/mg/ - Math General

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>>9086149
Is it true that a believing heart is your mathematics?

>>9076050

Continuing from last thread, how does one preform an induction proof on the 'real' numbers.

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>>9086157
Any property you would like to "prove" about the "reals" already holds if they "exist" in your system.

Frank Stenger, a mathematician from the University of Utah, claims to have proved the Riemann Hypothesis.

https://arxiv.org/abs/1708.01209

>>9086172
Hey there Frank.

>>9086159
this isn't about proving properties about the reals, but proving statements that are generalized over the reals.

>>9086174
As soon as you have the "reals" on the left hand side of an implication, you're done with the proof.

>>9086157
What do you mean? Prove something is true for all reals in a certain range by proving it's true for a single number, then proving it true for a series of numbers derived from that number?

That's impossible using standard induction. All Cross-Products of a countably infinite set are themselves countably infinite and thus can never reach Aleph-1 magnitude. You would have to find some clever way to subdivide the range of real numbers you're proving over into a countably infinite mapping (for example, prove that if something is true for x, it's also true for all numbers within a certain range of x and then prove that the range will tend towards covering the entire range you're proving over).

>>9086184
>That's impossible using standard induction
yes, I'm aware. suppose I wanted to prove that [math]\forall \alpha \in \mathbb{R}[/math]

[eqn]\frac{d^{\alpha}}{dx^{\alpha}}\left[e^{x}\right] = \sum_{n=\alpha}^{\infty}\frac{x^{n-\alpha}}{\Gamma\left(n-\alpha+1\right)} = e^{x} [/eqn]

this is what I'm struggling with.

>>9086188
shit, [math]\forall \alpha \geq 0[/math] **

they call them "reals" but they aren't
really makes you think

why the fuck is the Collatz conjecture so fucking difficult?

"Mathematics may not be ready for such problems." -Erdős

I wonder if we'll be able to ever have a thread where no one will fall for that retard's "what axiom do I need to disprove X" bait, and similar.
>the real numbers aren't real
^this right here, this is what cancer memes look like.

>>9086236
Part of what makes it hard is that it doesn't really directly belong to any field, so good ideas for even starting on it are scarce right now.
A concrete problem is that relating the factors of 3n+1 to n (which determines how much dividing by 2 will shrink the number) is extremely hard because prime factorizations are little fuckers when you start adding numbers to them.

>>9086149
It's like the z transform

EEs are brainlets

>>9086236
It is just one of those problems that shows how little we actually know about numbers. We have to think of this in relation to Fermat's Last Theorem. At its surface Fermat's Last Theorem looks a theorem about numbers but we found out that actually it is a theorem about elliptic curves. There is no way to crack Fermat without first cracking elliptic curves. Similarly there must be some unknown mathematical object that actually determines the behavior of Collatz' sequence and I think that is what Paul meant when he said the quote you posted.

>>9086296
fug. sucks to be a brainlet, this stuff really bakes my almonds.

>>9086188
I'm not really sure what summing using a non-Integer starting point means. I guess that's what you're asking. Maybe it means do a, a+1, a+2, ..... But then why write it like that? Why not write it as Sum from n=0 to Infinity and replace n with n+a? Which would just get rid of the -a's.

I think there's something wrong with the way that problem is written.

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>>9086188
>>9086193
You cannot use induction on the real numbers because [math] \mathbb{R} [/math] is not well-ordered (and you cannot define a well-order on it that is compatible with the natural ordering of real numbers) and no real number has a "successor" (and you cannot define a successor function that is compatible with all the other properties of the real numbers).
Seriously, fuck off with these stupid questions.
>>>/sci/sqt

>>9086236
Because it's a halting problem.

>>9086316
>sqt
they couldn't give a serious answer if you paid them you anime posting degenerate loser.

>>9086323
>they couldn't give a serious answer if you paid them
I very much doubt that. Either way, this sort of stupid shit doesn't belong here.

>>9086188
The exponential isn't an eigenfunction of the fractional derivative, these are though
https://en.wikipedia.org/wiki/Mittag-Leffler_function

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>>9086172
can a non-brainlet please peep this and tell me if its legit or bs

>>9086486
It's just the series expansion of sin(x)
https://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions

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>>9086493
please be joking

i was talking about the paper in the link i was replying to: >>9086172

>>9086149
what's the best number theory book for complete understanding?

