>>109752
10 seconds later, Tomo lost her virginity.
And most of her hair.
And a couple liters of blood.

I would love to see a version of this with, for example, the Banach-Tarski Theorem instead of 0.999...=1.
"Haha! This girl thinks that given any two bounded subsets A and B of a Euclidean space in at least three dimensions, both of which have a non-empty interior, there are partitions of A and B into a finite number of disjoint subsets, A = A1 ∪ … ∪ Ak, B = B1 ∪ … ∪ Bk, such that for each i between 1 and k, the sets Ai and Bi are congruent!"

I just remembered why I dropped maths.

Great.

>>109771 So do one. There's a version without writing in the 'Yukari teaches' thread.

The Azunonymous theorem:

Any fraction in which the numerator is composed of a number set, and the denominator is composed of a nine set with equal amounts of integers will produce a repeating decimal pattern of the number set.

4/9 = 0.444...

365/999 = 0.365365365...

5280/9999 = 0.5280528052805280...

in the case of repeating nines:

9/9 = 0.999... = 1

As someone ignorant of math, I found this very funny--yet, thanks to Chiyo-chan, I feel I actually learned something.

>>109747
The way I learned it...

1/3 = .3 repeating
1/3 x 3 = .3 repeating x 3
1 = .9 repeating

>>109729
muses Did I miss this episode...?

Tomo-chan falters before the power and majesty of infinite sets...

Osaka, on the other hand, actually looked at the entire infinite decimal expansion and knew it is 1. (And she used the Banach-Tarski Theorem to put the entire Universe in her bookbag... which she then forgot at home...)

Are there any uses for the concept of .999999 repeating in the real world? I mean, one can easily picture having one of something, but not ".999999 repeating" of something.

>>109771

Yeah, but if she knew enough about Zermelo–Fraenkel set theory to know what that even means, she'd probably say something like, "This girl accepts the intrinsically nonconstructive Axiom of Choice as a premise!"

To which Yomi would respond, "You always use the Axiom of Choice because it makes things easier!"

Tomo: "Well, yeah, but not when it causes a paradox!"

Yomi: "You can't recognize an axiom as valid only sometimes!"

And then they would go fight for a while while Chiyo more fully clarified the Banach-Tarski Theorem and its dependence on the Axiom of Choice to Kagura.

Tomo just needs to learn to pick her talking points. The essence she misses: Don't argue against anything already proven.

If Kagura had come up to clarify man's state of nature as a free willed creature unbound by morality forced into societal arrangement by external factors beyond their control, like Rousseau suggested, Tomo could have pointed and laughed and balked saying.

This girl never read Hobbes' "Leviathan!"

Then Kagura would be all like: "Yes, I did. I didn't buy it."

"Huh?" Tomo would say "Hey, Sakaki, did Kagura read Leviathan?"

"She did."

"Hey, Osaka, did Kagrua read Leviathan?"

"Purty sure."

Chiyo opens up.

"The presentation of state of nature, set forth in Leviathan, is an entity desighned with the capacity for good, which is corrupted by felicitous desire held fast in a miserable existance."

Kagura would be all like:

"Hobbes platform is based on the concept of society being structured around a deontological obligation to the commonwealth. I however opt for the social contract theory to be a suitable explanation for human society.

"John Locke" Chiyo chan would say, "Examined the approaches of Hobbes and Rousseau and established a very solid middle-ground based upon reason and tolerance. His concepts went on to shape the phrasing of the United States Constitution."

Then Tomo would be all like.

"What, really?! I'm not stupid, I'm not stupid!"

Woah, wait.

Aw, damn, I guess my argument breaks down here.

>>109833

>Are there any uses for the concept of .999999 repeating in the real world? I mean, one can easily picture having one of something, but not ".999999 repeating" of something.

If you're having trouble picturing .9 repeating, try picturing 1. They're the same thing.

Anyway, understanding why this is true is more important than the actual fact that it is. >>109737 actually shows a common misconception of the concept of infinity by which people commonly "refute" the claim that .9 repeating = 1. They imagine the statement ".9 repeating" as a process instead of an object, and say such ridiculous things as "999...999, where the ellipses represent an infinite number of 9s," when in fact no such construction is possible, as to say "the last 9" is akin to saying "the highest natural number."

A good sense of infinity is important for such basic math as calculus, which I'm sure you know has tremendous practical value. In fact, one of the more mathematically rigorous proofs that .9 repeating = 1 is that the limit of the converging infinite geometric series that can be used to write .9 repeating, is 1. To dispute that would be to dispute the concept of the limit, which some students sometimes do when first learning calculus based on a misunderstanding of the fact that a limit either does not exist, or is a number*. It is most certainly not some never ending process, in the exact same way that .9 repeating is a number (namely, 1), and not a process.

So, yeah, there are important real-world applications to being able to understand that .9 repeating = 1.

*Some calculus students may dispute this point by claiming that it often happens that the limit of something is occasionally equal to positive or negative infinity, which are not numbers. I must then point out that, like the improper integral, such statements are common but not technically true. A limit that does not converge does not exist; to say that the limit "approaches" positive or negative infinity is simply a way of specifying the particular way in which the limit does not exist.

