This is the Realm of Simply Beautiful Art

amWhy

00:00

what event have we, now?

Simply Beautiful Art

Hm, so I was thinking...

Perhaps we should have a story time thing when noone shows up to discuss large numbers

a chapter a week

And then I shall bookmark the stuff and you know

You guys (and gals) can say things while we go through this as you please.

**The Story of Large Numbers, a dive into the unknown**

Our story begins with a game. A simple one at first, but one that will get progressively more and more intriguing as well as complex. And so, we begin.

*Using only one sheet of paper, write the largest number you can, using common mathematical notation and/or any notation you can define on the backside of the sheet of paper. The dimensions of the paper are not important, and merely serve to set reasonable limits to what we can write.*

So, how did you do?

Hopefully, you didn't say "Hm, I'm gonna fill the entire sheet of paper with 9's and leave it at that" because that means you didn't really try hard at all. You missed the entire point of the backside of the sheet of paper and the allowance of any common mathematical notation. So if this is you, go back and try again.

Hopefully, unlike the previous group of people, you noticed that you could do something like...
9*9*9*9*9*....*9
and go all the way across the paper. Admit-ably, this isn't terrible... but its still not much. If you only got this far, want to try agai…

user21820

03:15

@SimplyBeautifulArt Yea I know but I thought you meant it would be difficult by purely elementary means. Like for example proving Stirling's approximation. Anyway is your solution similar?

shredalert

06:05

@amWhy apologies for the late reply; I dozed off. I don't think it's easy keeping track of time zones living in a country as big as the U.S..

Simply Beautiful Art

11:43

@user21820 i think trapezoid rule is enough

user21820

You mean the discrepancy between upper and lower trapezoids vanish?

Sorry not "vanish" but "have bounded total area". =)

@SimplyBeautifulArt: If so, then yea it's a much faster method. Lol.

Simply Beautiful Art

:-)

user21820

It's an interesting kind of question because I've always been fascinated with finding a generic way to do asymptotic indefinite summation.

I never found a nice way though.

Simply Beautiful Art

EMSF?

Euler Maclaurin summation formula?

user21820

11:59

Yea I saw that.

However I'm not sure how to use it effectively.

Though it's probably because I don't know much about it.

Simply Beautiful Art

???

Its just trapezoidal rule extended version

Take a few derivatives. Kind of like Taylor expanding

Abel's summation formula

In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in number theory to compute series. == Identity == Let a n {\displaystyle a_{n}\,} be a sequence of real or complex numbers and ϕ ( x ) {\displaystyle \phi (x)\,} a function of class C 1 ...

How about Abel's summation formula?

user21820

Hmm. I really don't know much about it. Never learned about it in school either.

Simply Beautiful Art

12:15

lol, Abel summation formula is mostly for number theory

user21820

13:29

@SimplyBeautifulArt: By the way, did you get round to posting a question about the backward bicycle?

Zophikel

14:43

Is this a bad way to go about learning Analysis currently I'm reading and working through a book on Real Variables(Intro), a book on Complex Variables by Stein, and another which focus's on the Set Theoretic knowledge required for Modern Measure Theory. I wonder if I should just learn one thing at a time(note I'm learning this on my own)

shredalert

@Zophikel learning one thing at a time is a great way to bore yourself to death.

Zophikel

@shredalert I have been finding gaps in my knowledge I want to make sure I have a good solild knowledge base I can't finish a book yet I read mutiple

shredalert

If it works for you and, more importantly, you're having fun and it keeps you motivated, by all means continue. If you have a gap just go and fill it. I tried reading books linearly a few times and it barely ever works when you're studying on your own. We've got loads of books to choose from so just go as you please.

Zophikel

@shredalert i'm going to spend some time filling my intial gaps

shredalert

@Zophikel well, you've got the motivation to fill them, so I don't see a problem. :)

Zophikel

14:47

@shredalert thanks

shredalert

@Zophikel No worries. :) If you're not doing a course then there isn't an exam at the end. If you find you don't understand something you can just read up on it. I think it's much more important to keep motivation levels high as that is the main driving force behind wanting to continue study.