>>9086501
It doesn't look right to me but I could barely get any sleep last night and I'm feeling pretty groggy so I didn't read it in detail.

>>9086509
No one has complete understanding of number theory, and consequently, no book that imparts complete understanding of number theory exists.

>>9086526
>>9086509
Also, >>>/sci/sqt

>>9086188
>>9086193

> real valued index of summation
> real valued degree of the derivative operator

If this problem made sense, you could solve it for a in terms of the other variables. Is that even possible? I don't know

>>9086501
In fairness, it wasn't that good of a joke to begin with. As for the paper I skimmed it and it looks to be the same as most "proofs" of the riemann hypothesis, I haven't looked at it closely but I think there are some issues with in some of the arguments about applying the fourier transforms tom some J operator don't seem rigorous to me

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w-what

>>9086651
What's the matter brainlet? You can't do a simple factorisation?

>>9086188
your notation is shit senpai

summing from n = a, then everywhere you use (n - a)? you can simplify it and write it in terms of standard summation notation.

you dont even need induction to prove that.

[eqn]\sum_{n=\alpha}^{\infty}\frac{x^{n-\alpha}}{\Gamma\left(n-\alpha+1\right)} = \sum_{n=0}^{\infty}\frac{x^{n}}{\Gamma\left(n+1\right)} [/eqn]

>>9086651
[eqn]\frac{e^x}{(1 + e^{x})^2} = \frac{e^x}{1 + e^{x}}\frac{1}{1 + e^{x}} = \frac{e^x}{1 + e^{x}}\frac{1 + e^x - e^x}{1 + e^{x}} = \frac{e^x}{1 + e^{x}}\frac{1 + e^x - e^x}{1 + e^{x}} = \frac{e^x}{1 + e^{x}}\left( \frac{1 + e^x}{1 + e^{x}} - \frac{e^x}{1 + e^{x}}\right) = \frac{e^x}{1 + e^{x}}\left( 1- \frac{e^x}{1 + e^{x}}\right) [/eqn]

>>9086316
slut

>>9086449
The "type" of naturals doesn't exist either though.

>>9086822
based anon, thanks.

What are the best available foundations which don't believe in fairy tales such as the "real" numbers and other similar fabrications?

>>9086188
>I wanted to prove that [math]\forall \alpha ∈ \emptyset[/math]
That's pretty simple actually.

>>9086895
brainlet

>>9086900
Excuse me?

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>>9086902
You're excused.

>>9086856
>>9086869
>>9086895
>>9086907
Kill yourself and rid us of your shit meme.
Fucking crank.

>>9086997
I don't expect you to understand.

why did we pick such a shit topology to be the "standard" topology on R

>>9087047
Any topology on "[math]\mathbb{R}[/math]" will be shit.

Can most rational numbers even be shown to exist? I'm starting to doubt the existence of most of them.

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Let's suppose I believe in the existence of the naturals but not the integers. How should I go about formalizing this?

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>>9087082
You'd need to refute the existence of Grothendieck groups.

>>9087093
Hm... This indeed seems like an obstacle. Can I at least "stop" at the integers, so to speak?

I'm doing real analysis + group theory right now but I seriously don't know what it's useful for

>>9087109
>real analysis
Garbage which should be completely ignored.

Can the mods fucking ban this real numbers aren't real fag already?
Please?

>>9087133
Reddit might be a better website for you.

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>>9087109
group theory is useful for improving your cubing speed

would a nonstandard topology be more appropriate for analysis on the rationals?

>>9086149
daily reminder that post 18th century math is brainlet cancer bullshit
prove me wrong
protip:
you can't

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>>9087530
>the rationals
It is not clear that there are any sets which are not finite.

>>9087684
Reposting from /sqt/ since there seem to be lots of group theory anons here

>>9086526
>No one has complete understanding of number theory
Speak for yourself please.

>>9087892
I have a believing heart and my believing heart believes you are a faggot.

You need help.

>>9087744
Think about what conjugate means

>>9086156
It's currently a conjecture.

