Mathematics - 2017-03-20 (page 1 of 5)

Ted Shifrin

00:00

With luck, Balarka is finally asleep.

Riemann-bitcoin.

so I need to prove that if $f: S^1 \longrightarrow S^1$ is a homeomorphism then deg(f)=1 or -1.

Ted Shifrin

What's your definition of degree?

Riemann-bitcoin.

If $f$ is homtopic to $n\theta$ then degree of $f$ is $n$

Ted Shifrin

So this is something about $f_*\colon H_1(S^1,\Bbb Z)\to H_1(S^1,\Bbb Z)$, right?

Riemann-bitcoin.

We have shown that all $f: S^1 \longrightarrow S^1$ has a associated $n$

Ted Shifrin

00:05

Well, you'll need to know how that behaves with regard to composition of functions.

Riemann-bitcoin.

And this is because of what ^^ @TedShifrin

Ted Shifrin

Because of what I said about the induced map on homology. I have no idea what you know and what you don't. It's hard to do exercises out of context.

But presumably you have to use $f$ and $f^{-1}$ somehow if $f$ is a homeomorphism.

Riemann-bitcoin.

^^ thats fine thanks for the hint I think I can do something with that

Ted Shifrin

OK ... have fun !

Adeek

so it is a functor

Simple

00:11

In numerical analysis, what is the formal definition of tolerance

Dragneel

01:05

$a, b, q, r \in$ integers such that $a=bq+r$. Prove or disprove $gcd(b,r)=gcd(a,q)$.

Here's what I did to prove it, and I would like someone's opinion:
Let $d=gcd(b,r)$. Then $d$ divides $b$ and $r$. So, $b+r=d(x+y)$.
Similarily, let $E=gcd(a,q)$. Then $E$ divides $a$ and $q$. So $a+q=E(j+k)$.
Putting both equations together, we get $d(x+y)=E(j+k)$.

Bernardo Meurer

01:15

Howdy

Quick question, is there a simple way to find an upper bound to the number of primes $\le x$?

Meow Mix

Hi @Akiva

@BernardoMeurer As in, the Euler totient function?

No, wait

Bernardo Meurer

@MeowMix No, as in just primes of absolute value $<= x$, $x$ some constant

Meow Mix

I'm thinking of a different function

Daminark

My mom today: "You're starting to look kinda Japanese... Maybe that's why you're good at math!"

Meow Mix

Approximately $x/\ln(x)$

Bernardo Meurer

01:17

That's not an upper bound

Meow Mix

$1.25506n / \ln(n)$ according to Mathworld

is an upper bound

Bernardo Meurer

Yep, I think I got it to work

Thanks mate!

Akiva Weinberger

@Daminark You're adopted

(jk)

Daminark

The consensus we came to was more that I was switched

Meow Mix

when will I get my first physical textbook

@BernardoMeurer No problem.

Constructing a basis is really tedious.

Oh and, hey @Akiva

Daminark

01:29

Soon, and then when you find how ridiculously expensive they are you will jump back to the pdf camp immediately

Jk some people very much prefer physical textbooks

The typical solution in that case is to buy the Indian editions, which are like, less than $20 in many cases

Many people I know did that for Hoffman-Kunze and Rudin

Tim The Enchanter

@Dragneel I don't think the statement needs to be true, for example if $r=0$, i.e. $q$ divides $a$. You have $gcd(a,q)=q$ and $gcd(b,r)=b$ which need not be equal.

Meow Mix

I'm actually using a rubik's cube to visualize the symmetries of a cube for one of Ted's exercises.

Adeek

02:07

@Daminark hi

Meow Mix

Hi @Adeek

Daminark

How's it going @Adeek?

Adeek

@Daminark would you like to check my argument for something ?

hi @MeowMix

Daminark

I can definitely try! :)

Adeek

good @Daminark solving some stuff in commutative algebra.

