Actually it will be more complicated than I stated, because you will have rules that allows you to end up at “.9~”, which, left as is, would be just as bad as starting from there.
“Math is hard” said Barbie.
Then explain all the new rules in your system, TATG. Mathematicians have built up a complete set of rules. You’re trying to add in a new one which isn’t consistent with the ones that exist. When shown an inconsistency, you say that you could just add some other new rule to make it consistent. Show us these rules. All of them. Otherwise, you got nothing.
That’s a good try.
But what are you basing your equation 10 x 1/∞ = 10/∞ on?
If you look at the decimal representation of 1/∞ as .000…1 you can clearly see that multiplying through by 10 (or any power of 10) has no effect on .000…1 much as adding more 9s on to the end of .999.. has no effect to it.
Try subtracting your equations again and you will see that you arrive at a consistent result.
Understanding the relationship between finite quantities and those that involve infinity, infinite precision, and infinitesimal amounts forces you to abandon rigor and think intuitively.
There are meaningful relationships involving infinity that we can express in arithmetic terms and say that we can demonstrate. But we can never achieve exact results in the finite world.
For example, there is demonstrable evidence that any finite quantity, when added to infinity results in infinity. The concept of an ideal heat sink can be demonstrated very easily…simply get a boiling cup of water and a thermometer, head out to the beach, dump it in, and see how much the temperature changes, try the same experiment at home on a bucket of water and you will be convinced that there is meaning to the notion that 2 + ∞ = ∞. One can still see the symmetry between reality and the conceptual.
Regarding you comments about intuition and “calculating” limits. Can you calculate a limit on a calculator? We do not “calculate” limits. We evaluate them. We can do things to change the way we see the patterns that are involved and make calculations to help us see how the pattern is evolving but at the end of the day we still, at some point, have to step back, and rely on intuition to
see the answer.
About patterns, it is precisely seeing the behavior of patterns that is intimately tied to our ability to evaluate limits and say what happens at infinity.
And again, when we say that even after an infinite number of terms the sequence .999… can never be greater than one we are make an assessment of what happens at infinity. Our ability to do so is entirely dependent on our ability to see and understand the pattern. We do not “calculate” our identification and understanding of a pattern…we either see it or we don’t (and if we don’t see it, we hope that someone else does and points it out to us at which point we accept it).
1/2 = .49999…?
Are you simply decreeing that this is the case or are you going to show us how you arrived at this equivalency? How did you arrive at this conclusion, what method did you use to establish the equivalency?
The same thing applies for your base 3 conversions.
Either show us the long division process that you employed to go from fractional to decimal values or show us the conversion algorithm that you used to convert decimal values from base 10 to base 3.
For what it’s worth, I’m pretty sure the original throwaway comment of TATG’s which started this whole weird debate was essentially a joke (and a funny one, too). But they haven’t said anything wrong in defending it either… you could make some weird system where the particular notation “0.9999…” was treated as whatever kind of ad hoc special case you like, signifying whatever particular random thing you like. It wouldn’t be very natural, but nothing’s stopping you.
Also, since it’s been a while, let us continue to flash the sign periodically for any newbies thinking of jumping in:
Have you read the entire thread first?
What I said borders on being trivially true. To represent the reals you can use decimal expansions. For each real, there is at least one decimal expansion; but there are more decimal expansions than real numbers. Ergo, some decimal expansions represent the same number. So say we have equivalence classes of real numbers, and functions from these equivalence classes, where, for example, +(1,2) is a function from the equivalence class of expansions representing 1, to those representing 2, where the output is the equivalence class representing 3. Moving one decimal expansion from the equivalence class representing 1 will not break anything, because we have other representations.
:smack: :smack:
Are you seriously telling us that you don’t know how to represent fractions as decimal expansions? Or that you can do that, but ternary expansions pose a difficulty?
Some argue, via some quaint metaphysics, that 0.49999… is not the same as 0.5, but even they do understand, whether they fully accept it or not, the trivial argument that they are equivalent. And you?
QFT.
Don’t fall for it, septimus! Don’t take that bait! I see where he wants to lead you.
He is challenging you to show, IN FULL, the conversion! Can you show, in any post of finite length, the FULL long division, out to the very last digit? The very last ∞’th digit? Oh, you can’t, eh? You’re just going to show enough to demonstrate the idea, or show the repeating pattern? And then stop after 5 or 6 digits? And leave the rest as an “exercise for the reader”? :smack: You can’t even show us, say, half way out to the ∞’th digit? Maybe you can at least get (1/∞)'th of the way?
