.999 = 1?

Thanks yes, checking, any comments on Lorentz equation?

Cognitive Tide, let me introduce you to 7777777, you’ll love talking math with him. You have the same style.

Re Lorentz. I can’t see anything special about this, and why there should be some form of problem escapes me.

  1. Physical constants are not defined over an infinite number of decimal points. The limited age of the universe places an upper bound on the accuracy they can be defined with with any meaning.

  2. Anyway, you seem to be reaching for some number that has an infinite - non-repeating representation.

  3. Decimal representation is just that - a representation. Don’t forget it is shorthand for a series. And that is all it is. There is no magical specialness about a decimal representation for a number. It is just shorthand.

Sorry, I can’t parse this. Trying to break it down.

Of course it is possible to take the square root of an infinite precision real number. √(1/9) = ±1/3.
Neither have finite decimal representations.

You mean a value to multiply the length by? Why not?

Sorry, I simply don’t get this. Why doesn’t 1 exist?
Anyway…

You seem to be searching for the irrationals. No big problem here. Irrationals don’t have a finite decimal representation, and don’t have any sort of repeating structure. √2 is a favourite. It caused the Greeks a huge amount of angst. Something that I suspect is a the root (Ha!) of your complaint.

A rational minus an irrational yields an irrational. Sure. But so what?

As above - physical constants do not exist to infinite precision. Although a very useful start is to define c = 1. Which is precise, but by fiat. But that is 1.0000…

But anyway. There is no magic about decimal numbers. They are shorthand for a series. That is all. There are some cute properties - that rational numbers have either terminating (for those rationals that are built from the prime factors of the representing base) or infinitely repeating representations (for those that contain prime factors that are not factors of the base). Irrational numbers have no repeating pattern and go on forever. Almost all real numbers are not rationals. That is just life. We can take the square root of any real number. How you choose to try to write one down has no bearing on the problem.

“Sorry, I can’t parse this. Trying to break it down.”
What is the square root of .75287999… ?

(without substituting 1 in for .999…)
The point is when you substitute .999… into the Lorentz equation for 1 you are introducing
the notion of a velocity measurement that was made with infinite accuracy…this is not possible due to quantum mechanical limitations. The speed of light IS known with infinite accuracy because that is the way physicists have defined things subsequent to Einstein’s relativity work. Everything is referenced to the speed of light which is an infinite precision number. The integer one does not contain the concept of an infinitely precise measurement and it is clear that any finite precision measurement could be subtracted from it.
This is the essence of this entire discussion which is that the concepts of 1 and .999… are
very different and cannot be seen as equal when using the Lorentz equation.

Show that α is less than .01

81[.0111…]^2 < .01 ?

Subtract 1 from both sides
81[.0111…]^2 - 1 < .01 - 1 ?
Expand via binomial theorem

(9[.0111…] + 1)(9[.0111…] - 1) < .01 - 1 ?
Simplify binomial expansion
1.0999…(.0999… - 1) < .01 - 1
Simplify
(1.0999…)(.0999…) - 1.0999… < .01 - 1

(1.0999…)(.0999…) < .01 + .0999…
Divide through by .0999…
1.0999… < .01/.0999… + 1
1.0999… - .01/.0999… < 1
Multiply through by 100
100.999… - 1/.0999… < 100
90.999… < 100

Neither math nor physics works the way you think it does. Please stop. There are a dozen valid proofs that 0.999… = 1 in this one thread; pick one and try to find where it’s wrong.

“Please Stop”?

Regarding the way math and physics works the last time I checked there was a common approach that has served us well which is the scientific method.
And based on that method…nothing is ever proven, we only fail to disprove.
So the law of gravity is a law only because it has never been dis-proven once.
All it takes to disprove it is one person demonstrating that the equation doesn’t
have symmetry with reality via an experiment.