And if that means not reading linearly so be it, but if you enjoy reading linearly, that's fine too.

Zophikel

@shredalert if it's a course then yeah I can go one after the other but i'm by myself

shredalert

@Zophikel just keep doing whatever works for you. At the end of the day it's important that you use materials and methods that gets you to understand.

Zophikel

all right @shredalert but i'll have to do some more basic real analysis

shredalert

@Zophikel Well you do it whenever you feel like you need it. All I'm trying to say is don't let a big list of prerequisites get in your way. Sometimes you need a lot less than you think you do to study something. Some topics, ofcourse, have very heavy dependencies on others, but imo the best way to find out is to dive headfirst in and then figure out what you need to know.

Zophikel

14:55

all right

shredalert

@Zophikel If you feel like you need to study some theorems from basic real analysis make a list and go learn them from somewhere. You seem to know quite a bit of analysis already, so I don't think you need to read a full book on it which might end up taking months anyway. I'm pretty certain that would bore you quite a bit.

Zophikel

thanks @shredalert

shredalert

Glad to help @Zophikel :)

Xam

Hello guys

shredalert

Hello @Xam

Simply Beautiful Art

14:59

@user21820 Oops nope

@Xam Hello

user21820

@SimplyBeautifulArt Ah well never mind then. =)

Xam

I was hoping youknowme would be here yikes!

@SimplyBeautifulArt hi

Zophikel

Hi @user21820 :)

Simply Beautiful Art

@Xam ping him

user21820

@Zophikel Hello!

Zophikel

15:00

@user21820 you dank

user21820

@Zophikel Sorry I don't get what you mean.

Xam

@SimplyBeautifulArt I don't know his complete nickname

Simply Beautiful Art

@Thelonewolfisbackbaby...

Zophikel

@user21820 your awesome that's what I mean

Xam

@Thelonewolfisbackbaby... hey

thanks @SimplyBeautifulArt

Simply Beautiful Art

15:05

these wolves in pixel dungeon are BS

I swear, when you enter, they smell you from a mile away and then the entire level's worth of wolves comes at you

user21820

@SimplyBeautifulArt I guess that's tautological for that level, right? =P

Simply Beautiful Art

Well, every time you beat a boss... the difficulty spikes...

but still, this is not cool

Xam

Well, I have to continue my searching xd

See you later @SimplyBeautifulArt

Simply Beautiful Art

See you @Xam

user21820

15:31

@Zophikel @shredalert: Since you're doing real analysis now, here's a fun problem if you like. =)

Does the infinite sum 1/4 + (1*3)/(4*6) + (1*3*5)/(4*6*8) + (1*3*5*7)/(4*6*8*10) + ... have a limit?

Prove your answer. =D

shredalert

@user21820 I put real analysis on hold to do more geometry, algebra, and logic. :P

I've got a slim book on nonstandard calculus which I'm more excited to read about than my analysis book. xD

user21820

@shredalert Aww. Are you taking analysis this semester?

shredalert

@user21820 No analysis until next year

user21820

I see. But you really should learn real analysis and logic proper before non-standard analysis, otherwise it is likely to muddle you.

shredalert

To be fair, I just studied some analysis to find out what a limit was. After that I lost most of my interest in it xD

I'm fine being muddled :P

user21820

15:34

Lol!

Logicians aren't fine with being muddled! =P

Simply Beautiful Art

haha

shredalert

But I'm not a logician. :P

user21820

Not yet. =P

@user21820 <- @SimplyBeautifulArt: Forgot to ping you; you may be interested in trying it too!

shredalert

I'm enjoying my logic book, no doubt. But I'm reading it more because I'm finding it interesting instead of as a prerequisite.

Simply Beautiful Art

I'm looking at it

And I'm fairly sure it fails the n-th term test

user21820

15:37

What's the n-th term test again?

