Mathematics - 2016-11-18 (page 2 of 4)

Null

02:06

anyways, the solutionset is the universal set containing everything. because there will exist probably infinitly many identities in infinitly many spaces.

(note that i dont have any clue whatsoever)

user228700

02:19

Ello everyone :-)

Ramanujan

Hello :)

Null

@Kaumudi hi (-:

Ramanujan

@Kaumudi how many days are left for IIT exam?

user228700

@Ramanujan Uhh, about 150, I'd say.

user228700

I've a quick question. When we write $x^y$, $y$ is called the exponent. What is $x$ called?

Ramanujan

02:28

Base

Semiclassical

@TedShifrin a (not entirely serious) one-line proof via complex algebra: $\dfrac{e^{i \phi}-1}{e^{i \phi}+1}=\dfrac{e^{i\phi/2}-e^{i\phi/2}}{e^{i\phi/2}+e^{i\phi/2}}=i\tan(\phi/2)$ is pure imaginary

Ramanujan

Y is also called power

user228700

Uhh. Is there another term, perhaps?

Ramanujan

@Kaumudi for x^y ?

user228700

Yes, for $x$ in $x^y$.

Null

02:30

base?

Ramanujan

@Null chat.stackexchange.com/transcript/message/33576489#33576489

Null

@Kaumudi you mean something like exponentee? like divident/divisor?

Semiclassical

for comparison: if $y=a^x$, then $x=\log_a y$ is the log with base $a$

referring to $a$ in $a^x$ as the base is consistent with that

user228700

Ah, never mind. I thought there was another term but it's probably just my brain screwing with me again. Thanks, guys :-)

Null

@Kaumudi well at least you can recognize it. some are not in that luxury ;)

user228700

02:34

:-P

Null

@Kaumudi at least i can understand why you thought there was a term. employer/employee and others would make that question arise.

actually there should be a term

@Kaumudi radix could be another term. But only in a restricted context.

Mike Miller

02:51

@PVAL The new Baldwin-Sivek paper looks interesting.

user228700

@Null Yeah...

Null

03:05

is "order(s) of magnitude" a superfluous term?

Ramanujan

I think "order of magnitude" is used at $10^x$ and we need to discuss only about x

Ted Shifrin

@Semiclassical Yes, of course.

Null

03:23

@TedShifrin can "$\forall x\in X:$ blub" be rephrased to "$x\in X\Rightarrow$ blub" ?

also welcome back?

Fargle

@Null They seem equivalent to me.

Null

okay my excercise is not completly out of the blue

0

Q: How to show that $\forall x\in \mathbb{R},\space \exists!n\in \mathbb{Z}$ such that $n\leq x < n+1$

How can I show that, $$\forall x\in \mathbb{R},\space \exists!n\in \mathbb{Z} \text{ such that, } \space n\leq x < n+1$$ which will be denoted later as $E(x)$ or $[x]$. In my textbook the the prof prove it like that: Let $A = \{k\in \mathbb{Z},k\leq x \}$ $A$ is a non empty part of $\mathbb{Z}$ ...

real-analysis

i just dont understand why it is that interesting

i.e. why would anyone prove that?

Semiclassical

'Interesting' depends on the context.

Null

at the moment we "build" $\mathbb{R}$

Semiclassical

I can't get myself excited about it, but evidently for that class it's important

Null

03:35

we made all the axioms for a field and now get started with R

Semiclassical

About the only thing in there that catches my eye is Bill Dubuque's reference to 'continuous mathematical induction.'

Null

@Semiclassical that indeed must be it

Semiclassical

which in turn reminds me of this MSE blog post: math.blogoverflow.com/2015/03/10/when-can-we-do-induction

Fargle

@Null It gets more interesting when it turns out that for each natural $k$ there is an $n_k$ such that $10^{-k}n_k \leq x < 10^{-k}(n_k+1)$, and then we get the standard base-$10$ representation for $x$

At least I think that's right...

Null

ah ok, so it is a preparing excercise for more to come like always, i see

Fargle

03:39

Not sure how that works with terminating decimals having two separate base-10 representations, but...