Suppose $J_{\alpha}$ is prime ideal chains. Ordered as follows $\alpha_1 \leq \alpha_2$ iff $J_{\alpha_2} \subset J_{\alpha_1}$.

Let $M = \bigcap_{\alpha \in O} J_{\alpha}$.Suppose $xy \in M$ and $y \notin M$. Pick arbitrarily $\alpha_1,\alpha_2 \in O$. Then by hypothesis $xy \in J_{alpha_1}$ and $xy \in J_{\alpha_2}$. Assume that $\alpha_1 \leq \alpha_2$. Then, by hypothesis $J_{\alpha_2} \subset J_{\alpha_2}$. Since $J_{\alpha_2}$ is prime so we have $x \in J_{\alpha_2}$ so $x \in J_{alpha_2}$. If we have $\alpha_2 \leq \alpha_1$ then by the symmetry of argument we can conclude $x \in J_{\alpha_1}$.

As $\alpha_1$ and $\alpha_2$ were arbitrarily, so $y \in M$ and thus M is prime.

@Daminark any critique ?

Meow Mix

02:16

I got Ted-cercises 15 through 19 completed :D

Adeek

@Daminark here ?

@MeowMix good

Daminark

02:51

Hey I'm back

@Adeek

Adeek

nvm @Daminark I noticed a mistake

I made the argument better now

Suppose $xy \in M$ such that $y \notin M$. Then $xy \in J_{\alpha} \forall \alpha$ and there exists $\beta \in O: \ y \notin J_{\beta} \ \ and \ xy \in J_{\beta}$.

Daminark

So, I'm only able to comment from the point of view of someone who knows nothing about commutative algebra, but this seems about right.

Well, at least no gaping holes

Adeek

If $y \in O$ is arbitrarily. As $(J_\alpha)_{\alpha \in Q}$ is a a chain. $\beta \leq y$ or $y \leq \beta$. Assume $\beta \leq y$ then $J_y \subset J_{\beta}$. Since $y \notin J_{\beta}$. Then $y \notin J_y$ since $xy \in J_y$ and $y \notin J_y$ so $x \in J_y$ as $J_y$ is prime. If $y \leq \beta$ then $xy \in J_\beta$ so $x \in J_y$. As y was arbitrarily x $\in M$.

@Daminark I guess my original argument works but I think this is more intuitive.

Daminark

03:11

I can see this better

Forever Mozart

is there a perfect length for a math article?

3-10 pages?

Daminark

I wouldn't say so, it's very context dependent

Adeek

yeah @Daminark brb making food

Daminark

Neat, see you in a bit!

Adeek

03:32

back time :D. I can't wait to understand more and more of commutative algebra.

Daminark

The subject looks pretty fantastic

The way things are in my school, the algebra sequence contains some amount of commutative stuff in second quarter

But that there's a separate class on it as well

So I'll get some next year, and in 4th year, either by way of the commutative algebra class or within second quarter of grad algebra, I'll get a more proper treatment of the stuff

Adeek

Yeah I want to understand many things in it so in the summer I can read algebraic geometry properly.

Planning to go through Hartshrone during the summer @Daminark

Daminark

Oh, yeah so a few people were talking about Hartshrone some time ago, can't remember if you were in that convo

I was wondering what it was all about so I read it

(I'm in this shared Google drive folder with t o n s of textbook pdfs)

Adeek

Hartshrone or the convo ?

Daminark

And I did not get far

Hartshrone

Adeek

03:37

it is probably convo between me and ali I am not sure

@Daminark Hartshrone is a algebraic geometry book. I guess it is the standard.

Daminark

Yeah, I have realized that

But it was harsh

Adeek

Yeah it is not for the faint of heart.

Daminark

I'll hold off on that until I actually learn algebra

Also commutative algebra

Adeek

Yeah you should this summer. I will dedicate it to geometry + algebra.

