Welp, I read the whole thing. It’s about like the previous conversation in this thread–the author has intuitions about quantity which could be made into a model, maybe even an interesting one, but the author doesn’t understand what it is that recommends the real number model for mainstream mathematical practice, and the author is confusing a discussion about a system of representation with a discussion about the things represented.
Same ol’…
And no one’s called me ‘grasshopper’ since I was knee high to a . . . pupa.
I’m amused to see this convo still going on.
Why is there confusion about .999… = 1, but agreement that .3333 = 1/3 ?
I know. I am not a math genius but the matter was settled to my complete satisfaction the first time someone pointed that out.
1/3 = .333333…
.333333… x 3 = ?
Well there you go.
I think that the problem is that there is only one decimal representation of 1/3, but there are two of 1. (I.e., 0.999… = 1.000… = 1)
Is there agreement? I thought the people who disputed the one would also dispute the other. They’d say the same thing: “It can get as close as you want, but will never actually equal it.”
Well they would, if they had thought it through. Then again, if they had thought it through, they wouldn’t say that about either.
That is what they say. I’ve never seen someone who affirms the .333… equation and denies the .999… one.
Seems to me there are an infinite number of irrational values between any two integers (say, zero and one, for example). By contrast, with rational numbers, there are only an infinite number of those between any two integers (say, 41 and 17,159,327). The repeating-9s fraction is merely the number that is infinitely close to its neighboring integer, but it is still not an integer. For practical purposes, it might as well be equivalent to that integer, but it is still not actually equal to it.
This thread is long but interesting and I recommend you read it. It covers your argument in detail.
The short version is: You’re suggesting a model for something other than the real numbers. Your model might be of an interesting structure, but this does nothing to in any way contradict or counter the standard interpretation of decimal notation as representing the real number line.
Consider this thread, for example, as a countably infinite sequence of posts (analagous to 0.99999…), but having only a finite length at any particular point in time. Let the completeness of this thread be designated by a real number in the range from 0 (that is, 0% complete) to 1 (meaning 100% complete).
Once the topic is discussed to death and beyond, it’s completeness resembles something like 0.9999… (for some finite number of digits). Each discrete post adds another 9 to the sequence, which thus converges upon completeness = 1, but is always some infinitesimal distance short of that. This post itself, for example, increases the completeness of this discussion from 0.99999…9 to 0.99999…99
By implementing an auto-zombie-resurrector event handling function and attaching it to this thread, we can have zombie-resurrection events queued automatically on some scheduled basis, which I suspect is what is happening here. Thus, this thread can grow infinitely in completeness of its own volition, asymptotically approaching complete completeness, but still always falling infinitesimally short.
According to ancient Hindu prophecy, when this thread reaches the state of perfect completion, then all posters, computers, the SDMB message board (indeed, ALL message boards), and infinitesimal numbers will all vanish, and in a great thunderclap the universe will end.
You certainly speak with the confidence/arrogance of someone who comes across as if he feels that paradoxes are merely something that minds weaker than your own are susceptible to.
I did not state that the website discussed this topic in depth in terms of mathematical rigor. My point, was that it does discuss it in depth but from the approach of understanding what cannot be understood through rigor.
It is abundantly obvious that:
“You can, as all the real mathematicians here have said, design a mathematics that uses infinitesimals and defines 1/infinity.”
Every high school calculus student knows that…it doesn’t take a “real mathematician” as you put it (whatever Ivory tower connotation that carries with it)
Your major disagreement appears to be:
“That’s not what he says.”
You need to cite specifically where it is that the author states that you cannot design a mathematical system that uses 1/∞. His point, as I read it, is that the Real number system, as defined, does not allow for inclusion of Newton’s original definition of the infinitesimal which is 1/∞, which is why the so called real number .999… is equal to the real number 1. That is, because the necessary term to complete the equation is not an element of the set.
“the way infinity is normally defined and used accords perfectly with the real world”
Oh does it? If that is the case why would there be the need for things like Quantum Calculus? What is the most likely explanation for resolving Zeno’s paradox? I suggest you cite some examples of this “perfect accordance” by first of all naming just one single example of something that can be pointed to and deemed as a real world example of infinity. One can easily do this for integers.
As far as we know infinity is nothing more than a concept which happens to have use in mathematical systems but has absolutely no direct correspondence with reality. Much in the same way complex numbers or the concept happy have no direct real world symmetry.
Finally, you seem to have a problem with the notion of a number “at infinity”
The author is not presenting this as a mathematically rigorous concept, but rather a label for what can be seen and understood intuitively but not proved rigorously.
Seeing and understanding what happens “at infinity” is what calculus is all about.
The fact that the best we can do rigorously is say that a limit exists and that a convergent sequence never traverses that limit does not preclude us from discussing and attempting to see what happens at the limit with whatever means we have available whether they be mathematical rigor, our understanding of physical reality, or our interpretation of the last episode of our favorite TV show.
You made little mention of the fact that the long division of 1/3 when done using the correct method and when including the remainder always yields the correct answer. If, as far as can be shown, the correct answer always requires the addition of a remainder, why should we expect that it would ever not require the addition of a remainder to yield the correct answer?
No, they don’t. I’m pretty sure he was talking about something like Abraham Robinson’s Non-standard Analysis, which most high-school calculus students are unaware of.
The trouble is, Newton (and Leibniz, and mathematicians of the early days of calculus) didn’t have a definition of the infinitessimal—at least not one which made sense and was rigorous enough to withstand criticism from people like George Berkeley—which the linked article actually reminded me a bit of. Newton also didn’t have a rigorous definition of the real number system—that came hundreds of years later, thanks to mathematicians such as Dedekind.
Also later came the more precise formulation of the ideas behind calculus as it is normally taught today (to high school and undergraduate college students), in terms of limits.
Every high school student “knows” that, but the intuitive view of infinitesimals taken by students at that level inevitably leads to trouble. There’s a reason nearly 3 centuries passed between Newton/Leibniz and the formalization of non-standard analysis.
Infinitesimals were dropped nearly the moment that limits appeared, since limits could rigorously show proofs that mathematicians had been hand-waving for the previous century and a half. If the infinitesimal approach were a good one for use as a teaching tool, I can assure you it would be regularly used.
Yep. The phrase is nonsensical. You don’t get “to” infinity. You keep moving down the road, long as long can be, but you don’t ever “get there.”
Intuitive mathematics is a field of psychology, not of mathematics. If you can’t prove it rigorously, it isn’t math.
Nope. Calculus is all about showing that differences can be made arbitrarily small. The concept of the limit has nothing to do with the behavior of a function “at infinity,” but, rather, in any arbitrarily small neighborhood nearby a limit.
So what would be bad about using Robinson’s hyperreals to teach calculus? Be specific, please.
Valmont314, can we assume that you are the author of the page you linked to?
It’s baaaack. The thread that would’t die.
It makes perfect sense if you understand the concept.
I like the example in Valmont’s link of taking the decimal representation of 1/3 and multiplying it by 3. I didn’t read it just skimmed it a bit.
No one would dare question the fact that 3 times 1/3 is 1. And yet, by some strange magic, if you convert 1/3 to .333… and multiply by 3 to get .999… somehow that can NEVER be one. WITCHES! We must burn them! :smack:
Technically, .999… = 1/1. 
Infinity: the journey is the destination.
We need a huge sign that pops up whenever anyone tries to post here that reads:
Have you read the entire thread first?
It isn’t as if this hasn’t all been beaten into the ground already.