>/mg/
>where continuum doesn't exist but gender is a spectrum

>>9088000
Gender is actually a discrete spectrum, not a continuous spectrum. Learn the difference brainlet.

>>9087109
For doing other mathematics. Take more statistics and computer science if you want something useful.

>>9088019
>statistics and computer science
Seems like the wrong thread for your kind. Please use something else for this type of garbage.

lebesgue integration for calc undergrad brainlet?
can I understand it or even just compute lebesgue integrals without measure theory prereqs?
I'd like to be able to compute the riemann integral of f(x), and the lebesgue integral of f(x) and be like "huh it's the same" or different, if it's different. I can't find anything about the topic that addresses brainlets, it's all grad level stuff.

>>9087082
>I believe in the existence of the naturals
what do you mean by this?

>>9086316
https://math.stackexchange.com/questions/4202/induction-on-real-numbers

>>9088061
If you don't know what an integral is beyond "take the antiderivative", it's not worth your time.

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>>9087532
/sci/, be honest, are you a classical math chad or a modern math cuck?

>>9088061
Do not waste your time with lebesque if you do not even know riemann. Learn the riemann integral and all its nice properties and tricks. It is actually more than enough for any elementary application. Lebesque integration is only necessary when you want to generalize your theory and be able to do integration on weird sets which you won't need for literally all of undergrad.

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>>9088129
source?

>>9088116
>has never seen an irrational

Euler never saw his number?

>>9088061
read garden of integrals

>>9088141
I made it some years ago

>>9086316
Every set can be well-ordered, so there is well-ordering of [math]\mathbf{R}[/math] and therefore we can use induction on reals.

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>>9088150
Choicefags aren't welcome here

>>9088061
>>9088061
>I'd like to be able to compute the riemann integral of f(x), and the lebesgue integral of f(x) and be like "huh it's the same"
the thought process goes as follows: you learn what an antiderivative is. then you approach the "area under curve" issue which is a completely unrelated problem. you define the riemann integral as an abstract tool for measuring the area under the curve. now you have the definition which you think makes sense, but you have no idea how would you actually compute the value. turns out that you can do it using the antiderivative. then you find out that the riemann integral has some issues which you think it shouldn't have, so you come up with a better tool for measuring areas, the lebesgue integral. now you have your better tool, but once again, you have no idea how to compute it. turns out, that the lebesgue integral equals the riemann integral provided both exists. so the lebesgue integral is also computed using the antiderivative.

never in your life you're going to compute a riemann or lebesgue integral using the definition (that might be an overstatement)

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This is a really wonderful book if you're interested in this topic. Full formal treatment of the subject, exercises, well written, good notation, an all around pleasure.
I bought it last summer, went into it a bit and am just returning to it soon, after some work is finished this month. I can't wait!

>>9088166
I see, thanks a lot for the clarification.

>>9088166
Actually in calc 2 my professor made us compute the riemann integral of f(x) = x^3 from 0 to 1 and then asked us to compare with the antiderivative approach.

It was tedious but it was kinda fun.

>>9088170

lots of good fap material in this. recomended

I have noticed the funniest, most original posts happen from people who add extra spaces between their lines. I wonder if there is some kind of correlation.

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>>9088129

Russel was BTFO so hard by Godel, he spent a fuck ton of time writing his meme book and then BOOM *super theorem*.

Also, where did he get the idea that Godel was jew? Fucking anglos.

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>>9088129

this:

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>>9088432
truly marvelous

>>9087109
>I don't know what group theory is useful for
>I don't know what the study of symmetry is useful for
You sir are a brainlet and a philistine.

>>9088106
>https://math.stackexchange.com/questions/4202/induction-on-real-numbers
Point to where a well-order on [math] \mathbb{R} [/math] is constructed or an analogue to the successor function is defined on it. All that guy does is construct a model for transfinite induction on intervals of real numbers that are bounded from bellow. [math] (0, \infty) \neq \mathbb{R} [/math].

>>9088019
kek good luck doing anything of worth in stats without having knowledge of real anal and measure meme.