Daminark

This spring break I'm doing part 2 of a crash course on group theory because the hope is to take group algorithms next quarter

Adeek

03:43

nice

Daminark

First round was in an REU last summer because the plan was to write a paper on the Sylow theorems

But I didn't have any time to dedicate for basically the first 5 weeks because Laci's linear algebra/combo homework was heavy

So I had to blast through groups really quickly in order to get to group actions and Sylow

Now, I've got about a week to just go through DF

Laci said in particular to learn about equivalent definitions of solvable groups, nilpotence of p-groups, and Jordan-Holder

This summer I've got an analysis bootcamp going on

Adeek

very cool.

Daminark

We're doing diffgeo of curves/surfaces (Ted's book), complex analysis, probability, and dynamics

Next year is gonna be when algebra dominates more

Adeek

what type of complex analysis ?

Daminark

Introductory stuff, it's using Titchmarsh's Theory of Functions

Adeek

03:49

I see.

Daminark

Aside from the complex, this will luckily be mostly non-redundant

Like, I'm not likely to take probability or dynamics (the latter is in the stats department and is done in a way I wouldn't really like)

And I don't believe we have a class on curves/surfaces

Like, our only geotop classes are point-set topology, algebraic topology, and smooth manifolds for undergrad

And the grad sequence is algebraic topology first quarter, differential topology second, and Riemannian geometry third

Adeek

@Daminark do you have an idea which grad school you would like to join ?

Daminark

I haven't thought much about that yet

Whichever one takes me

Adeek

which year are you in again ?

Daminark

2nd year

Adeek

03:55

cool

Daminark

Where do you go?

Adeek

at university of alberta

The algebra/geometry department here is strong

Daminark

Nice

I imagine I'd probably be pushing more in that direction anyway

Not much of an epsilonics fan :P

Adeek

haha me neither

Daminark

This quarter in analysis, we started with a couple topics in linear algebra we didn't do in last quarter (spectral theorem, polar and singular value decomposition), then did an intro to differential forms, ODEs, manifolds, and then Banach/Hilbert spaces

In class, everything is nice and shiny, but I found that the problem sets I enjoyed most were those on forms and manifolds

As well as the first pset on linalg

Adeek

04:01

yeah

Daminark

In grad school, how much time do you tend to spend taking classes and all?

Adeek

yeah

Daminark

Like I definitely want to get the breadth of coverage before narrowing down to whatever I'll be specializing in

Adeek

yeah that is good idea

Daminark

So right now you're doing commutative algebra and what else?

Adeek

04:11

@Daminark we are gonna touch on algebraic geometry intro soon and in topology we finished with higher homotopy groups we are gonna start covering space theory tomorrow.

In my third class which is reading course I am reading some publication and gonna present about it soon.

Daminark

Nice!

Adeek

04:23

@arctictern I was wondering if you could explain to me why is the isomorphism theorems imply that the map factors ?math.stackexchange.com/questions/2188886/…

Daminark

Hey @Semi!

arctic tern

@Adeek You mean, given any homomorphism $M_1\to M_3$, there is a corresponding faithful homomorphism $M_1/\ker\to M_3$ for which $M_1\to M_3$ factors as $M_1\to M_1/\ker\to M_3$?

Simple

suppose that for $n\geq 1$, $X_n$ is uniformly distributed on $\{0,1,2,\dots,n\}$. Show $\lim\limits_{n\to\infty}P(\frac{X_n}{n}\leq y)=y$ for $y\in(0,1)$

Adeek

yeah ?

yeah @arctictern ? I just want to know why is this true ?

arctic tern

what I just said is pretty to easy to verify, and is more or less one version of how the isomorphism theorem is stated

Adeek

04:41

oh right @arctictern the way you phrase it this way it is trivial. Here is the argument Suppose $\psi : M_1 \rightarrow M_3$ and $\phi : M_2 \rightarrow M_3$ with $\phi$ surjective and $ker(\phi) \subset ker(\psi)$. Then define $\eta : M_1 / ker(\phi) \rightarrow M_3$ as follows $\eta([x]) = \psi(x)$ this is well defined by the fact $ker(\phi) \subset ker(\psi)$

arctic tern

yes

copper.hat

anyone remember the name of the theorem that says $[f'(a),f'(b)] \subset f'([a,b]) $?

