The phrase “for practical purposes” doesn’t have any definite meaning here, but that doesn’t matter. What those who understand algebra and calculus are saying in this thread is that (given the normal definitions) .999… and 1 are exactly equal: there is no difference in the value, despite the different representation. They are equal in the same way that 1+1 and 2 are exactly equal.
Not really. In fact just plain no. Decimal representation, or representation in any base is simply a notation for a mathematical expression. For numbers that require an infinite number of digits, the notation represents in infinite series. I would have hoped that anyone reading this thread would have grasped that point.
0.999… is shorthand notation for the expression 0 + 9 * 10[sup]-1[/sup] + 9 * 10[sup]-2[/sup] + 9 * 10[sup]-3[/sup] + … + 9 * 10 [sup]-n[/sup] + …
The question is and has always been what the values of these (possibly infinite) series are. To reverse the process and somehow restrict the discussion to series that have certain patterns or other properties in the coefficients (such as zero past a certain point) makes no useful sense. We can switch the base, and that will change the patterns, infinite series in some bases will become finite in others and vice versa. So what? The actual real number represented remains the same. Heck we can use a non-integer as the base. None of it changes the useful mathematics.
In fact, you can think of .9999… as just being an infinite geometric series and sum it like other infinite geometric series:
Finding the value of .9999… is thus just calculating the summation of n from 1 to infinity of 9 times (1/(10**n)). You can see from the formula given in that Wikipedia entry that the sum from 0 to infinity of that series (i.e., the value of 9.9999…) is 9 times (1/(1-(1/10))), which is 9 * (1/(9/10)), which is 10. We want the value of the sum from 1 to infinity though, so we subtract the 0-th term, which is 9, and we get 1. In the same way, if you use base 8, the summation of .7777… is 7 times (1/(1-(1/8))), which is 7 * (1/(7/8)), which is 8, and subtracting the 0-th term, which is 7, we get 1. In base 2, the summation of .1111… is 1 times (1/(1-(1/2))), which is 2, and subtracting the 0-th term gives us 1.
Well, let’s recap.
You came into this thread with this quote:
That was an amateur way of saying that you wanted us to read something you had written and tell you how wonderful it was. Unfortunately, when we did read it we found errors at the conceptual level and in particular details. Indistinguishable, a real, working mathematician, found it to be crankery. As a whole you could say that every single sentence in it was incorrect.
However, you continued to repeat those errors here, even though you were told that reading the thread would point out that what you wrote was a mash-up of very common errors made by countless others who thought that they could decide for themselves what pieces of math were acceptable and which weren’t, just because they didn’t understand them.
You have not varied from that original set of errors, but you have along the way managed to make a new error with every statement about math. You have demonstrated literally no knowledge or understanding of any formal mathematics or concept or procedure above the level of arithmetic. A large number of people have found these mistakes and attempted to correct you without you acknowledging error. I find it fascinating that your irrationality on this subject parallels to a startling degree what I wrote about conspiracy theorists in another thread. While correcting your individual errors is important, the bigger issue is getting you to realize that your larger world view is so skewed and flawed that nothing you say can possibly make any impression on anyone with every basic comprehension of the subject. That’s an infinitely more difficult task, pun intended, and therefore probably impossible.
So, all in all, I have to say that I’m really not the problem here.
I just reviewed some ancient history.
I see that mandrake started this thread with his OP in July 2000, and that this thread subsequently seemed to have died a well-deserved death in less than a month, after only 15 posts . . .
. . . only to be exhumed this year (2012) by erik150x at post #16. (I myself entered the swamp at post #37.)
I don’t see any mention anywhere that Cecil Himself took up the matter on July 11, 2003. Having just discovered that fact, I am of course morally obliged to mention it here.
What erik150x and Valmont314, et several al (as relative newcomers here) need to understand is this: If Cecil settled it, that settles it!
Even Wikipedia has a page devoted to just this question – including sections discussing the history of the question; various genres of proofs and/or definitions; and the psychological questions about why some math students have such a hard time wrapping their heads around it.
That Wiki even includes a sub-section titled “In popular culture” which refers to Cecil’s discussion (which, in fact, is where I just discovered it).
Those of us who tried to convince erik150x of something-or-other might do well to study that, and try to understand the psychological or pedagogical barriers he may have had that gave him such a hard time with it. Maybe we weren’t addressing the blocks in the best way.
Valmont314 is a different case. We, lacking pit bulls, venemous snakes, and vice grips, just aren’t equipped for the challenge. In this case, as several frustrated respondents above seem to be discovering, aren’t we all just wasting electrons here?
I have some frame-making clamps I could lend, if it’d do any good…
The actual value of these threads is at least two-fold: it is educational to those who may read it and be deterred from error. It’s a kind of prophylaxis against ignorance. And…I’ve learned stuff from it, and I know I’m far from alone. These threads, tedious though they may be, really are educational.
Thanks Sen. I love the part on Zeno’s paradox as it’s reminiscent of my failed attempt to introduce Eric to the concept of an asymptote and thus recast the problem. Unfortunately Eric saw a great deal more potential in it than I did - as many valiant posters will attest.
