My initial argument was explicitly (thought perhaps not clearly described) a proof by contradiction, in which I start by assuming that E can be treated like an algebraic variable (in which case 10E - E = 9), then end up on a nonsensical or contradictory result, proving the assumption is wrong.
The gist is that the moment the expression 10E appeared, the logic swerved into the twilight zone and all bets were off.
But I don’t see how you obtained a contradictory result. I only see how you obtained E = -1.
To reach a contradiction, you claimed also to derive E = +1, from 10 E - E = +9, but I couldn’t follow how you derived 10E - E = +9 in the first place. You said this was “clear”, but it’s not clear to me.
It seemed you said it was clear because you thought this was what you would get by digit-by-digit subtraction, but when I carry out the digit-by-digit subtraction, I get a -9 in the units place (where 10E has the digit 0 and E has the digit 9) and zero contribution from every other place (where 10E has the digit 9 and E has the digit 9). This comes out to -9 overall, which is no contradiction.
We end up deriving 10E - E = -9 by digit-by-digit subtraction. How do you derive 10E - E = +9 instead?
Dang it - I can’t believe I didn’t catch that and I did proofread it!
Anyhow, thanks to you and all the other learned folks for sharing your knowledge. I might not understand some of the really high end stuff you all get into, but I do try.
For example, last night I composed a post asking for your clarification on why you linked to the p-adic wiki article when talking about 10-adic numbers since 10 is not prime.
But before I pulled the trigger on that I thought I would google it to make sure I wouldn’t be asking a stupid question. Needless to say, since I didn’t post my question, I figured out that what you had written was valid, even if I didn’t really understand the math.
Oh damn - time to bow out of this thread for now on a high note and go grab a beer!
If it helps, Bryan Ekers, suppose I’m not arguing against you. Suppose I actually really like the idea of your proof by contradiction, and I want to use it to convince other people in just the same way that you intended it. But one of the people I show it to stops me at the “10E - E = 9, clearly.” step and says “Wait, I’m not following. How did you get that?”.
What should I say to them?
All I can say at this point is… sure, whatever. The explanation I would give would just be a repeat of the explanation I’ve already given. I will happily recognize the existence of ways to interpret the issue that involve mathematical approaches with which I am not currently familiar.
I would suggest something lacking in exclamation points and invocations of children making arithmetic errors.
<Opens file labelled 7777777
<Adds footnote “Bryan Ekers, possible accomplice.”
Having walked away for a few minutes to be distracted by another problem, I realized my error regarding this one and will bow out of the thread.
Ah, yeah, I thought someone might ask this. I see that you’ve resolved the matter for yourself, but for anyone else wondering: The p-adic numbers make sense for any base p, even when p is not prime. However, people often only consider the case where p is prime, because the resulting system has particularly nice properties in that case.
Specifically, when p is prime, the resulting system (by which I mean, base p numbers where we allow digits to extend infinitely to the left but only finitely to the right) is what’s called a “field”: it has the property that you can divide by any nonzero value. As a corollary to this, a product will never be zero unless one of its factors was zero.
When p is not prime, the p-adics will not comprise a field. For example, in the 10-adics, …87109376 * …87109375 = 0, even though neither factor is zero.
In fact, the structure of the 10-adics is essentially this: a 10-adic number amounts to a 2-adic number along with a 5-adic number (this amounts to the observation that knowing the last [however many] digits of an ordinary number in base 5 and its last digits in base 2 lets you determine its last digits in base 10 and vice versa).
The situations in which a 10-adic number is nonzero but can’t be divided by are when its 2-adic component is zero but its 5-adic component isn’t, or vice versa.
Similarly, the situations in which a product of 10-adic numbers is zero even though neither individually is are those in which one factor has a zero 2-adic component and a nonzero 5-adic component, and the other has a nonzero 2-adic component and a zero 5-adic component. (Like our previous example: …87109376 is …0000 in base 2, and …0001 in base 5, while …87109375 is …1111 in base 2 and …0000 in base 5)
So, the 10-adics aren’t a field, but they’re very close to one. They’re what some people call a “meadow”, which is to say, a structure which would be a field, except its “internal logic” corresponds to a Boolean algebra which may a number of elements other than 2. [In the case of the 10-adics, we can think of the answer to a question like “Are you zero?” as fundamentally given by two Booleans instead of one: one for the 2-adic component, and one for the 5-adic component. This means there are 2^2 = 4 such truth-values possible.]
In general, an n-adic number is given by a combination of p-adic numbers, for each prime p dividing n, so understanding the structure of the prime p-adics is enough for us to understand the general n-adics. [This is just like how positive integers are are uniquely built up as products of powers of primes, so for many problems, understanding the behavior of the prime powers is often sufficient to understand the behavior of positive integers in general]. That may be another reason p-adics gets so much more attention for the case when p is prime.
Missing word reinstated in bold.
that math looks fuckin nuts
Who’s doing this math? Who are you people, how can you follow that shit? Are you guys professors or something? I love it, the wikipedia article’s got a weirdo freaking color graph of weird colorful shapes. Reminds me of the paintings by the autistic guy who “sees” numbers as shapes and colors
What Wikipedia article are you talking about? Yes, some of us have Ph.D.'s in math. Some, like me, have a master’s degree in it.