Daminark

I think this is just referred to as the derivative satisfying the intermediate value property?

copper.hat

@Daminark: Thanks, that is what I needed, its Darboux's theorem.

Much appreciated.

Good night.

Daminark

Neat

And good night!

CausingUnderflowsEverywhere

04:54

Am I forgetting something or is there a way to say that the limit doesnt exist or break that down into 2 different limits?

Hey arctic! :)

2017

@CausingUnderflowsEverywhere It should just return f(P(A)). All polynomials are continuous functions.

@CausingUnderflowsEverywhere Why do you think the limit doesn't exist ?

Daminark

I mean the problem is that $f$ might be discontinuous at $P(A)$

2017

@Daminark Oh right. There is too less information...

CausingUnderflowsEverywhere

Cause im dumb

2017

Wut? ^

CausingUnderflowsEverywhere

05:01

Why does it return f(P(A)) insetad of P(A)?

Function of function of A?

2017

@CausingUnderflowsEverywhere It is a composite function...

khanacademy.org/math/ap-calculus-ab/continuity-ab/…

This has a good explanation...

CausingUnderflowsEverywhere

Ohhh cause f(p(x) is part of the question right

Akiva Weinberger

Too much history work :(

I want to go back to thinking about math but I have so much history to do for this project

CausingUnderflowsEverywhere

History is in the past just forget about it and move on to the future where you become a mathematician

2017

@AkivaWeinberger What type of project?

I am happy that I no more have to study history. History classes bored the hell out of me till 10 th grade :-P

Actually, I don't hate history. I hate the way they teach history in schools!

Mug mug mug mug mug mug uuuuuuupppppppppp...lol XD

Akiva Weinberger

05:15

I have an essay due in a few months, but for Tuesday I need to hand in like seventy notecards (each containing a piece of relevant information gathered from the books and articles I've been reading) so that they know I've been doing all my research

and seventy note cards is a huge amount!

2017

On what topic is the project?

Akiva Weinberger

The annexation of Texas

Well, my project, anyway

2017

wow...70 is huge....hire somebody to do it :-P

I know that feeling man, I remember struggling to make my history projects in 10 th grade on the world war

At the end...me and some of my friends collaborated and copied bits and pieces of each other's work...:D

The teacher never even realized

Anyhow, good luck :-)

Fargle

05:50

@AkivaWeinberger That seems...excessive.

Christian

06:11

Hey guys, I have a question regarding signals and systems.
Let's say I have b_k = a_k H(e^jkw0)
does that hold only for frequency response,
or can that be used in time domain also?

SoumyoB

06:49

@AkivaWeinberger wait are you telling you've got to annex Texas as your project? :o

good luck getting the required military and political support!

I'm ready to support you with finances a bit if you agree to let me become the Prime Minister or Vice President if we win

ALannister

0

Q: For a ring $R$ with a single proper ideal $I$, show that $I$ is prime

Let $R$ be a ring with a single proper ideal $I$. I need to prove that $I$ is prime, but all I have at my disposal to do it with are the definitions of prime and maximal ideals as stated below: An ideal $I$ of a ring $R$ is prime if $I \neq R$ and the following condition is satisfied: for all i...

I hope somebody answers that before the morning.

Ali Caglayan

07:21

@Daminark Hartshorne is by no means a starting book in Algebraic geometry. Try some lighter reading like Reids Undergraduate Algebraic Geometry if you want to get a small flavour of the subject.

And books like Shafervich introduce the classical viewpoint quite nicely whilst working with the commutative algebra you would have developed.

But Hartshorne is technical and it knows it, its really to get into the good bits.