Ted Shifrin

@Null: It means that the greatest integer function $f(x)=[x]$ is defined :)

Null

@TedShifrin would that be the floorfunction? :D

Ted Shifrin

@Fargle: You're right, mod .99999...

Yes @Null

Fargle

@TedShifrin It seems like you'd have to lose uniqueness, though, or maybe I'm just being hard-headed.

Ted Shifrin

You do get uniqueness mod .999999... = 1.

@Null, did you forget to sleep?

Null

03:43

maybe yes, maybe no. im feeling sick, i probably OVERslept!

Ted Shifrin

Aww ...

Null

man

the OP of the questioner feels like cheating already

Ted Shifrin

Huh?

Null

$A = \{k\in \mathbb{Z},k\leq x \}$ with recognizing this has a maximum

Ted Shifrin

Without assuming the lub axiom?

Null

03:47

eh. mmh.

doesnt this follow from the ordering of Z?

i mean if i take all integers that are lower or equal than a real number x. There has to be one integer that is the biggest or not?

otherwise ordering makes no sense

Mike Miller

h

Fargle

@MikeMiller h to you too!

Null

$h^{h^{h^{h^{h}}}}$

Fargle

$h^{h^{h^{h^{h^{h^{h}}}}}}$

I was about to say lrn2bracket but then I forgot the dollar sign, so...here we are.

Null

haha

actually

Fargle

03:51

Poetic justice.

Null

h^{h}^{h}^{h} doesnt work either

Fargle

No, they have to be nested.

It's a bit of a weird convention but it definitely drives home to the writer that exponentiation is right-associative.

A big reason why I like writing in LaTeX is those little things--elements of the language that actually can help you think more mathematically as you type.

Null

without this h^{h}^{h} could be interpreted many ways

Fargle

Right. Is it $(h^h)^h = h^{h^2}$, or is it $h^{(h^h)}$? Modern convention is the latter, and that's why I like how LaTeX does it--the brackets line up perfectly with the understood parentheses.

Null

$h^{h^{\ddots}}$

well

enough of h!

Fargle

04:00

Anyway, I'm out. G'night chat.

Null

@Fargle you too, sleep well.

user228700

I've a quick question again. It's a bit of a homework-tsy question so please bear w/ me...

Null

@TedShifrin so basicly i need well ordering if i don't assume lub?

user228700

I've been asked to prove that $$\mathbf{OB}-\mathbf{OA}=\mathbf{OC}-\mathbf{OD}$$ for a hexagon ABCDEF with center O.

user228700

I've been trying to do this but it doesn't look like this is correct :-|

Sophie

04:11

that looks false. Isn't there any more information about the hexagon?

user228700

Nope, nothing at all.

Sophie

try to come up with a counterexample

user228700

Well, I mean, except that the hexagon is regular. Apart from that, no, nothing else.

Sophie

those distances are all the same

user228700

Yeah.

user228700

04:15

Anyway, alright, I just wanted to make sure that it isn't correct.

user228700

Thanks @Sophie :-)

FreshAir

Elementary math question: why does 2^log_3(n)=n^log_3(2)?

ie you can switch "2" and "n"

It's been a while for me...I guess I am a little rusty on log rules

Sophie

@FreshAir take the log(in any base) of both sides and then use the base change property

FreshAir

04:30

@Sophie Thanks for looking in this - by base change do you mean the property log_x(y)=log(y)/log(x)?

I'll give it a try

Okay I got it

Thanks!

I never knew this property existed - ie x^log_y(z)=z^log_y(x)

It looks pretty useful though I couldn't find it listed on any of the elementary math tables

Ramanujan

@Sophie OB - OA is AB and OC - OD is DC so equation becomes AB=DC which might be correct for a regular hexagon

Sophie

@Ramanujan if the hexagon is regular all the vertices are symmetric to the center, so OA=OB=OC=OD=OE=OF

Ramanujan

Then?

Sophie

then OB-OA=OC-OD=0

user228700

@Ramanujan We're talking vectors, people!

user228700

04:39

The direction matters too. So $$\mathbf{AB}≠\mathbf{DC}$$

Semiclassical

is OB supposed to be the vector from O to B, or vice versa?