>>9088129
Russell is a known charlatan among philosophers. He lied like crazy because he was a politically motivated loon. A proto-SJW so to say. See for example his book on the history of philosophy. It's full of crass inaccuracies that he inserted there for pretty much social justice reasons. "muh Aristotle justifies oppression so I'll paint him as an idiot by misrepresenting what he wrote and read his works in the most uncharitable way I can manage".

>>9088150
Read what I wrote again. Something important might dawn on you.

Is a function still considered well-defined if one output can have multiple inputs? Example, sin(x).

>>9088668
By definition, all functions are well defined. Well defined only means that when you write down an expression, it has one and only one value or interpretation. If you write in(1) it is equal to one and only one real number, there is no ambiguity and therefore it is well defined.

Guys, how to show that [math]F(A ⊔B) = F(A)*F(B)[/math], where [math]F(X)[/math] denotes free group over [math]X[/math], [math]⊔[/math] is a disjoint union, [math]*[/math] is a free product. I.e. I need to show that free groups preserve coproduct.
I can only show that they are isomorphic by messing around with universal property, but I have no idea how to show they are equal. Any hints?

>>9088693
You can't prove a non-theorem. They're not equal.

>>9088758
>>9088693
It is enough that they're isomorphic. In a categorified setting you only care about structures up to isomorphisms.

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Is this normal for 2nd year homeworks?

>>9088818
Is it normal for brainlets to post here? <- rhetorical question btw

>>9088693
Fairly sure the exercise should read [math]\cong[/math] instead of "=" since [math]A \sqcup B[/math] is defined only up to an isomorphism.

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>>9088959
A german exchange student always wrote stuff like [math]\pi_1(S^1)=\mathbb{Z}[/math]. Some people do this, tacitly use a quotient category.

>>9088761
>>9088959
Thanks. I had strong feeling that I should really prove isomorphism instead of "=", now I'm sure

>>9088965
yeah, my algebraic topology teacher wrote stuff like [math]\mathbb{R}^n \subseteq M[/math] (for coordinate patch), in my opinion there's really no problem with that but I totally understand that it can drive some autistic people crazy.

>>9088835
>Is it normal for brainlets to post here? <- rhetorical question btw

First time in /mg/?

>>9088987
>in my opinion there's really no problem with that
then you don't understand what subset means

>>9089007
>what is a subobject

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>>9088987
>[math]\mathbb{R}^n[/math]
>in my opinion there's really no problem with that

>>9088432
this is fantastic

>>9088524
that's interesting

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I was reading about modal logic at work. It looks really fun and I'm really excited to learn more.

I also found this neat page: http://logitext.mit.edu/tutorial to teach me about the sequent calculus. Because I've seen stuff like pic related and never known how to read it.

>>9088101
I believe in the existence of natural numbers.

Since there wasn't much activity today I'll make another shitpost by asking you to verify my proof.
So, an exercise:
Let [math]G = \mathbb{Z}^{(\mathbb{\mathbb{N}})}[/math]. Prove that [math]G \times G \cong G[/math].
[math]\mathbb{Z}^{(\mathbb{\mathbb{N}})} := \{f:\mathbb{N}\rightarrow \mathbb{Z}|f(x)\ne 0\ \text{for only finitely many elements} x \in \mathbb{N} \}[/math].So it's a free group over natural numbers.

My solution: let [math]f:\mathbb{N}\rightarrow H[/math] be a set-theoretic mapping to some group [math]H[/math]. It induces a unique group homomorphism [math]\phi: G\rightarrow H[/math] such that [math]\phi[/math] extends [math]f[/math](because [math]G[/math] is a free group over naturals). Next, I define a function [math]j:\mathbb{Z} \rightarrow G[/math] such that [math]j\phi =f[/math]. This function induces [math]j \times j: \mathbb{N}\rightarrow G\times G[/math] and a natural projection [math]\pi _G:G\times G \rightarrow G[/math]. So, what we get in the end: a unique group homomorphism [math]\pi _G \phi: G \times G \rightarrow H[/math] such that [math]\pi _G \phi = f[/math] meaning that [math]G\times G[/math] satisfies universal property of a free group over naturals. Hence, [math]G \times G \cong G[/math]

>>9089324
>for only finitely many elements [math]x \in \mathbb{N}[/math]
No need to explicitly mention this. It's clear from the use of "[math]\in[/math]"

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>>9089324
There's a tex sandbox so you can make sure your post doesn't look like dogshit before posting

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>>9089324
>free group over natural numbers

>>9089324
Can't you just map the pair (f,g) to the function that maps x to f((x+1)/2) if x is odd and g(x/2) if x is even to get an isomorphism?