Daminark

I see

Ali Caglayan

Algebraic Geometry is essentially the realisation that parts of algebra and parts of geoemtry are actually the same thing. So when you do stuff in one part, you automatically do stuff in the other.

It opens up very beautiful relationships between structures

And is highly applicable if you are interested in that. Especially in number theory

Daminark

Well there's a lot to look forward to then

SteamyRoot

07:36

I always felt like Hartshorne was the perfect book to use once you've already learned most of what it's in there

But for a first course in Algebraic Geometry, it's rather awful.

Daminark

I can see that

I'll say it's kinda inconvenient that we have commutative algebra in the winter and algebraic geometry in the fall

SteamyRoot

Yeah... that definitely sounds like the opposite order

Daminark

The undergrad algebraic geometry class apparently doesn't rest on commutative algebra, but it'd prob make more sense if they swapped the two quarters and made commutative algebra a prereq, building the algebraic geometry course around it more

They do build some of what they need in class

affine and projective varieties; coordinate rings; the Zariski topology; Nullstellensatz; Hilbert basis Theorem; the dictionary between algebraic geometry and commutative algebra; rational functions and morphisms; smoothness; theory of dimension

From the description

SteamyRoot

07:51

I didn't have an algebraic geometry course as undergrad... First term of master's had Commutative Algebra, second term had Algebraic Geometry; the latter really building on the former.

The things you describe are pretty much the first half of my algebraic geometry course :P

onderwijsaanbod.kuleuven.be/syllabi/e/…

Daminark

Lmao, maybe it's because it's somewhat lighter that the prereqs aren't too high

Full year of analysis + 2 quarters of algebra, recommended the full year of algebra + complex + point-set

In the grad case, the year of algebra starts with rep theory in the fall, commutative algebra/algebraic geometry winter, and general topics in the spring

Niclas Jonsson

08:11

What's the meaning of "priceout" in the context of linear programs: "In each iteration of the simplex method we look for a non-basic variable to price
out and enterthe basis".

user129412

10:46

Stupid question. I have $\int_{-\infty}^{E_c - E_g}dE \ \sqrt{E-E_c -E_g} \ (1-H(eV-E))$ with H a heavyside step function (its 1 up to eV and 0 after). But I'm not entirely sure how to evaluate an integral of the step function. Intuitively I'd say I get $\frac{2}{3} (eV-E_c - E_g)^{3/2}$ which is only valid for $eV > E_c - E_g$?

Sha Vuklia

Anyone here who could help me with the following problem?: math.stackexchange.com/questions/2194830/…

Martin Sleziak

Somebody with some free time to spare and willing to give some basic explanation about ideals to this user? chat.stackexchange.com/transcript/message/36152002#36152002

SteamyRoot

@ShaVuklia Your question starts with "Let $m,n \in \mathbb{Z}$, $n>0$ and $d=\gcd(m,n)$." I don't see why you still try to prove that $d$ is the gcd of $m$ and $n$, then?

Sha Vuklia

11:01

yea that is so confusing, I know

I should have used a different symbol

I am introducing a new $d$

and later on I would like to show that this is the same $d$ they're talking about

but maybe I should have written $k$

and then show that $k=d$

I'll edit it

Learning user

Hi

can anyone answer this question

0

Q: Mixture and Allegation Problems

A 10 litre container holds a mixture of water and sugar, the volume of sugar being 15% of total volume. A few litres of the mixture is released and an equal amount of water is added. Then the same amount of the mixture as before is released and replaced with water for a second time. As a resul...

algebra-precalculus

SteamyRoot

@ShaVuklia Do you know the concept of a generator of a group?

Because, if I read your question correctly, you have the subgroup $\langle \bar{m} \rangle \leq \mathbb{Z}_n$, and you need to show it's equal to the subgroup $X = \langle \bar{d} \rangle$.