>>109839

Hmmm...well, I'll take your word for it. I do at least understand that calculus made modern technology possible, even if I don't understand calculus itself.

>>109832
Based on the Xmas episode where Tomo doesn't believe reindeer exist.

Well, regarding an earlier post...

DOWN WITH LOCKE!
DOWN WITH HOBBES!
DOWN WITH ROUSSEAU!
LONG LIVE ST. THOMAS AQUINAS!

sits down exhausted Whew, that felt good.

>>109943
Spinoza > all

>>109915
Oh, I know, I finally watched the series a few months ago. I was joking a bit over the idea that a comedy manga/anime series would have a sequence centering on such a technical point...

(apart from,say, Cromartie's occasional excursions into social or moral philosophy... until the gorilla shows up...)

As a Pure Math grad, I'm often the one trying to get this point through people's heads. The same method doesn't work for everybody. For me, the most compelling argument (due to not fiddling around with multiplying infinite strings by 10 or whatever) is this: any two numbers which are different have some number in between them. But it only takes a moment's thought to realize you can't possibly make something bigger than 0.999... and still smaller than 1. (If it's less than 1, it must have 0 before the decimal, but then the first digit that isn't a 9 makes it smaller than 0.999...) So they're not different, and the only other possibility is that they're the same number.

Failing to understand this picky mathematical point doesn't make Tomo, or anyone else, dumb. What makes you dumb is seeing a whole bunch of proofs, not finding any flaws in them -- and still insisting that you know better.

>>109987
Another way of looking at this is to consider already having .9999... out to some number of digits already and looking at the difference between that number and 1. Adding one more '9' digit makes that difference 10 times smaller than it already was. Then imagine repeating this process forever: the difference gets closer and closer to zero; doing that forever makes the difference zero. (Analysts call such processes "infinite tasks" or "super-tasks".)

Infinity is a very strange "place" because it is the "destination" of an imaginarily unending process. All sorts of results occur which sound bizarre when considered in terms of humanly finite activities or collections of objects, yet are logically inescapable...

'Z', eh? Is this to say you're mathematically "complete"?

(Good Lord! choke /azu/ has been invaded by a buncha chrome-vc: domes!!)

>>109987

OP here: That's one of my favorite proofs for demonstrating this fact to people as well, but it doesn't fit as well on the chalkboard with a single line of dialog. And what makes you think I'm saying Tomo is dumb because she doesn't know this? I was just been making an observation about Tomo in general. Actually, I wasn't, but I also wasn't saying she is dumb because she didn't know this fact, but because she asserted an answer to the question with confidence for the purpose of making fun of someone despite not knowing what the hell she was talking about.

>>109989

That is another way of looking at it, but it requires understanding the concept of the limit, which isn't a whole lot easier for those who don't already understand calculus.

>>110001
Theory: Tomo is mostly flat chested because most of her boobs went to her brains.

Either that, or Sakaki acts not just as a height and bust vacuum, but also to certain brains. (Chiyo and Yomi are clearly immune to the brain effects, or maybe they too are mind-suckers?!)

And let's face it, Yukari is basically an older, evil Tomo. Except Tomo is evil too. And Yukari isn't the sharpest lightbulb in the crayon box.

>And Yukari isn't the sharpest lightbulb in the crayon box.

A comment worthy of a rap upside the head with a rolled up magazine from Yomi.

>>109987
>>109989

I actually understood it better when you explained it this way, rather than with proofs using symbols. I guess what's more difficult for me to understand is what the necessity of a concept such as ".99999 repeating" is in the first place; that is, if it can be demonstrated in multiple ways that it's equivalent to 1, why bother to even use it, then? That is, if it's equal to 1, why not just use 1?

I like to think this whole thread actually happened in the Azu story. There's plenty of room for such outtakes in a series that only showed us twelve hours out of three years of time.

>>109943
>>109946

Wittgenstein is better.

>>110110
Wittgenstein is a philosophical catastrophe. It's the very type of a "school", a regression of all philosophy, a massive regression. They imposed a system of terror in which, under the pretext of doing something new, it's poverty introduced as grandeur. There isn't a word to express this kind of danger, but this danger is one that recurs, it's not the first time that it has arrived. It's serious especially since the Wittgensteinians are nasty and destructive. They are assassins of philosophy, and because of that, one must remain very vigilant.

>>110046
It's like the way tan is still useful, even though tan(π/2) is undefined (and approaches both ∞ and -∞ !): an abuse of an otherwise perfectly useful mechanism.

0.333repeating isn't equal to any integer, but is an interesting number, and one you'd otherwise be unable to express as a decimal. (It's the rational number 1/3.)

>>110046

>...if it can be demonstrated in multiple ways that it's equivalent to 1, why bother to even use it, then? That is, if it's equal to 1, why not just use 1?

Well, in fact, we do! Or we use 3/3 or 9/9. The fact that we get this endless decimal expansion .999999... from working with 3/3, 9/9, 11/11, etc. is due to the incompatibility of decimal notation with such numbers in the denominators of the fractions. (All rational numbers [fractions] has such "non-terminating" decimal expansions, except for those whose denominators contain only factors of 2 or 5 -- the factors of 10.)