You cannot separate mathematics from physics. The integers were created for the purpose of identifying quantities of things in the real world…not abstractions. The real world comes first, the abstractions come later. You can dream up any system of understanding you want and it may be internally consistent but it would be pointless to discuss equations and operations because those things have direct symmetry with the real world.

So the “real” number system is used for applied math. The Lorentz equation is an example of that…furthermore all it takes is ONE example of how .999… = 1 leads to a contradiction in the system to take the whole thing down.

I fail to see why you are pleading with me to stop here…simply point out the errors as others have done and make your case unless you want the discussion to grind to a halt (why you would want this I know not).

Well, to start with:
(a) People above have already pointed out your arithmetic errors;
(b) The uncertainty principle does not somehow invalidate Lorentz invariance;
© You can certainly take the square root of any non-negative real number;
(d) If you want to argue that putting finite-accuracy numbers into infinite-accuracy equations is somehow impossible, you should probably explain how everything else about physics works;
(e) The imprecise value to which we’ve measured or know a value is not the same as the real value itself. Imprecisely-measured real numbers are still real numbers; we just can’t write down all the digits;
(f) Math is not an experimental science, and the idea of performing an experiment to determine whether 0.999… = 1 is patently ridiculous;
(g) The idea that 0.999… = 1 has nothing to do with physics. It’s just an immediate consequence of the very definition of real numbers, for the same reason that I can make the corresponding statements in the 10-adics, various contructions in non-standard analysis, etc.;
(h) You are not going to disprove relativity with your thought-experiments.

And so on.

“(f) Math is not an experimental science, and the idea of performing an experiment to determine whether 0.999… = 1 is patently ridiculous;”

“F”

Its not?

“patently ridiculous”
So let me see if I understand all of the wisdom you are sharing with me here.
Math is essentially just an art form. There are no constraints…AND it governs reality.
So I dream up some system via set theory and reality just follows along.

YOU decide how reality works and project your mathematical ideas onto reality and not
the other way around correct?

Math came first and then physics after right?

“Please”…seek help from the nearest High School science teacher at your earliest convenience…good night.

Oh well. I suppose I shouldn’t have bothered anyway; it’s not like page 32 of a thread on a fact of high-school level math with a one-line proof is a great battlefield in the fight against ignorance. “It’s taking longer than we thought,” indeed.

You are the one who is not leaving.

If you are so exasperated why not just get rid of me by pointing out the error in the alpha proof above. You would be following along a great tradition of people who believe in the iterative process of the scientific method whereby people correct one another in order to whittle things down to something which is very difficult to disprove.

When are the mods going to move this thread to Comedy Central … or BBQ Pit?

I thought 7777777 was the ultimate, but now:

:confused:

Yes.

Well for one thing look at the last 2 steps;

definitely something missing there.

1.0999… times 100 is 109.999…, not 100.999…

Right. I’m going to make two astonishing predictions here: (1) Pointing out the flaw above won’t convince you that you’re mistaken (and, indeed, nothing will); (2) You’re unfamiliar with mathematics and mathematical proof in general, and despite your talk about the scientific method, you’re not a scientist.

Yeah, the part Septimus just pointed out is complete nonsense as well.

No argument from me there at all.

That’s right. It’s a fact I wish more people would understand.

Also right. But we generally agree to apply certain constraints because they are useful. We call those axioms. An example would be the axioms that describe the real numbers.

No. Math is a tool for describing reality. This may sound pedantic, but it’s important. God did not invent math. Man invented math to try and make sense of the world.

I encourage you to think seriously about this idea. A lot of discussions about this topic (and other unintuitive results in mathematics) would be over a lot sooner if people stopped thinking mathematics was some kind of mystical knowledge passed down to us through the ages. Math is a hammer, used to sloppily bludgeon something that vaguely resembles reality.

Math in no way “governs” reality. Nothing in math governs the way the world works, and the way the world works doesn’t govern math.