Sha Vuklia

11:16

officially, no

I've read it because it was mentioned as an answer

but I can work with it I guess

@SteamyRoot

SteamyRoot

Well, for a cyclic group it's easy enough, it's just all "multiples" (including negatives)

So for your question you just have to show that $\langle \bar{m} \rangle \leq \langle \bar{d} \rangle$ and vice versa

Sha Vuklia

the generator of a group is just the set of elements that give us the entire group right

SteamyRoot

Yeah

Sha Vuklia

what do you mean by $\leq$ though?

SteamyRoot

Subgroup

Sha Vuklia

11:18

ah ok

SteamyRoot

So it suffices to prove that $\bar{d} \in \langle \bar{m} \rangle$ and vice versa

Sha Vuklia

ok let me see, I need some time because this notation is new to me:p

SteamyRoot

I'll just finish typing since my lunch break is, well, now

Sha Vuklia

ah sure

SteamyRoot

For this, use bezout's identity (and then take it mod n)

the other way around, well, m is a multiple of d so that's rather trivial

Sha Vuklia

11:20

ahhh yea, I think I'm slowly getting it now

oh this is so easy now:l

why have I been struggling with this for an hour? XD

haha but thanks a lot, now I can finally move on :P

Parth Kohli

hey guys I made another problem up

this one is a little less obscure and probably has been solved but I don't know what to call it

never mind. it's the simplest problem ever.

Sha Vuklia

hahaha, ^ mathematics in a nutshell :P

Mary Star

Hello!!

Let $u, v\in \mathbb{R}^n$. I want to show that the vectors $a=u+v$, $b=u-2v$, $c=v$ are always linearly dependent.

We have that $$\lambda_1 a+\lambda_2 b+\lambda_3 c= 0 \\ \Rightarrow \lambda_1\left ( u+ v\right )+\lambda_2\left (u-2v\right )+\lambda_3v=0 \\ \Rightarrow \lambda_1 u+\lambda_1 v+\lambda_2 u-2\lambda_2 v+\lambda_3 v=0 \\ \Rightarrow \left (\lambda_1+\lambda_2\right )u+\left (\lambda_1-2\lambda_2+\lambda_3\right )v=0$$

How could we continue? Do we get from here that the two coefficients must be equal to $0$ ?

Parth Kohli

11:36

@MaryStar Wait, you have to show that they're linearly dependent, not independent.

And they're clearly linearly dependent because $a - b - c = 0$.

Sha Vuklia

no

$b-a-c$ right

Parth Kohli

ehhh sorry

b - a + 3c

Sha Vuklia

oh yea

you're right :P

Parth Kohli

oh boy i kinda feel proud about my latest problem now

Mary Star

Ah ok. Thank you very much!! :-)

Sha Vuklia

11:41

haha what is the problem? (I don't know if I'll be able to follow it though)

Parth Kohli

oh, this one.

Sha Vuklia

oh, combinatorics

I'll skip that one XD haha

I'm immersed in algebra now :P

Parth Kohli

someone added the ramsey-theory tag and I don't even know what that is

Sha Vuklia

hahahaha

Alessandro Codenotti

It does have a Ramsey theoretic flavour

Ramsey theory deals with problems like "how big must this structure be for some pattern to be present" if you want a very broad and unprecise description

Parth Kohli

11:47

ah, I see

Akiva Weinberger

I realized that some of the literature says an arc is wild if there's no ambient homeomorphism of it to the standard unit interval, and other parts of the literature say it's wild if there's no ambient isotopy of it to the standard unit interval. Are these equivalent?

Maybe "equivalent up to ambient isotopy" and "equivalent up to ambient orientation-preserving homeomorphism" are the same in $\Bbb R^n$.

LegionMammal978

Hmm

If I'm doing this right, then the structure $(S,+\!+)$ of strings and string concatenation should be a monoid but not a group

Because it's closed, associative, and has an identity, but no inverse elements in general

Right?

