Strange

And you still think it is a plausible project. I am surprised. Could you share your preliminary results with us?

Excellent summary. Although, I suspect the response will be, "yeah but ..." The Wikipedia article describing the axiomatization of the reals, linked to Tarski's approach: https://en.wikipedia.org/wiki/Tarski's_axiomatization_of_the_reals I think the first three axioms by themselves make it pretty clear that 0.999... = 1. We can leave the complete proof as an exercise for the OP....

Have you even thought about how much energy it takes to launch a satellite!? You are a scientist. Do a quick calculation. It would be more efficient to just burn the rocket fuel to generate electricity.

There is no interval between them. That is why it called the continuum.

You are using "imaginary" in a completely different sense than it is usually used in mathematics, which really doesn't help. The OP is not looking for a "school-level" or simple proof. He is looking for a proof from the axioms of number theory.

Is that rather like the modern euphemism "Bless [your heart]"to imply that someone isn't too bright?

All mathematicians and a large number of other people (who are familiar with mathematics) do deny exactly this. They are the same number. I'm sure this is possible although it may not be practical. And seems a little pointless. Mathematics works as a hierarchy of proofs and derivations. You don't need to start from number theory to prove the Taniyama-Shimura conjecture. You build on what is already known about elliptical functions, etc. No, because this doesn't apply to reals. Although, the fact it doesn't apply to reals is a good proof that 0.999... = 1 (as I think someone already pointed out).

Indeed. It is a word that is very widely used in engineering and other fields. I don't see that changing just because it is also used as an insult.

In terms of crankdom, it seems to be physics, biology(1), environment(2), mathematics and then everything else. No idea why. (1) Apart from the obvious evolution "debate", there are all sorts of things like "all disease caused by X", anti-vaccination, cryptozoology, etc. (2) As well as the obvious climate change denialism there are "chemtrails", "weather manipulation" and various other weird theories out there.

It is not a matter of protecting them from non-mainstream ideas (some non mainstream ideas go on to become mainstream, after all). It is important that people (e.g. students) know what is mainstream and what is new, cutting edge and possibly wrong. Or even just wrong.

1. The number 666 is a mistranslation. 2. Those titles are invented to make the numerology work.

As the Speaker of the UK Parliament (Bercow?) once said: "They are, of course, entitled to their own opinion. However, they do suffer from the not inconsiderable disadvantage of being wrong."

I haven't looked back at the thread(s) in question but if I said anything inappropriate then I certainly apologise for it. I am going to suggest the mods close this post now as I feel rather embarrassed by having our "private" disagreement discussed in public (even though it was public to begin with!)

Well, thank you for the apology. I should say that I wasn't particularly upset or offended by anything you said, but it is pretty brave and decent to make a public statement like this (I'm not sure I would have the guts!)

By itself, I can't see that this would ever lead to anything but an integer n being represented by (n,n). So for example, if we make A = 2 = (2, 2) and B = 3 = (3, 3) then, using your rule for multiplication: A*B = (2x3) = (2x3) = (3x2) = (3x2) And, presumably, based on the initial axiom, all of these are equivalent to (6,6). Even division leads to the same thing: A/B = (2/3) Which can, presumably be represented as (0.666, 0.666) B/A = (3/2) Which can, presumably be represented as (1.5, 1.5) So, a couple of questions: 1. Are all numbers of them form (n, n); i.e. where z1 = z2? If not, how do we get a number where z1 and z2 are not equal? 2. In your multiplication and division rules, why don't we end up with a pair of numbers as the result? For example, I would have expected something more like: A x B = ( z1forA x z2forB , z2forA x z1forB) ... or something similar Also, just as an aside, the "z1forA" notation isn't very easy to read. I would go for something like A[z1] or A.z1 (both notations used in programming, not sure about pure math). But not important. I'm not sure this is a problem. If the table is just an aid to understanding the concept, it could probably be extended to reals without needing an explicit table. Good question. That might be one way of understanding it. But then, wouldn't this be like complex numbers (except the multiplication rules are different). Does this mean that, unlike the other numbers, 0 is represented as (0, 1) ? Ah, yes. Sorry just noticed that is explicitly stated in your table. So if we have A = 3 = (3, 3) then A * 0 = (3 x 0) = (3 x 1). But that seems to imply that multiplying a number by 0 can give two possible results, 0 or A. Is there a rule for choosing which to use? When you say "z1 by default" do you mean always z1?

https://en.wikipedia.org/wiki/Archaic_human_admixture_with_modern_humans (Is this so you can decide who you should marry? :))

I thought that was a great answer. Why on Earth would anyone give it a down vote!?

It is for the sake of other users that it is not in the Physics section.

Energy is a party of "things" (from subatomic particles to complete systems). I don't know if the ability to do work definition is complete, but it is pretty good. Other definitions are likely to be a bit vague: "it is a property we can measure that is conserved when things interact". And, strictly, energy isn't conserved, but mass-energy is because we can convert energy tombs and mass to energy.

It would look like light.

And then Einstein's explanation of the photoelectric effect. Also, photons are not particles or waves, they are ... well, photons. They obey their own rules. Sometimes that looks a little bit particle-like to our human intuitions and sometimes it looks a little bit wave-like. It depends what you measure and what model you use to describe light. (There is no point asking what light "really" is. All we can do (with anything) is model how it behaves.)

It wasn't quite sure what you were looking for. I was going to comment on John's very first reply where he referred to "elementary school maths" to point out that that was what you mean by "mid-level mathematics" but I wasn't sure if that was what you meant. Anyway, obviously it is possible to derive all the properties real numbers from first principles. It would however, be a huge amount of work. You might want to use an automated theorem proving system to help. But there really isn't any point. 0.999... is the same as 1. We are not dealing with infinities, so this isn't really relevant.

I'm not sure that is true: https://en.wikipedia.org/wiki/Myth_of_the_flat_Earth Do you mean epicycles? But these examples are not really relevant because we are talking about mathematics. If something is true in mathematics then it is always true.

And therefore, because your h doesn't exist (or, at least, it is zero), there is no successor (or 7 is its own successor).

Or are you confusing a number and its representation? Well, we have a formal definition of infinity. Although, even an informal definition of infinity will show that the two represent the same number. As has already been pointed out, the difference between 1 and 0.999... is an infinite sequence of zeroes followed, according to some, by a 1. But if there are an infinite number of zeroes, then there cannot be a 1 at the end - wherever you attempt to append the 1, another zero can be inserted. Therefore there is no 1 and therefore the two numbers are equal.