By the way, only elementary methods are required. =)

Simply Beautiful Art

the terms must go to zero

user21820

Um the terms do go to zero...

Simply Beautiful Art

or do they?

oh, the denom starts at 4

hehe.... thinking of Wallis products

user21820

The question is; how fast do they go to zero and can you solve it without high-power tools?

Simply Beautiful Art

Well, it certainly converges

shredalert

15:39

so products of odds over products of evens

Simply Beautiful Art

I think

No, hm...

user21820

Yeap n odds starting from 1 over n evens starting from 4.

Think more. =D =D

shredalert

This is a very naive guess but it looks like the denominator is growing faster than the numerator

user21820

@shredalert You can compare each term with the previous term. That tells you that the terms do go to zero.

shredalert

Looked like that from the numbers

user21820

15:42

But since we have things like 1/1+1/2+1/3+..., we still can't tell about this one so easily.

Simply Beautiful Art

it diverges by Raabe's test

user21820

I've never even heard of it, but let me look it up.

What's your sequence of constants?

Simply Beautiful Art

Huh?

oh, u is your fractions

user21820

Oh I misread the Wolfram page.

Let me see.

Zophikel

@user21820 it could be possible summiablity methods could be used

user21820

15:47

@SimplyBeautifulArt I don't think it gives the conclusion you claimed.

@Zophikel I only used extremely elementary methods, available to most high-school students. =D

Of course, with full rigour.

Zophikel

@user21820 here's a viable appoarch we can conjecture that the series converages if it's partial sums tend to a limit

Simply Beautiful Art

Ratio test

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }a_{n},} where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. == Motivation... ==

I think it does work

oh, fractions upside down

XD

user21820

@SimplyBeautifulArt What does work? Does it converge or not?

Simply Beautiful Art

nvm. It still diverges I think

shredalert

I'll tell you what. I've never seen an analysis text with lots of applications in it for each chapter. Any of you know of one?

user21820

15:51

@shredalert That's a tough one. I'd still recommend Spivak's even though it doesn't have a high 'obvious real-world applications per chapter' score... But generally real analysis is crucial to subsequent fields, including numerical algorithms, differential equations and whatever else relies on real numbers.

shredalert

I've been trying to find one for ages. It's easier for other topics like geometric algebra because there are so many things you can do on a computer with it, or fractals and dynamical systems.

user21820

But dynamical systems are going to require in-depth knowledge of real analysis.

shredalert

I've got a book that goes over chaos and fractals using just algebra

it's pretty elementary, but I only want an introduction anyway

A lot of the stuff in it is used in real world applications as well. Without needing any real analysis.

But I've been trying to get a hold of an analysis book which emphasises physics and geometry, instead of just drowning me in epsilon-delta exercises. I might have to go back to Zorich

Simply Beautiful Art

@user21820 Did I win?

user21820

Ah. Well Spivak's exercises are not repetitive at all, but still mathematics-oriented so not much physics.

@SimplyBeautifulArt No you lost terribly. =D

Simply Beautiful Art

15:58

Aw dang it!

shredalert

I think I'll actually be better off just looking for a video lecture course.

with a homework problem set

user21820

Perhaps.

But sometimes, it's good to get the basics firm.

So when you actually take the real analysis course, you might want to get Spivak's, if it's not too expensive.

shredalert

Even my logic textbook is more entertaining than the analysis texts I've come across, and that's saying something.

user21820

Unless your university uses another reasonably good alternative.

shredalert

I've found Rob Ash's book good. I'm on metric spaces right now, but I haven't got around to reading more because I've been having more fun doing other stuff.

The lone wolf is back baby...

16:05

Hey @Xam

user21820

@shredalert Then you could stick to his book. The fact that I have his abstract algebra book in my computer implies that I must have found it good.

shredalert

Yeah, I think I'll just stick to Ash. He's a good writer.