Mary Star

We have the system $$2x_1+ x_2 + ax_3=0 \\ -2x_1+(4-2a)x_2+(a-5)x_3=0 \\ 4x_1+(3a-4)x_2+2x_3=0$$ and I want to find for what $a$ the system has infinitely many solutions.

By the Gauss Algorithm we get $$\begin{bmatrix}\begin{matrix}
\ 2 & 1 & a\\
-2 & (4-2a) & (a-5)\\
\ 4 & (3a-4) & 2 \end{matrix}\left|\begin{matrix}0\\ 0\\ 0\end{matrix}\right.\end{bmatrix}\longrightarrow \begin{bmatrix}\begin{matrix}
2 & 1 & a\\
0 & (5-2a) & (2a-5)\\
0 & (3a-6) & (2-2a) \end{matrix}\left|\begin{matrix}0\\ 0\\ 0

samjoe

12:08

Hello everyone! I have a question regarding Binary operation: is * defined as a*b = a - b + ab commutative or associative and find inverse and identity.

I did the question, it neither was comm, nor associative. now if it is not commutative, it wont have identity element right?

Parth Kohli

not necessarily

@samjoe on an unrelated note, are you from India?

samjoe

hey yes! today was my boards paper!

Parth Kohli

same here

samjoe

@ParthKohli you are in 12?

Parth Kohli

yes sir

samjoe

12:10

@ParthKohli This question was on paper, set 1 a six mark question.

Parth Kohli

Delhi?

samjoe

yeah!

you guessed it so right lol!

Parth Kohli

Great, because I don't recall a question like this in outside Delhi, and I got scared for a moment.

samjoe

haha happens with me too! Where are you from?

Parth Kohli

Gurgaon

samjoe

12:12

oh! But for identity I to exist, we have aI = Ia = 1

sorry a * I = I * a = 1

Parth Kohli

Yes, correct. In this case, the property doesn't hold.

There is no identity element indeed.

samjoe

Yeah very nice!

Parth Kohli

I had a question in my paper where the binary operation wasn't commutative, and yet there was an identity element.

samjoe

So no inverse either right? As A * A' = I and no indentity

how can that be? @ParthKohli

Parth Kohli

(a, b) * (c, d) = (ac, b + ad)
the identity element is (1, 0)

I hope your question wasn't a misprint. I've normally seen that problem stated in the form a * b = a + b + ab.

samjoe

12:18

yes it is a * b = a - b + ab, checked it again!

Parth Kohli

oh boy... did you read that wrong in the exam?

samjoe

nope its a - b only!

Romantic Electron

0

A: Division of a negative number by a positive number

On a number line all addition operations in multiplication should go in the same direction.The remainder is what's left to go in that direction not what you have already left behind. To me the first answer is acceptable and the second is not.Because that will be like going in one direction and th...

Is anyone sure about the answer to the above question?

Sorry for posting the link to my answer about which I am uncertain.Here's the question again
http://math.stackexchange.com/questions/2194820/division-of-a-negative-number-by-a-positive-number/2194864

samjoe

Sorry Parth but how do we evaluate identity in your question? I am not good in maths .. so only know basic binary operation (ie RxR -> R), yours is i guess ((RxR)x(RxR) -> RxR)

Parth Kohli

the question says (QxQ) * (QxQ) -> QxQ but that doesn't make much of a difference

Just do the same thing you do in R x R -> R

a*e = e*a = a.

samjoe

12:25

but i am stuck , i got its not commutative. but how to find identity?

do we do (a,b) * (I1, I2) = (1,1)

?

Parth Kohli

we do (a, b) * (I1, I2) = (a, b)

and then check if that also holds true for (I1, I2) * (a, b) = (a, b), which it fortunately does

samjoe

aaaaaah

i am so stupid

thanks.

Parth Kohli

a fun fact was that it's associative

samjoe

hah fortunately lets hope :P

Parth Kohli

@samjoe how much are you expecting?

Balarka Sen

12:30

Whew. Math done.