Null

doesnt really matter for the excercise does it?

Semiclassical

matters for not getting confused in communication

Null

agree ;)

Semiclassical

@kaumadi At the level of direction, I'd agree with you

It would true if they asked about OD-OE, but as stated---nope.

you'd also have OB-OA=AB=OC as vectors

Null

04:51

@Semiclassical =FO? just for understanding

Semiclassical

Yeah.

Null

do we need to define a plane this hexagon is on tho?

Semiclassical

I don't see why the geometry would be different on one Euclidean plane than any other.

Null

(or: do we need to say where A,..,F are?)

Semiclassical

could also do this in terms of complex numbers, of course. e.g. $a=1$, $b=e^{i\pi /3}$, etc.

Ted Shifrin

04:57

@Null, doesn't it follow from that floor fn exercise?

Null

i think of a noneuclidian space where A=D, B=E, and C=F

@TedShifrin do you mean lub? otherwise pls say what exactly :D

user228700

@Semiclassical Agreed.

Ted Shifrin

I mean the exercise you mentioned an hour ago.

Null

@TedShifrin well, i'm only allowed to use stuff that is already proven in the seminar. I doubt floorfunctions where part of that.

Ted Shifrin

The previous exercise is what I'm referring to.

Null

05:01

also since functions are a purely immaterial object, ... eh no sleep and all -.-

@TedShifrin you mean find maxima/minima/sup/inf?

Ted Shifrin

Which I rephrased in terms of the floor fn.

Agh. No.

Null

$\forall x\in\mathbb{R}:\exists n|n\leq x<n+1$

Ted Shifrin

Yes.

Null

well, then i have to prove that the floorfunction exists. not practical (but interesting ;) )

Ted Shifrin

No. Use directly what you just typed.

Null

05:04

you mean "doesnt this follow from the ordering of Z?
i mean if i take all integers that are lower or equal than a real number x. There has to be one integer that is the biggest or not?
otherwise ordering makes no sense"?

sorry for being dumber then dumb hehe

Ted Shifrin

I mean that you should think about $n$ From that previous exercise.

Go sleep!

Null

can't, i go to the university soon

i think i just have to wait for his "part 1.2" ...

stupid excercises imo

Ted Shifrin

Well, when you wake up, you'll figure out what I said.

Null

that's the same like saying "use only words without vocals to communicate to me"

Ted Shifrin

I'm gone.

Null

05:08

@TedShifrin thanks! until later then, i think i sleep

ffahim

@Null . (a+b)^2=a^2+2ab+b^2 ... What's the solution set here?

And what's the solution set here (x+1/x)^2=x^2 + 1/x^2 + 2

Null

@ffahim pls use $-signs before and after equations

@ffahim $(a+b)^2=a^2+2ab+b^2$ is kinda a tautology imo

the second too, because its the first

user228700

05:32

I've another quick homework-tsy question :-P

ffahim

Okay.. I shall try to keep this in my mind next time... @Null.. now just tell me pls...

user228700

If we have a rational expression of polynomials in $x$ and we're asked to find the limit of this expression as $x→\infty$, you know how we divide the whole thing by the $x$ raised to the highest degree of $x$ in the whole expression and then any term with $x$ in the denominator just reduces to 0?

user228700

And then we have our limit? Well, if the limit is $x→-\infty$, then do we get another answer? Following the above steps ^, no we don't but are we s'posed to get a different answer?

Topologicalife

$\int$

Is not working latex?

user228700

Hm?

arctic tern

05:38

@Topologicalife see "$\LaTeX$ in chat" in chatroom description --->

Topologicalife

I did.

arctic tern

@Kaumudi Huh?

@Topologicalife then follow the instructions

Topologicalife

It usually works to me but isn't working right now dunno why.

user228700

@arctictern What's not clear?

arctic tern

okay...