What math is is a language. It’s a descriptive mechanism, like English, Japanese, or French. Where it differs is mainly that it has extra prescriptive requirements and baggage that English, Japanese, and French don’t. This makes it much less flexible, and removes nuance. But this terseness and precision allows it to do some things extremely well. It just so happens to be that one of the myriad things this makes it well suited for is expressing fundamental facts about our universe.

In fact, the feuds between pure/theoretical mathematicians and physicists/engineers over what the former consider flagrant abuses of notation is a well known joke. Some mathematics people with inclinations towards physics will practically bend over backwards to “prove” that what the physics people do “works” within the standard mathematical axioms of the spaces they’re working in (though it’s often far more complex than the simple intuition that led the physics people to perform the useful abuse of notation in the first place).

All this means is that math is a language, and slightly different variants are used to express different things. My mom is an accountant and they do some things with basic chains of arithmetic that clearly work, but in a very indirect and arcane way if you were to consider it from a purely mathematical standpoint. Other variants of mathematics are used to control computers, draw pretty pictures, recommend movies to you. All of these have their own weird soups of abuses of notation, weird constants (see fast inverse square root, also an example of “pure mathematicians bending over backwards to prove it” as mentioned above), descriptive crutches, and idiosyncracies.

And sometimes math isn’t even the best tool. Some (most) real world systems are so complex that we can’t, tractably, describe them mathematically. So we talk about them in English, Japanese, or French, or make extremely simplifying assumptions so we can describe them mathematically. Even then, we usually write some sort of translator in the form of a computer program to help simplify the math into a more succinct form we can understand. It’s fundamentally no different than asking a Japanese translator to tell you what Tanaka-san is saying.

.999… = 1 is a truth in the Real number system. Whether such a process that would cause the limit .999… to really be evaluated with physical objects and be subject to experimentation is immaterial. Consider a quirk of the language, if you must. Like collective nouns in English or gendered nouns in German or French.

It’s not arbitrary, though. The point of math is that you can set your own rules, but you have to play by them once you set them. Mathematicians have defined a real number by the constructions mentioned in this thread: Dedekind cuts, Cauchy sequences modulo equivalence, and so on. Once that definition has been set, answers about the real numbers like, “Is 0.999… = 1?” are set in stone.

It’s nice when math can solve problems in science, but that’s not necessarily a point or requirement. (Besides, real numbers are so basic that they can’t be classified as pure or applied math.) There’s no requirement that real numbers model any real-world phenomenon. Real numbers happen to be useful in physics and elsewhere, but a lot of perfectly valid math isn’t, or at least isn’t currently and wasn’t developed for that purpose. My own background is in a field of theoretical math that has no practical application as far as I know, except possibly for some speculative bits in esoteric quantum field theory. But so what? It wasn’t developed to model reality, it’s not connected to any real-world phenomenon, and I don’t care about any practical applications it might have.

I think this might be a bit naive, or at least taking the idea too far. It’s relatively clear that math developed in lockstep with a couple of things: a need to describe reality (mostly for engineering, architecture, ballistics etc) and counting money. I think it’s a bit naive to state that the real number system “just so happens” to describe physics. That is in no way an accident, it was more or less engineered to do so. In fact, the axiomatic construction of the reals using the method of your choice is, more or less, a post-hoc formalization of the less rigorous mathematics used up to that point. Certainly there were “pure” math advancements mixed up in there, but even the likes of Pythagoras was busy arguing whether or not irrationals “existed”, a statement about objective reality if I ever heard one.

.999… = 1 is a quirk. Yes, you have to play by the rules, and that’s why we say that the quirk is, well, a fact, but let’s not act like it’s a magic coincidence that reals, vectors, complex numbers, and calculus just so happen to be good at describing physics. That shies away from a very clear history of the relationship between the advancement of math and physics. Sure, modern theoretical math often develops in a way with no (clear) application to reality, but I wouldn’t consider that the norm until fairly recently.