I can take it you agree with it, and are indeed just talking about some other set of numbers than those we commonly call ‘real’. Since they’re not real, let’s call them ‘virtual’. So what, do you think, is the interest of your ‘virtual’ numbers?
Bah, stupid five minute edit limit…
That’s what you said, but ‘∞=-1/12’ does not follow from Ramanujan summation—basically, you can view it as the operator ‘+’ being understood in different ways: in ordinary summation, 1 + 2 + 3 + … does not have a value—the ‘+’ operation is not defined for a divergent sum. But you can introduce a new kind of ‘+’ operator, +[sub]R[/sub], which is connected to, but different from the original plus in that the sum 1 +[sub]R[/sub] 2 +[sub]R[/sub] 3 +[sub]R[/sub] … now is defined, and yields the value -1/12. You’re effectively doing different kinds of things with the summands, so the endproduct of both procedures isn’t equivalent.
Not everything that sounds strange is wrong. That in special relativity, two mutually moving observers will see each other’s clock going slow sounds strange, but is right. That (in the ordinary real number system) …99999 = -1 sounds strange, and also happens to be wrong. That 0.999~ = 1 may sound strange, but it’s right (again, in the ordinary real number system). Bottom line, sounding strange is not a reliable indicator of truth.
You want to make it infinitely as long?
I did not have time to read the post you are referring to. I will try to read it
as soon as I can. I am a little busy today, so I don’t know exactly when I have
time. Your post is lengthy, meaning I am necessarily not able to follow your
way of thinking, at least it would require much time. So I don’t guarantee I can
reply in any sensible way.
7777777, nobody treats infinity as a negative number. It’s possible to calculate with it as though it were a number, but following very different rules; however, nobody treats it as though it were a number. You’re the only one pretending that infinity has a negative value.
Please point out the flaw in this proof that 0.333… = 1/3
There’s no infinitesimal difference between 0.333… and 1/3. They are EXACTLY equal.
It’s not. Your “proof” involves multiplication by infinity. That’s the same as dividing by zero. A “proof” that includes such a step can be used to demonstrate that ANY number equals ANY OTHER NUMBER. You can “prove” that 1=2 or 1776.45 = -992.14 . It’s nonsense.
Not to criticise, but I prefer the construction:
Stop! It is not true. Didn’t we go over this already?
…999 does not equal negative one.
And it’s misleading to say that …9999 equals infinity. I mean, you could say that. but the problem is that infinity is not a number. …9999 just means 9 + 90 + 900 + 9000, and you keep going and never stop. That just means the expression keeps getting bigger and bigger and the farther you go the biggerer the biggerings get. We could just say “therefore it equals infinity”. But that’s sloppy, since infinity is not a number that you can use with equals signs. A better way is to say that the series diverges without limit.
You cannot use the word “infinity” in arithmetic and treat it like a normal number. An equation like 1/0=∞ is not a good equation.
Nobody except you, and a character in a book thinks that …999 = -1. So stop asserting that we believe it does. I mean, maybe it does, but the proof offered by the character in the book you’re referencing makes a simple error of thinking you can multiply infinity by 10, and then subtract infinity. That’s not allowed in normal arithmetic, because if we allow you to do that then you can prove anything equal to anything. And that’s not a very useful way to do math.
I think …999 = -1, in many contexts. The proof offered by the character is a very good one, in many contexts, where it does not lead to proving everything is equal to everything.
That having been said, I’m also happy to say …999 = infinity in many other contexts. And what’s wrong with saying that? You are clearly itching to say it yourself. It’s very natural. What is accomplished by flagellating yourself over saying “9 + 90 + 900 + … = infinity” instead of “The limit of ‘9 + 90 + 900 + …’ is infinity”, when usage shows they mean the same thing? “infinity is not a number” sayeth the textbook, but why shouldn’t we call it one? It has lots of numbery properties. Are you unhappy with infinity + 5 = infinity + 6? Perhaps someone else is just unhappy with zero * 5 = zero * 6. Are you unhappy with the difficulty of finding an object infinity meters long? Hey, at least infinity counts things (the number of primes, say), like numbers were invented to do; someone else might be just as unhappy with the fact that no set has 0.5 many elements, or -1 many elements, or i many elements.