Parth Kohli

@BalarkaSen How was it?

samjoe

@ParthKohli paper was fairly easy. i guess saying 90+ would be safe. You? I got it now, the point that you made! lacking commutativity doesn't imply absence of identity, as is case with DIVISION.

Balarka Sen

I think I messed up a coordinate geometry a little bit, but should still get marks for it because I did the other bit of it right. Other than that, fine.

Parth Kohli

@samjoe Haha, 1 is unfortunately not an identity in division though.

But yes, you're getting close.

samjoe

coordinate geometry?

Parth Kohli

12:31

he's in a state board

samjoe

Yea except that!

Balarka Sen

I am also not talking about Higher Secondary exams.

Alessandro Codenotti

Good job @Balarka, how many exams do you have left?

Balarka Sen

This is 11 to 12

@Alessandro Three more.

samjoe

oh sorry! they have tougher papers I have heard. This years paper was very easy but lengthy.

@BalarkaSen Yea right in 11 we had coordinate geom!

Balarka Sen

12:34

Ah, so we don't have coordinate in 12th grade?

Parth Kohli

CBSE doesn't.

samjoe

@BalarkaSen We are lucky haha!

Balarka Sen

Gotcha. Interesting. Is this, like, all filled with differential/integral calculus?

@samjoe Yeah... sometimes coordinate is a pain in the neck.

Parth Kohli

Yeah, with a little bit of matrices, determinants, vector algebra and 3D geometry.

samjoe

Hmm some 3D stuff using vectors!

yea and others.

Mahmoud

12:35

Hello

Balarka Sen

Very nice.

samjoe

much more interesting than coordinate geometry, imo

@ParthKohli Hey how was your paper :)

Parth Kohli

Just hoping I didn't make any calculation errors.

samjoe

Hmm I guess you score full, because most of my classmates knew some mistakes or other that they made. best of luck for other exams!

@BalarkaSen All the best to you too!

Mahmoud

Can anyone who has studied Math in french tell me what do we call the term ''Dénombrement'' in English ? The symbol of ''Le cardianl d'un ensemble $E$'' is $\operatorname{card}(E)$

It's basic Set theory, but I couldn't find the terms to start googling.

Balarka Sen

12:39

Thanks, @samjoe. I suppose you and @Parth are taking HS? Best of luck.

samjoe

@Mahmoud Have studied french, have studied maths, but not math in french lol.

@BalarkaSen Hmm

Mahmoud

LOL, NeverMind then @samjoe.

Parth Kohli

@BalarkaSen Thank you!

@samjoe plans for chemistry?

Tim The Enchanter

@Mahmoud Google translate says its 'enumeration'

translate.google.co.in/translate?hl=en&sl=fr&u=https://…

Mahmoud

@TimTheEnchanter Hmm thank you !

Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends...

That is, a much more convenient way to name a Mathematical term, I think.

And a much nicer notation too, $|S|$

Sha Vuklia

12:56

Show that there is a groupsmorphism $g\colon(\mathbb Z/n\mathbb Z)^*\to(\mathbb Z/d\mathbb Z)^*\colon a\mod n\mapsto a\mod d$, for all $a\in\mathbb Z$ for which $\gcd(a,n)=1$.
Why do we need $\gcd(a,n)=1$? Let $\overline a=a\mod n$ and $\tilde a=a\mod d$. We have $g(\overline a\cdot\overline b)=g(\overline{a\cdot b})=\widetilde{a\cdot b}=\tilde a\cdot\tilde b=g(\overline a)\cdot g(\overline b)$.

Akiva Weinberger

@ShaVuklia Are there any restrictions on $d$ and $n$?

Is $d$ a factor of $n$?

SteamyRoot

If it's not coprime to n, then it's not in the unit group at all?

Sha Vuklia

@Akiva Oh right, I forgot to add that, but indeed, we have that $d\mid n$

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$Mathematics$ Mathematics