@Kaumudi if you know that a/x^n tends to 0 as x->infinity, then surely you also know it tends to 0 as x->-infinity

so you know very well you aren't "supposed" to get a different answer, although I am not quite sure what you mean by "supposed" in this case

Topologicalife

05:40

Oh I was bugged.

user228700

@arctictern Yes, I do, but I'm doing a problem and this method isn't giving me the correct answer-I'm off by a negative sign :-|

arctic tern

you got the wrong answer => you did it wrong

Topologicalife

Is there more rooms of math chat with people? Or is this the 'main room'?

arctic tern

this is the main room on the SE network. the more advanced one is the homotopy theory one associated with mathoverflow. then there are #math channels on various IRC servers.

user228700

@arctictern I did the thing with dividing by the highest power of $x$ but since I'm getting the wrong answer, I dunno what I did wrong.

arctic tern

05:44

@Kaumudi how is anybody supposed to point out the error in your work without being able to see your work?

in any case, this should not be hard. if you divide by the highest power of x, all of the coefficients up top and down below are literally the same numbers

then all but the leading numbers go away

user228700

^ I was waiting to see if anybody wanted to see my work.

arctic tern

so you end up taking the ratio of what were originally the leading coefficients

just make sure that this happened in your work

user228700

OK, what if one of the leading coefficients is inside a sqrt?

arctic tern

then you're not taking the limit of a rational function like you originally stated...

user228700

It's not rational, okay. What steps should I follow in this case, then?

arctic tern

05:46

I don't know what "this case" is, so...

user228700

"this case"=the numerator is in a sqrt and doing the thing with the division isn't giving me the correct answer. Shall I upload the problem?

arctic tern

yes

vague descriptions are just wasting our time

user228700

arctic tern

okay. keep in mind sqrt() is always going to be positive

so (1/x)sqrt(blah) = - sqrt(blah / x^2) for negative values of x

notice the minus sign in front of sqrt on the right side of my equation

user228700

@arctictern Yeeah, that's what I got wrong. Thanks :-)

Null

06:32

@arctictern what is the solutionset of $a=a$? otherwise asked: does it even make sense here to talk about solution sets?

arctic tern

as long as you have a particular context you're working in (like real numbers, complex numbers, elements of some set), sure it makes sense.

Null

so $\frac{1}{x}=\frac{1}{x}$ for example

does this have in R all of R or R\{0} as a solutionset?

arctic tern

what do you think?

Null

i think whole R

arctic tern

does 1/0 make sense?

Null

06:39

no, but nonsense=nonsense

which is perfectly fine? :D

arctic tern

no

Null

well, then obv R\{0}..

arctic tern

yes

Null

but without a given set to work with, this is basicly unanswerable

as i might plumber me a set where 1/0 is defined..

Secret

Nonsense actually don't always equal to nonsense. Take the indeterminate form $\infty -\infty$ as example. While initially it seems the two symbols will cancel, you can, in fact, generate this indeterminate form via limits of some suitabley chosen functions f and g. Then depending on what f and g are, you can have this indeterminate form to equal to anything you want at the limit, hence why it is indetermiate

A simple example is to consider $f(x)=x$ and $g(x)=x^2$, then $f(x)-g(x)$ as $x\rightarrow \infty$ will give the above indeterminate form, but it is clear that as seen from the graph, there's a nonzero unbounded area between $f$ and $g$

Null

06:48

@Secret l'hoptial is not unheard of for me ;)

*l'hopital

so in $\mathbb{N}$ the following makes no sense either: $0.5=0.5$?

Secret

yes, as $0.5 \not \in \mathbb{N}$

in $\mathbb{N}$ you cannot even say anything about 0.5 because it is not an element of $\mathbb{N}$

to $\mathbb{N}$ 0.5 is just a symbol for some element that does not belong to it. It can potentially have any property it want as long it does not have all the properties that allow it to be in $\mathbb{N}$

Null

is there some crazy set where $a\not=a$?

and a is obv IN the set

Secret

Russell's paradox

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl, and other members of the University of Göttingen. According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself...

You will end up with a proper class, and not a set

Null

so, f(f(X)) will never make sense, at least in math?