(I’m not the one saying “go,go,go” here)
If you scroll up you will see both my response to you in regards to the topic of this thread (which you did not address in your reply) and my reply to you in regards to continuing this discussion.
LOL
Seriously, are we talking about Geopolitics here or math? What does my larger world view (which I have not so much as even referenced in the most indirect way) have one iota to do with anything? How could you possibly know my world view or anything about me based on a handful of posts in this thread? You seem more obsessed with trying to formulate subtle ad hominem attacks rather than formulating explanations that will help people understand what is being discussed in this thread. Your tone comes across as that of a Bully who seeks to convey their point via intimidation which seems rather odd to me for a discussion of tricky mathematical nuances. Of course with nearly 20,000 posts to my handful I doubt that I am equipped to match you in your skill (and available free time) in the art of internet message board debating.
You are not? The above statement conveys that you exactly are the problem because you just stated that an impossible task is possible. An impossible real world task (yes my world view is one of the real world) is by definition one that requires an infinite number of steps and is therefore exactly impossible. Not probably impossible. There is no probability any more than there is probability of a die coming up 1 if there is no die. You have just contradicted yourself in the most fundamental way that a person can contradict themselves.
I have already posted what is for me the logical terminus in my mind of the topic of this thread. That terminus involves a conclusion and some assertions.
If you would like to refute them via citing an example that contradicts the assertions please do so. Otherwise my only response to the topic of this thread will simply be this:
Since nobody has done so, I will take the ellipsis to be defined as the logical terminus of this thread which is equivalent to the assertions I have made above and will simply respond with an ellipsis whenever someone fails to cite an example which refutes the assertions I have made. So here is my first (and what I hope will be the last) reference to my assertions with the ellipsis:
…
Silence!
Ahh sweet silence.
Well, since nobody has stepped up to the plate to try to strike down the ellipsis all that is needed is a conceptual terminus.
What is needed is a conceptual definition of 1/∞.
As the Scarecrow (who is still working towards his Diploma) I henceforth degree it that:
1/∞ is the symbolic representation of an infinite series of recursive Necker cubes. That is, it is the symbolic representation of the recursively iterated representation of fundamental self reference among the Integers (which is one) which also conveys the uncertainty in measurement via the concept of irreducible interval.
Night All
Your lovely envoi on infinity reminded me of some of Leo Bloom’s (the original) thoughts upon seeing a constellation in the night sky:
His (Bloom’s) logical conclusion, having weighed the matter and allowing for possible error?
That it was not a heaventree, not a heavengrot, not a heavenbeast, not a heavenman. That it was a Utopia, there being no known method from the known to the unknown: an infinity, renderable equally finite by the suppositions probable apposition of one or more bodies equally of the same and of different magnitudes: a mobility of illusory forms immobilised in space, remobilised in air: a past which possibly had ceased to exist as a present before its future spectators had entered actual present existence.
Thank you, whoever you are, for the missing piece of the puzzle, an example of self-reference coupled with self-expression. I believe that no computer will ever be able to speak poetry where we can’t eventually (and probably very quickly) tell that the inflection in the words was programmed in. The operating system would require an infinite number of lines of code to prevent this and even then it may still be incomplete. We are not the body, and we are not the cognitive mind.
I am <=> I feel
I wanted the thread to end at post #913.
#999 would be so much neater, but #914 was certainly where it went off the rails again.
999 messages is an attainable goal without reopening the discussion (until someone else needing some vectoring arrives)
All that is required is shifting the discussion from what you think about ∞ to what you feel when you think about ∞. Is infinity just another cognitive association no different from potato or widget or is it so personal that it should never be discussed in the way we discuss our feelings about our favorite painting or piece of music?
If anyone cares I will offer up a fragment of a memory of something I read which I cannot place but I know was real:
It was an 18th or 19th century description of Newton’s Calculus as:
“God working through the hand of the infinitesimal”
The Scarecrow (who moonlights as a telemarketer) has taken position in the field and is
trying to convince the crows that there is nothing to be afraid of.
Frylock slowly backs away from Valmont.
The scarecrow has little patience for those that cannot see the difference between the symbol and the meaning behind the symbol. You have not passed the test.
The Scarecrow observes Frylock, acting on learned behavior, reinforced by rigor, flying off into the distance asymptotically approaching the vanishing point.
I no longer have any idea whether you think you’re making serious points anymore, but in case you do, it should be noted that the distinction between symbol and meaning is something I have a strong appreciation for. If you think you’ve got a reason to think otherwise, there’s something wrong with your diagnostic criteria.
Are you a crow or not?
If not, I’m not sure why you were backing away…what were you doing out in a cornfield and are you really afraid of harmless scarecrows?
And please, no more logic, you yourself said that you wished that part of this discussion was over up above, do you really think you can distill your feelings down to logic? At least answer that question because that’s all I am discussing on this thread from here on out. (I will leave it up to the heavy hitters here to bark out the vectors). If you want to veer off into things like collective conscious, pattern recognition, synchronized associations we could do so but it would be off topic here.