Well its a free country and you can do what you want, but it will cause problems with other mathematicians understanding you if you don’t use standard notation. In the same way that a capital Sigma implies a sum and an elongated S implies an integral All mathematicians understand […] in the context of decimal notation to imply a limit. What you are doing is the equivalent of someone arguing that 2+3 is undefined because he rejects the idea that the plus sign indicates addition.
Its a non standard use of notation so its hard to determine what it means, just like if I asked you what the value of 2+ is? Without a value on the other side of the plus sign I don’t know what you mean.
See, this is why people (including me) are getting bored with this thread. I’ve explain the definition and construction of real numbers and shown why it follows immediately from that definition that 0.999…= 1— and you respond by posting without further commentary a 30-minute YouTube video that happens to contain a keyword in my post.
You don’t get to define what real numbers are: They are Cauchy sequences of rationals modulo sequences that converge to 0. This is unarguable; it is literally the very definition of real numbers. If you want to talk about schooner26 numbers that are real numbers with only finitely many nonzero decimal digits, or computable numbers, or whatever other condition you want, fine. But the real number 0.999… is well-defined and exactly equal to 1. We’ve proved that many times and in many different already in this thread, and replying with some crackpot’s Youtube video or some nonsense about infinity is utterly irrelevant.
In each pair here, the two statements are compatible (and, indeed, might as well be considered equivalent, except for the = vs. ≈ pairs, where the former is just a stronger version of the latter).
When in the mood to make sense of “…999 = -1”, I will happily sign on to every statement in this list except for “…99999.99999… = ∞” and “…99999.99999… ≈ ∞”. In another mood, I would endorse those two instead, and of course abandon the “…999 = -1” type statements.
But never would I be inclined to take each pair as having one correct and one incorrect statement, as you seem to think I should.
Yes, if you want to use this notation (i.e. …ddddd is defined as a Borel sum) then
…9999.9999… = 0
and its the same story for
…8888.8888… = 0
…7777.7777… = 0
and “shockingly” enough stuff like
…15151515.15151515… = 0
so the rules of multiplication and addition still work. So we now have the dreaded situation where 0 has multiple decimal notations. In fact an infinite number of them…
And even so far as = vs. ≈ goes, if we interpret the latter as “differs at most infinitesimally from”, as apparently is the intention, and decide to work modulo infinitesimal differences (as in the “real numbers” we have spent so much effort trying to explain), we get the former. So there’s a sort of equivalence here as well.
We can never know what the base-10 “number” actually is. Using […] again, is lame to me to say “the rest of the number which we do not know, and will never know completely”
No one has refuted my 2 exact definitions for 0.(3) and 1/3 and how they are different.
You didn’t refute what I said though.
You construct 1 with the sequence of partial sums. Attempting to write the sequence of partial sums and claiming it as a number is not how you constructed 1, and therefore, not equal.
ONE is the number that is constructed using the sequence. Thanks.
An irrational number is an irrational number irrespective of the base used. We could even use an irrational base if we wanted to be awkward about it. Rationality/irrationality is a property of the number and has nothing to do with the base used and indeed has nothing to do with the way that the number is represented at all. Greek mathematicians used irrational numbers all the time only their representation was the distance apart of compass points.
And on this point, we part company. I got nothing for you. Enjoy your deviant little path – you are not going to find many fellow travellers. Nor will you find many of the benefits of mathematics at your destination.
Oh, that? Back on Page 37? I assume people ignored it, rather than refuted it, because it was idiotic. For me it was gibberish, but for me a lot in this thread is gibberish and I don’t know why I keep trying to read it. However, even I find .333 repetend 3 = 1/3 intuitively obvious, and therefore it follows that .999 repetend 9 = 1. If it’s glaringly obvious to a mook like me you should find it obvious, too, and I can’t understand why it is not. Unless you are that married to the idea that it’s not true because, I dunno, maybe it makes you happy to be smarter than us. Usually I’d give you that you’re smarter than me, but you’re testing my charity.
Oh, that’s easy! 1.414213562 and so on. It’s irrational, but that number is close enough for what I do. I might even round it up to 1.4375 if I’m going to the closest sixteenth. Easy peasy!
Right, though one might object that (for positive d) the Laplace transform here should actually come out to +∞ (being the integral over positive t of d * e[SUP]10 t[/SUP] * e[sup]-st[/sup] dt at s = 1). This would, of course, be just in keeping with the position that d + d0 + d00 + d000 + … goes to +∞ as well.
So it’s worth noting that there is actually an element of meromorphic continuation involved here: we actually observe that the integral converges to a finite value for sufficiently large s (specifically, converging to -1/(10 - s) for s > 10), and then meromorphically extend this function of s to a finite value at s = 1.
In fact, we can observe term by term that the contribution of a[sub]n[/sub] to the function of s will be a[sub]n[/sub] s[sup]-(n + 1)[/sup]. So we can think of Borel summation as a distorted form of (perhaps more straightforward) Abel summation (in which we take the sum of the a[sub]n[/sub] to be the limiting value of the sum of a[sub]n[/sub] x[sup]n[/sup] as x approaches 1 from below), and similarly for these summation methods extended by analytic or meromorphic continuation.
Oh, FFS! The further you take out 1.414213562 and so on, the closer its square closes in on 2. That’s the way math works, and I’m embarrassed for you that you refuse to understand it. It is as if you are being wilfully stupid or trolling, but dumb people and trolls usually don’t put this much effort into things. I will give you the benefit of the doubt and assume you are really, really stuck on this, almost like a mental illness. Being monumentally wrong is bad enough in the privacy of your home, but here you should be the one who is embarrassed.
Wait, so now you’re rejecting that the limit of that convergent series is 1? Oy vey. Not sure how to help you with that one.
Similarly, I can’t help you with this. Just because a number has an infinite number of digits to the right of the decimal point does not mean it does not exist or cannot be reasonably defined or portrayed. I could never write down all the digits of sqrt(2). But it is trivially easy for me to calculate with it. And virtually all of mathematics that has anything interesting to offer us goes beyond this. You seem to be rejecting irrational numbers. I’m sorry, but math went beyond where you’re stuck 200 years ago.
200 years? The Pythagoreans figured out there were irrationals 2500 years ago. He’s rejecting all math from the point where it started to get interesting.