(have read a little on that article)

crazy minds haha :)

Secret

07:04

I am not really sure what that means (because my logic is not as good), I think someone else is needed to help you to understand that

Null

i mean " a function cannont be argument of itself"

Secret

I know, but I don't understood his argument on how FF(x) can dispose of the russel paradox

Null

me neither to be frankly

i think the "List of all lists that do not contain themselves" makes more sense to me.

to make the problem clear

altho as its stated, its not directly Russel's

mmh

is $x\notin x$ the same as x does not equal to x?

and is the second x to be understood as a singleton?

Maks

08:50

Hello Darkness my old friend, I've come come to talk with you again...

Alessandro

09:07

Anyone familiar with filters? I want to prove that a nonprincipal ultrafilter can't have a countable basis, but I'm a bit at a loss on how to do so

Balarka Sen

hi Alessandro

@Maks good song

Alessandro

Hi @Balarka

Maks

Anyone knows how to use matlab ?

Alessandro

A bit

Maks

I want to plot 3d vectors in order to see them, if they are orthogonal parallel etc..

Alessandro

09:12

I don't know how to do 3D plots, sorry

Balarka Sen

I wish I had rigorously read the standard existence & uniqueness theorems for ODE's from somewhere.

The Substitute

Hello. I have an off-topic question for anyone familiar with mathematica. I'm taking a limit as $n\to \infty$ but the expression has a variable $p$ in it. How can I include the constraint $0<p<1$?

Alessandro

09:39

Are you still doing chemistry @Balarka?

DHMO

@Null the point is that division is not an operation per se. it is just a shorthand for the multiplication of the multiplicative inverse (a/b = a•b^(-1)) and the field axioms only require inverse to exist for non-zero element. In fact, in a field, zero cannot have an inverse. Therefore, x cannot be zero.

Balarka Sen

Nope, @Alessandro

DHMO

@BalarkaSen why not?

Balarka Sen

I ran out of my weekly chemistry quota.

DHMO

lolwut

Alessandro

09:46

What are you working on then?

Balarka Sen

Reading Milnor's Morse theory.

Alessandro

I just looked up what that is...

Maks

Hey , could use a little help here

I have the equation of the plane $ 3x + 3y + z = 1 $

That is a normal equation, and I need to express it as a parametric

I know the normal vector is (3,3,1)

Now I need to find two vectors and a point that fulfill the equation right ??

Balarka Sen

@Alessandro 'tis the cool stuff

SteamyRoot

10:03

@Maks You don't need the normal vector at all. You need a point on the plane and two non-collinear vectors in the plane

Or, equivalently, $3$ non-collinear points will also do

Alessandro

I trust you @Balarka! I'm still dealing with topology (ultrafilters and that kind of stuff in particular)

Maks

So I have to find the vectors so that $ 3x + 3y + z = 0 $ And a point that fulfills $ 3x + 3y + z = 1 $ ?

That's what I understood from the theory

And the vectors should be non-collinear as you said

Balarka Sen

@Alessandro I don't really know Morse theory, so you shouldn't trust me.

I don't really know about ultrafilters, etc

Secret

I cannot believe it, it seems I have at least 70% of the stuff about parametric planes in my memory have drifted away. Hopefully when I get back to differential geometry they will return and will stick permanently

SteamyRoot

@Maks yes

Maks

10:07

@SteamyRoot Is there some system of equation I can create to find the vectors and the point ? Or I have to do it manually ?

I want to justify from where do the vectors and the points come rather than just pointing that a specific vector fulfills the equation

SteamyRoot

@Maks I would simply find $3$ non-collinear points and use those to construct the vectors

Alessandro

Me neither @Balarka, but apparently they are useful in topology (the stone-cech compactification of N can be thougth of as the set of ultrafilters on N or so I read, but I have yet to get there)

SteamyRoot

Given the equation of your plane, $3x + 3y +z = 1$, you can easily find non-collinear points: $P = (0,0,1), Q = (1,0,-2), R = (0, 1, -2)$

Balarka Sen

@Alessandro Let me know when you get there, because I'd like to listen to that.

SteamyRoot

And then you consider the vectors, say, $\vec{PQ}$ and $\vec{PR}$

Alessandro

10:13

Sure

Maks

@SteamyRoot and in order to find the angle between to planes I have to do $ \dfrac {<n_1,n_2>} {|n_1||n_2|} $ ?