Don’t even try to go there, septimus. In that way lies madness!
HAHAHAHAHAHA!
Senegoid…I want you to think about a recursively defined algorithm. Now relax, breath deep.
You can feel your eyes slllllloooowly closing.
Wait <insert violins from Psycho>…stop…Senegoid, you are falling…noooooooo!
It’s the infinite regress!
Somebody, please, help Senegoid!
Oooooops…he’s gone <insert Chamber music>
He is now drifting off into recursive oblivion.
Stay tuned for next weeks episode of “Help me…I’m stuck in a Hofstadter Mobius Loop and I can’t out of it!”
Dont worry septimus, I haven’t forgetten about you <insert diabolical laughter>
Reply coming shortly (I have other victims to tend to you know)
erik150x, Valmont314, and TATG, let me repeat what I said back in post #657
> [Y]ou have failed to convince us. Even if you did convince us, what do you think
> we could do? Do you think that we’re the secret council of mathematicians who
> control the world of mathematical definitions? We control nothing. If you want
> to convince the world of your theories, get a Ph.D. in math and become a
> big-name mathematician. Publish your theories. Maybe then you could convince
> the mathematical community. Or if you want to bypass the mathematical
> community, publish your theories in ordinary books. Maybe you can convince the
> people of the world of your theories and ordinary people will unite to kill all
> mathematicians because they refuse to accept your theories.
Nothing in your posts has been convincing to the mathematicians who have spent an enormous total amount of time trying to explain to you why your ideas won’t work. When we point out mistakes, you make up some new rule out of nowhere that isn’t any more consistent than the ones you’ve already explained. There’s no point in you wasting any more of your time and ours.
And it is wasting our time. Do you honestly think that we really get a big thrill from having people ignore our careful attempts to explain your mistakes? Do you get a big thrill from this thread because you enjoy proving that we’re just too stupid to understand your ideas? Look, either you’re so vastly smarter than us that we will never figure out what you’re talking about, or you’re wrong in your claims. Try convincing someone else rather than us.
How 'bout if we guarantee we will believe you AT INFINITY?
Nothing I have said should be controversial. You can use the symbol “1” to mean 2, or “2” to mean 1; if you balk at that, you are confusing your notation of the real numbers with the structure, and doing exactly what erik and valmont have done.
I really don’t see why people are having so much trouble with this.
Assume we have, >(x,y), =(x,y), +(x,y), .(x,y), defined, and that our structure is the real numbers. Assume our relations are defined on equivalences classes (as will be the case if we use decimal expansions, because we have redundancies). Assume x1,y2 are two equivalence classes, now take the members, shuffle them up, put them back in x1,y2 as you please, but ensure that each has at least one member. Call the new classes x1’,y2’. Define your new relations such that they are as above, except where the class x1’ is substituted for x1, and likewise for y1.
At the end of this you still have exactly the same structure. Moreover, that structure is the reals. And if you disagree with this, then you are the one engaging in revision of maths.
Deal, the .999… cent beers are on me. 
I don’t know. But let me try my hand at saying what you’ve been saying, in the hopes that it clears it up for others:
You could set up a notation system where all decimal-strings were interpreted the same way they standardly are, except for “0.9999…”, which would, by ad hoc fiat, be interpreted as the number 2. This wouldn’t be a very natural or useful notation system (far from it!), and its bizarre definition would make it a pain to work with, but it would be perfectly legitimate, and describe the same arithmetic structure (the real numbers) as the conventional interpretation of decimal notation does, with each real number continuing to have at least one representation (and some having two, and the number 2 having three representations).
Where are you going that beer costs a penny?
Oh. Oh no. No no no no no. Don’t tell me. It all makes sense now.
If you pay for a $0.999… beer with a $1 bill, do you get $0.000…1 in change?
How much would two such beers cost? (You’ll have one for yourself too, no?)
Well. Clearly, 2 * 0.999… = 1.999…
Oh wait a minute. I’m thinking 2 * 0.999… = 0.999…8
Pay for that with a $2 bill and your change should be 0.000…2
Valmont314, I think I want to change my user name to Senegoid271828.
Then (me)[sup]i(you)[/sup] = -0.999… (within limits, of course, to the nearest ∞ decimal places).
.9999… bottles of beers on the wall, .9999… bottles of beer, take one down and pass it around, um . . . what would be the next line?
Next line!
0 bottles of beers on the wall, 0 bottles of beer, take none down and don’t pass it around…