Ok, I could:
1/2.9 ≈0.3
1/2.99 ≈0.33
1/2.999 ≈0.333
1/2.9999 ≈0.3333
1/2.99999 ≈0.33333
1/2.999999 ≈0.333333
1/2.9999999 ≈0.3333333
.
.
.
1/2.999999999… ≈0.33333333333…
Assume next that 3=2.9999999…
so that 1/3 = 1/2.999999… ≈ 0.333333333…
because of the assumption that 3=2.9999999…
The result, 1 divided by 3:
1/3 ≈ 0.333333333…
Conclusions and final remarks: this is a proof by assumption. It is assumed that 3=2.999999… is true. Similarly one can assume that 1=0.9999999…
As a newbie, you may not be aware that this site has a lot of folks with advanced degrees in almost any field you would care to name, including not only mathematicians but also (literally) rocket scientists and brain surgeons.
This is particularly clear in the construction of p-adics (p-adic integers, at least) as the inverse limit of the Z/p^n Z, with the Chinese remainder theorem.
Yup; that construction also makes it particularly clear why the p-adic rationals comprise a field for prime p: each nonzero p-adic rational is a power of p times a p-adic integer ending in a nonzero digit. So it suffices to establish that p is invertible (which is trivial) and p-adic integers ending in nonzero digits are invertible (for which, by your indicated construction, we just need to establish the corresponding fact in Z/p^n Z, where it follows from the primality of p by Bezout’s lemma).
I don’t think this is really as ‘proven’ a theory as you think, but suppose for a minute it’s true. The problem is then your inference that this ‘innate math’ is somehow ‘more right’ than the math that seems too abstract to you.
Consider the case of physics. There, a good case can be made that we have an ‘innate physics’, a kind of cartoonish, Aristotelian everyday-physics in which things need a constant force to take going, there is an absolute notion of rest and simultaneity, objects with a different weight fall at different speeds, and so on. This is a good approximation for many everyday purposes, but in general, it’s simply wrong, and the abstract physics in which two observers moving relatively to one another see each others clock going slow, etc., is right.
That’s not to say math involving things like Ramanjuan summation and other nonintuitive notions is ‘more right’ than more intuitive notions, as long as the latter are consistent, but it shows that relying on intuition can be severely misleading. Just because it appeals to our intuitions does not mean it should be considered more plausible.
I wouldn’t call it definitely ‘proven’. No one is arguing that there isn’t some inherent biological requirement to language (as opposed to mere communication) that humans have and other animasl lack; nor is anyone arguing that there are linguistic universals. It’s a nontrivial jump from that, though, to some sort of language acquisition device and some sort of universal grammar. Besides, if you want to posit a universal grammar, you have to give some sort of details to make it falsifiable (or proven). Chomsky proposed a couple of possibilities (X-bar and government and binding are the ones I’m familiar with), but none has been rigorously proven. It’s a convenient framework, and thus most basic ideas are uncontroversial, but it’s hardly conclusive. (Er, the link goes to a webcomic with profanity in it. I don’t think it comes close to the two-click rule, but keep it in mind if you’re at work and are bothered by such things.)
More to the point, what does this have to do with math? Mathematicians are certainly capable of handling math that isn’t rooted in physical intuition, and physicists have been amazingly successful with relativistic and quantum theories. What does biological hard-wiring about counting, if it exists as you describe, have to do with being unable to accept proofs that 0.999… = 1?
Do you know how long division works?
EdwinAmi, there are a considerable number of linguists who think Chomsky’s theory of an innate special ability in humans for language is wrong:
They believe that language learning in humans comes from general mental abilities in humans that can be applied to all sorts of different mental processes that humans have. They don’t believe that there is some structure in the human brain that evolved simply for language. There is a lot of controversy about how close the “language” that animals use comes to human language. How much have you read about animal language?:
In various animals there are uses of structures similar to human language. It’s possible that humans simply have a little more general mental ability than any other animals and thus are able to use language better than animals. Similarly, there is no evidence that there is any special mental structure in humans that enables us to do counting. As I said, some languages and cultures don’t find it necessary to count beyond three. Apparently some don’t count at all. As hard as you may find it to believe, there are societies in which it’s not necessary to count (or not necessary to count beyond small numbers).
This isn’t because such peoples are stupid. They just don’t have any necessity to count. The fact that some cultures have no one in them who does quantum physics doesn’t show that the people of those cultures are inherently more stupid than those of cultures where there are quantum physicists. Someone taken from that culture shortly after birth who was raised in a society with quantum physicists could learn to count and perhaps become a quantum physicist themselves. It’s entirely possible that counting is something that can be done by general human mental abilities, just as it’s entirely possible that language is something done by general human mental abilities. There is no evidence for special structures that have evolved in the human brain just for language or just for counting.
And, incidentally, not only are there posters to the SDMB who have master’s degrees in math, as I do, but there are ones who have master’s degrees in linguistics, as I do.