The book was very cheap too. It's Dover.

user21820

And I'm surprised you're doing metric spaces before even basic real analysis, since metric spaces is something like the generalization of sequences.

From sequences from R to sequences from a metric space.

shredalert

I did some basic real analysis in R. P. Burn's book. But as soon as I finished the chapter on sequences I literally couldn't keep reading because I found the topic so mind-numbingly boring.

I guess it's more a problem on my side than any book xD

user21820

@shredalert You know he has a lot of his books available online right?

math.uiuc.edu/~r-ash

shredalert

16:08

Yeah. I just like having a paper version I can carry around with me.

user21820

Ok.

shredalert

It's a dover book so it only cost me £6. :P

that's including postage haha

user21820

Nice! I must remember Dover next time I want to find a cheap good book!

shredalert

I find it interesting that the triangle inequality is an axiom in metric spaces

user21820

It's because it's an axiom for Euclidean space.

Metric spaces generalize Euclidean spaces, but keep many of the nice properties.

shredalert

16:12

Ah, I see.

user21820

Do you know why triangle inequality holds for Euclidean space?

shredalert

I do not

I've proved the triangle inequality before, even for complex numbers, but don't think I ever thought about why.

user21820

I'm now trying to recall the cleanest proof I once knew...

Probably you first translate one vertex to the origin..

shredalert

I'll be happy with just a rough sketch of the proof

user21820

So you want |u| + |v| >= |u-v|.. Then square both sides and use dot product.

shredalert

16:21

@user21820 What's the axiom in Euclidean space?

user21820

|x-y| + |y-z| >= |x-z|.

shredalert

Ah. It makes sense then. I can imagine a lot of proofs in metric spaces are simplified if we can just quote the triangle inequality axiom.

the basic ones at least

user21820

In metric spaces all you have is the metric satisfying triangle inequality, and indeed a lot of things rely on just that property.

shredalert

I've proved the one dimensional one |a+b|<=|a|+|b|. So it's interesting that the one you give needs to be an axiom.

user21820

Consider any graph with each edge having a non-negative length. Then shortest-path distance is a metric on the vertices.

And indeed many graph algorithms are tailored to work for such graphs.

So it's indeed useful to have the general notion of a metric satisfying triangle inequality abstracted out from Euclidean spaces.

shredalert

16:30

Speaking of graphs, I need to get around to devoting some time to Chartrand and Zhang's book on Graph theory.

Also Dover btw.

user21820

Haha.

shredalert

I only read the first section and did a few problems. Never got around to more.

user21820

@shredalert The proof I recalled is indeed along the lines I sketched. It goes as follows.

We wish to prove |x-y| + |y-z| >= |x-z| for every points x,y,z in Euclidean space.

Since each term is unchanged by translation we can assume that y = 0.

Thus we need to prove |x| + |z| >= |x-z|.

Both sides are non-negative so it is equivalent to prove ( |x| + |z| )^2 >= |x-z|^2.

By expanding and dot product this is just x.x - 2*|x|*|z| + z.z >= x.x - 2(x.z) + z.z.

So we need to prove |x|*|z| >= x.z

shredalert

Isn't the axiom for an arbitrary set of points?

user21820

Yes. Any 3 points in n-dimensional Euclidean space.

shredalert

16:36

Then why are we proving it?

user21820

Because distance in Euclidean space is given by sqrt of the sum of squares of differences in coordinates.

It's not given by, you know, intuition...

It works in 4-dimensional, 5-dimensional, Euclidean space too.

shredalert

I'm confused here. Why does it need to be an axiom if we can prove it?

user21820

It is an axiom for metric spaces. We chose those axioms so that Euclidean spaces are instances of metric spaces.

shredalert

Oh

user21820

The example I gave about arbitrary graphs is an example that is not a Euclidean space but is a metric space.

shredalert

16:38

@user21820 When you said that I thought you meant it was already an axiom for Euclidean space.

user21820

I understand. That's why it's good to know logic so you know precisely what are axioms for what and why they are chosen like so.