>>110001

>That is another way of looking at it, but it requires understanding the concept of the limit, which isn't a whole lot easier for those who don't already understand calculus

I guess I don't agree that it isn't easier. I find people understand the idea of a limit all right; it's proving that such a process has a particular limiting value that's not so easy. (In any case, it's kind of tough to deal with the question without being something infinite in somewhere...

>>110163
Argh! That's "...bringing something infinite in..."

>>110163
Meh. A limit is easy enough to explain (though I'm in calculus III, so, I may be biased).

The best way to explain it, I've found is this:
Where is a function (a line on a graph) going? Not where is it, but where is it headed to? The easiest example is y=1/x. As x gets bigger, y gets smaller. But x can never get so big that y is zero. So, the limit of y=1/x is zero. Not because y will ever actually BE zero, but anyone can see that the line is eternally getting closer and closer to it.

tl;dr version: Sometimes a line can't actually get to a point but for all intents and purposes it's heading there. That's basically a limit. (And it doesn't have to be at a place where the line can't reach. The limit of y=x at x=1 is 1. The line IS headed there. It is also a point on the line. It's just that there's really no reason to ever worry about the limit at a point that's actually on the line).

>>109837

>...then they would go fight for a while while Chiyo more fully clarified the Banach-Tarski Theorem and its dependence on the Axiom of Choice to Kagura.

I think I'd pay money to see that... (Explaining it to one of the Bourbaki, all right; but to one of the Bonkuras...?)

>>110166

lol. Your tl;dr version is longer than your original. In any case, "all intents and purposes" doesn't cut it in math.

>>110163

Decimal notation is not incompatible with such fractions as 1/3, as the means to represent them is easy. Generally, this is done with a line over the repeating digits. It can also be written with parenthesis: 0.(09)

Thus, all rational numbers have a finite representation in decimal notation; it is irrational numbers which do not. Putting a line over the repeating part is not cheating the systems; it's PART of the system.

this is bullshit
1=/=.999(indefinate)

>>110185
Not in chemistry. It's very common to calculate a small loss vs. the original whole than to calculate a small loss vs. the new value.

It's like... if you wanted to see how much one of your hairs was in relation to your body mass, would you factor in the weight lost from one plucked hair, or just use one hair vs. your body mass?

Don't feed the troll.

>>110169
Indeed. I like to trick people who don't like to read. It's like hiding an entire book in crib notes.

Also, "all intents and purposes" may not cut it in math, but this is just a quick, simplified explanation to people who have never seen this sort of thing before.

>>110119

..."Assassins of Philosophy"? Fuck yeah, that's the coolest thing I've ever heard. Does this mean I get to wear a ninja costume? And if it does, can I wear it whenever I like or only when I'm destroying philosophy?

>>110264
I think he means a person who targets philosophers

>>110001
OP, eh? Excellent thread. But I think if we continue to argue about precisely what you meant by "Tomo is dumb" and precisely what I meant in my reply, we will simply prove that Tomo is smarter than both of us.

>>109989

>'Z', eh? Is this to say you're mathematically "complete"?

Heh, hadn't thought of that. Z is actually just the first letter of my usual online handle.

I appreciate >>110181 making my point, btw.

This thread has taught me things. Or I am the victim of a forum-wide joke. =[

Nah, you're not.

Honest.

.9999~ is something that has to be overcome.

>>110519
I thought you'd appreciate the eternal recurrence of the number nine.

>>110539
"Number Nine... Number Nine... Number Nine... Number Nine..."

(The trouble with most diets is that after losing a bunch of weight, one tends to vc: repudge...)

>>110570 "You are Number Six."

>>110539
Sure. But every appreciation is based on premise of contempt. I appreciate it exactly because it paves the way to beyond 1.0

>>110571
LOL
Osaka, so you offed John, too? So it was you in his tree.. And for what kind of reason?
Whimsical, upside down, as clever as ever.
He'd been a fine chap, tho'.

>>110571

Osaka: the new No. 2?

>>110587

Didn't The Prisoner say in "Once Upon A Time" that he was "very good at sums"?

so in the same sense, would 2.999... = 3?

>>110604

Yes, and 0.77(9) = 0.78, or 4.(9) = 5.

In general, any number that can be written with a finite number of digits can also be written as a non-terminating decimal ending in repeating 9s.

>>110605
Pie arguement still stands, the arguement against it is the equivilant of "No cuz my god says dat isnt truez!!!11!!1

>>110628
Actually it doesn't.

Seems our mod is stronger than your god.

>>110629
my god? lol wut?

Does this mean that 1/2*1/2*1/2... = 0? I'm not trolling...

... also, that's a great parody. Azuma's comedic setups are so complex and fluid, yet his jokes are so simple... but very outside-the-box, as to where they have about the same effect that your parody has (though I found your parody funnier... it's hard to appeal to such a small audience). Reindeer not existing because they're most often thought of being related to Santa Clause (who is dead)... brilliant.

>>110830

Yes, lim( (1/2)^x ) as x -> infinity is equal to 0.