Being $n_1,n_2$ the normal vectors of the planes

The arccos of that*

SteamyRoot

Sounds about right, yeah

Fargle

G'morning chat.

Balarka Sen

Morning, @Fargle

Maks

And how can I find the dom and img of an f(x,y,z) ?

or an $ f : R -> R ^n $

DHMO

11:14

@Maks the domain is R by definition

what is the function?

@Null f(X) is a value not a function. Perhaps you meant f(f).

Ramanujan

11:32

@DHMO hi

DHMO

hi

Hey-men-whatsup

hi guys

DHMO

hi

Hey-men-whatsup

help me understanding limit

i1.wp.com/www.giface.com/wp-content/uploads/2016/06/102/…

Maks

Wish me luck ! If I pass I'll have to thank all of you for your help

Balarka Sen

11:42

@Krijn Hi there.

Krijn

@BalarkaSen Eyooooo!

Balarka Sen

sup man

Krijn

Doing lots of mathematics

Mostly number theory

Balarka Sen

cool beans

Krijn

Very cool

Hey-men-whatsup

11:45

cool

Krijn

You should some more algebra and number theory still

Balarka Sen

it's too hard for me

Krijn

:(

15 year old genius doing university mathematics quoted as saying "its too hard for me"

Balarka Sen

Wrong number there. I find "genius" to be an odd choice of word.

Krijn

Ahh, I missed your birthday?

Balarka Sen

11:55

Eg, does one really have to be of a specific age or a genius to do mathematics? Or rather, to like doing mathematics? (I personally think I am not really good at doing math, but I do love mathematics)

@Krijn heh, I suppose

Krijn

It depends.

But if we don't call people who do mathematics at a high level genius, then who can we still call genius

6

Balarka Sen

To me, you do the kind of mathematics I'll probably never be able to grok. Shouldn't that be considered mathematics at a high level?

Krijn

If someone at 16 told me I would be able to grok this today I would have declared him mad

So imagine what you are capable of in 5 or 6 years

Balarka Sen

@Krijn I don't think I'd be capable of learning algebraic geometry/number theory from the algebraic point of view, ever. I tried; as in actually tried to work through Hartshorne with a person. Had to switch to Shafarevich because I didn't like. So I say this from experience.

but we're better off discussing math instead, I suppose.

Krijn

@BalarkaSen Oh Hartshorne is hard as balls for me as well, so don't take that as an indication

Alessandro

12:10

Is it Hart-shorne or Harts-horne?

Krijn

But yeah, your better of doing what you're good at

Balarka Sen

i think the former

Krijn

I've always heard it as Hart-shorne

meow-mix

12:21

good morning

isnt reflexivity a consequence of both transitivity and symmetry?

Alessandro

No

meow-mix

that is, if x ~ y, then y ~ x by symmetry and thus x ~ x by transitivity

Balarka Sen

what if there is no other y?

Alessandro

Transitivity is stated for $3$ distinct elements

meow-mix

oh

Balarka Sen

12:25

@Alessandro Um, that's not right.

It works for any 3 elements.

The reason meow-mix's logic does not work is because there may not be any element related to x.

meow-mix

@BalarkaSen cant y just be x in that case?

oh nevermind

Balarka Sen

:)

Alessandro

Actually Balarka is right, I misremembered, sorry

meow-mix

what do you guys think about times new roman like fonts?

Krijn

They look like Times New Roman

meow-mix

12:32

i mean, for books

SteamyRoot

12:48

Comic sans should be mandatory

Krijn

Combined with wingdings if you are making a bluff argument in a proof

cfp

Would I have been better off asking this question in Math Overflow? math.stackexchange.com/questions/2015288/… If so, could a mod possibly move it there for me? (I'm not entirely sure of the process.)

Semiclassical

@cfp I'd say it's still better off here than MO, but it's closer than a lot of questions

cfp

OK, I'll be patient. Thanks.

Semiclassical

I think it's a nice quetion, though.

cfp

12:59

Thanks. Let's hope someone can answer it!

Transcript for

$Mathematics$ Mathematics