@user21820 <- Continuing, if the right side is negative we are done, and otherwise it is equivalent to proving (|x|*|z|)^2 >= (x.z)^2.

Which is |x|^2 |z|^2 >= (x.z)^2. This is a well-known inequality called Cauchy-Schwarz, and I temporarily forgot the simple proof just now.

shredalert

I already proved this before

user21820

Great. Then we are done.

shredalert

I've proved the triangle inequality as well

I was waiting for you to tell me why it was an axiom for Euclidean Space xD

user21820

For other people who may not know, the easiest is to start from | x - (x.z/z.z) z |^2 >= 0, expand and cancel and immediately get the CS inequality.

shredalert

16:42

There's book called the Cauchy Scwhartz Master Class, I saw it while browsing for other stuff. Some of you might be interested in it.

user21820

Equality happens when x is parallel to z, since (x.z/z.z) z is the projection of x onto z.

@shredalert I'm curious whether you proved triangle inequality in this fashion or in another way?

shredalert

I'm pretty sure I proved it in at least two different ways for real numbers.

One using CS and one without.

user21820

No for Euclidean spaces.

Real numbers triangle inequality is just trivial case analysis.

shredalert

I can't remember if I proved the general CS.

user21820

I see. Okay so the proof I gave above works for any dimensional Euclidean space.

Simply Beautiful Art

16:45

T^T Farming stuff in games is boring

user21820

Lol..

shredalert

I do remember doing CS once, at least, for real numbers and it was just case by case.

@SimplyBeautifulArt Then do something else. lol

user21820

Ok I'm going off soon! See you all next time!

Simply Beautiful Art

@shredalert I want something really badly

shredalert

@user21820 Have a good night. :)

@SimplyBeautifulArt What do you want?

Simply Beautiful Art

16:46

It shall take me at least 3 more hours of clicking monotonously

shredalert

@_@

Simply Beautiful Art

paragon shogun pet

shredalert

Which game?

Simply Beautiful Art

Must have!

AQW

user21820

@SimplyBeautifulArt Have you heard of cookie clicker? If not, don't start it. It consists entirely of clicking.

Simply Beautiful Art

16:47

lol

@shredalert If you want to play, I have accounts you can use that I'll give away

user21820

Ok good night all!

shredalert

@SimplyBeautifulArt I spent a looooot of time on MMOs quite a few years ago. Those days are behind me now. xD

Simply Beautiful Art

@user21820 Good night!

xD

I don't play often

just occasionally

shredalert

I just occasionally hop on TF2 or some M&B Warband. That cures the itch.

Simply Beautiful Art

Yeah

I mean, that paragon shogun pet is once in 10 years type thing...

so a couple hours of this should be worth it...

shredalert

16:50

Yeah, but will the game even be around in that long? xD

Simply Beautiful Art

probably?

I want to use a bot so bad

but I'm already cheating the game xD

shredalert

Going to try to get this assignment done by tomorrow so I can have the next week and a bit for studying whatever I feel like. :P

Simply Beautiful Art

nice

shredalert

I had to ditch my old geometric algebra book, because a lot of the stuff in it was not well-defined. :/

Simply Beautiful Art

dang

well, it happens

Like me

I'm not well-defined

shredalert

16:56

Yeah

xD

wasn't 'Yeah'ing that xD

Simply Beautiful Art

XD

shredalert

I've found two that look good. Might have to wave the jolly roger because the library doesn't have them. heh heh

Simply Beautiful Art

:P

shredalert

17:23

@SimplyBeautifulArt study for the exams going well?

Simply Beautiful Art

I suppose

shredalert

19:18

Awfully quiet round here

Simply Beautiful Art

21:37

@shredalert Yeah

Sunday business

Feeds

22:49

Simply Beautiful Art has added The lone wolf is back baby... to the list of this room's owners.

Transcript for

$Mathematics$ This is the Realm of Simply Beautiful…