Can the so-called "real" numbers be shown to exist?

>>9089324
>.So it's a free group over natural numbers.
No it isn't.

>>9089451
Define "exist".

>>9089627
Define "define".

>>9089523
Indeed. The notion of "natural numbers" isn't currently known to be well defined.

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>>9089671
https://iocdf.org/about-ocd/treatment/meds/

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>>9089702
OCD is about as real as the "real" numbers.

>>9088170
This is a safe for work board damnit.

>>9088500
>group theory
>""the study of symmetry""
Spotted the brainlet.

bourbaki was a mistake

>>9089725
>Spotted the brainlet.
Wrong.

They call them complex but they're pretty simple
They call them real but they aren't

it really causes me to ponder

>>9089957
Mathematicians should be jailed desu

>>9089968
>"real" "numbers"
>mathematicians
Idiots should be jailed.

guys please let me use real numbers, it makes my work so much easier just please, it's too much work to construct everything

This is the worst thread on /sci/ and it's approaching SC2 /vg/ threads on badness.

>>9090012
>"real" numbers
Why do you still believe in fairy tales?
>it's too much work to construct everything
Then try out >>>/lit/

>>9090021
>/vg/
And why would I trust a g*mer to be able to accurately tell if a thread is bad?

Let's say I would like to postulate the existence of God as an axiom. What's the best way to formally do that?

>>9090264
Okay. I'm taking the bait. Why do the real numbers not exist?

>>9090355
i dont know go find a philosopher, this general is about math

>>9090359
Formalization of intuitive notions is a part of math. A mathematician would understand.

>>9090358
>Why do the real numbers not exist?
They can't really be shown to exist.

>>9090391
Sure they can. The continuum exists. See space.
>inb4 Planck distance
Fuck off with that neo-Pythagorean bullshit. Just because you can't make a rod shorter than a certain threshold doesn't mean shorter distances don't exist. Furthermore, you can prove that distances that are not commensurate to the Planck distance exist.

I'm willing to bet that the >real numbers aren't real shitposter is some proxy-ban-evading avatarfag that got mad people can't stand him so he's been shitting up the thread now. These sort of shitposts become much more frequent after the yukarifag got shunned and banned.

>>9086316
Can you tell me who this semen demon is please? Inquiring minds must know.

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>>9090536

/mg/ is there an idea that is to the reals as axiomatic systems are to the naturals?

as in, is there a continuous logic.

where you could, for instance, advance along a proof by 0.01 steps, or prove an interminable proof is valid by computing the finite limit of its inference as the quantity thereof diverges.

>>9090568
>or prove an interminable proof is valid by computing the finite limit of its inference as the quantity thereof diverges.
kek
in other words, a formalized notion of "well i mean you know what i'm sayin"

>>9090509
>The continuum exists
It is not yet clear if it even can exist.

>>9090568
What are you even on about? The "reals" are provably discrete.

>>9090585
Okay I didn't know that but regardless is there a continuous set and if so is there a system of reasoning based on it? Like fuzzy logic but instead of the truth values being fuzzy the set of valid inferences is fuzzy instead

>>9090585
what?

>>9090509
Two points in space are indistinguishable from each other at one Planck distance. It does not make sense to consider things that in theory are unobservable.

Or, if you like, we could say the existence of these things is "undefined". Can't be proven one way or another.

But what you CAN'T do is suppose possibility for reality. Continuity in the universe is an assumption. As we have discovered, and built upon those discoveries, that even something as basic as distance will cause actual mathematical contradictions. The Planck length can be derived from wave mechanics, or by an operator approach. Yes, it is related to the famous Heisenberg uncertainty relation.

Of course, the relationship between math and physics is subtle. Physics statements are models, if you want to suppose that your current models are inaccurate at small scales then you have your work cut out for you.

>>9090509
>The continuum exists
So I'm guessing you're getting a Fields medal soon?