What was Cantor doing with all his lauded work on transfinite arithmetic, if not showing some useful ways to treat infinity as a number? Granted, cardinal arithmetic is different from the sort of ways you would treat infinity as a number if you wanted to study limiting behavior of real arithmetic, which is different from the sort of ways you would treat infinity as a number if you wanted to study ordinal arithmetic, which is different from… This is probably the problem, then: people don’t like to think of infinity as a number because they find different formalizations of it clash; they aren’t happy confronting the idea that infinity is not just one number, but a multitude of different numbers, in different number systems. But this is actually how it is, and the same is true for every other number too; just more easily ignorable.
Or “The limit of ‘9 + 90 + 900 + …’ doesn’t exist, specifically because it grows and stays larger than any finite value”, or whatever rewording of the same concept you’d prefer.
Could be worse. I used to TA students who were told “1/0 is undefined” and would write “1/0 = undefined”, as if “undefined” was a number.
…999999=-1 is true.
-1 is a number, …999999 is a number
Therefore …999999 is not infinity because infinity is not a number.
Also -1 cannot be equal to infinity, and neither can -1/12 no matter what
Ramanujan said, -1/12 is a number.
I admit, infinity needs sometimes be treated as a number in various formulas
to work with it. But how can one manipulate with a symbol ∞ if one does
not know what it is? Can numbers transform into a symbol ∞?
And if they can, how? Can addition transform numbers into ∞?
Because, it is usually thought that, for example
1+1+1+1+…=∞
There are infinite amount of number 1s transforming into ∞
Therefore numbers have this property, although numbers themselves are not
infinities themselves. Perhaps it is the property of addition transforming numbers
into infinity.
Can you tell me what number is …999999? I am certain that it is not
infinity, so that it not a correct answer.
In many contexts. Not in all.
In the context of the standard real numbers, it doesn’t exist.
Math isn’t decided by fiat. You define your context, and statements are proven in that context. “Truth” isn’t the same as fact or proof. “Truth” vs “proof” is basically philosophy vs math.
In the context we wish to use (our standard set of real numbers), …999 doesn’t exist.
It’s certainly possible (as repeated ad nauseum above) to have a context in which it does exist. But it is not generally appropriate to use statements in one context in a different context. That’s the crux of the failure of your arguments.
You don’t think we can find the decimal equivalent of a fraction by dividing the numerator by the denominator?
Do you think .5 = 1/2? If not, why not? If so, how do you provide a proof for it?
…
Wait, what? No, …999 is not a number in the conventional sense! It’s a construct which cannot possibly be resolved to any real or complex number (your failed proof which does something about as viable as dividing by zero notwithstanding).
Well, your certainty doesn’t help anyone. You might as well ask “What number is Zebra”. The only way you could possibly define …999 is by means of the limit of an infinite, divergent row. Unless you have something else, by which I mean a proof which doesn’t multiply by infinity.
A) Hammering 7777777 about “…999 = -1” isn’t particularly salient (or, I think, fair); the role this plays in their rejection of .999… = 1 is non-essential.
Specifically, the same reasoning which led to them to …999 = -1 has a natural analogue which would also lead to …999.999… = 0 , perfectly compatibly with .999… = -1. In fact, against the backdrop of their acceptance of …999 = -1, the two claims …999.999… = 0 and .999… = 1 are equivalent. But that’s fine; someone could endorse them both.
They refuse to accept …999.999… = 0 on the grounds that the digits are different, and therefore conclude that .999… must be distinct from 1. This is how they’ve phrased their “counterproof”, but they could just as well raise the same objection more directly: that .999 is distinct from 1 because the digits are different. The detour through …999 = -1 is a red herring. The fundamental issue is the refusal to entertain the idea of distinct decimal series representing the same value; changing their perspective on …999 will not do anything to accomplish the goal of changing their perspective on uniqueness of decimal series.
B) That having been said, nothing will accomplish this goal anyway. Why are people still trying arguing with 7777777? Once things turned to weirdly aggressive witnessing, any hope this would be a non-futile endeavor was quashed. And it’s fine; what’s one more mistaken person in the world? Ignore them. Let it go. Forget it; it’s 7town.
I’m missing an ellipsis, and there are various other things I’d edit given the chance, but whatever.
I hope that isn’t too painful for you. Missing ellipses can cause all sorts of problems.
It’s happening already - I just want to cut to the chase.