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Is there any reason whatsoever to restrain myself from postulating the negation of AC?

>>9086188
if the real summation and derivation make sense this quite possibly is not true?

>>9090645
considering you don't do anything anybody gives a shit about anyway it doesn't really matter what axiomatic system you choose

>>9090716
Really now? I have quite a few people who give a shit about what I do. That's why I'm asking.

>>9090607
>if no one is there to observe it it means a tree that falls in the forest makes no sound
This is how retarded you are.
Being unable to measure something has no bearing on whether it exists or not.

>>9090568
>is there a continuous logic.
Yes, it's called a topos.

>>9087099
what do you mean "stop"?

>>9090607
The biggest irony here is that the validity of the mathematics of QM depends on the validity of the continuum. "Distances smaller than the Planck length are undefined" is as self-refuting a statement as one can get.
Fucking Neo-Pythagoreans. This explains why a crank like Wildberger is so popular on /sci/. This place is filled with physicists and engineers.

>>9090745
I mean stop increasing the level of fairy-tale retardation. The next step up would be the "rational numbers".

>>9090607
>>9090749
>physics
Use some other thread to discuss garbage such as "physics" and "real numbers". We don't need fairy-tale bullshit here.

>>9090749
>The biggest irony here is that the validity of the mathematics of QM depends on the validity of the continuum.
Which instantly shows that QM is invalid.

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>>9090751
>>9090753
>>9090756
Physishits rub me the wrong way too but you should seriously consider suicide. Worst /mg/ meme.

>>9090645
Is there a reason why you should not kill yourself? No, I believe there isn't.

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>>9090769
>you should seriously consider suicide
Very rude and uncalled for.

>>9090645
>>9090722
Kek you literally do not matter.

>>9090751
have you studied any algebra at all?

>>9090797
You're replying to a spammer. Just hide and report all his posts.

>>9086149
Okay so my kid is in 2nd grade and is already failing in maths. I as a kid sucked in maths too. How do I help him?

>>9090885
You help him by understanding that aptitude for mathematics is inborn. Try to be patient with him and push him to do the best he can, but don't expect much.

>>9088432
kek

Is Von Neumann the greatest mathematician of all time?
Is he the smartest person of all time?

>>9090980
no to both

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What is your dumbest math related joke ?

>>9091212
[math]\mathbb{R}[/math]

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>>9091212
Nothing beats the snake lemma. It's an inexhaustible source of shitty jokes.

>>9090980
> Euler
> Leibniz
> Fisher
> Lovelace
> Abel

You honestly think Von Neumann was greater than them all?

>>9091212
Frequentist statistics.

>>9091279
Definitely greater than Fisher and Lovelace.

>>9091283
>t. compsci brainlet
GTFO with that Bayesianism bullshit. This is a math thread.

Guys, I'm new to mathematics and seriously stumped on the numeracy being used in this worked example of the induction hypothesis. Can anyone explain to me what is going on in these first two highlighted lines? Why are we adding 6/6 infront of (k+1)^2? Also, how does 6/6(k+1)^2 =6k + 6?

To me it would equal:
6/6(k+1)^2
=(1)(k+1)^2
=(1)(k^2+2k+1)
=k^2 +2k + 1

I just don't get it, help me math wizards.

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>>9091325
Forgot to post the picture.

>>9091325
>Why are we adding 6/6 infront of (k+1)^2?
it's just to bring everything to a common denominator, the first term has a 1/6 and so if you want to group something of the form (1/6)*x + y into one fraction you write
(1/6)*x + y
= (1/6)*x + (6/6)*y
= (1/6)*(x+6y)

>Also, how does 6/6(k+1)^2 =6k + 6?
it doesn't, one (k+1) is factored out of (1/6)k(k+1)(2k+1) and also factored out of (6/6)(k+1)^2, look at the bracketing

>>9091325
You're factorising the expression by taking out
[math]\frac{1}{6}(k+1)[/math]
This puts it into the correct form with the k replaced by k + 1.

>>9091325
>>9091327
[math] = \frac{1}{6}k(k+1)(2k+1) + \frac{6}{6}(k+1)^2 [/math]
[math] = \frac{1}{6}k(k+1)(2k+1) + \frac{1}{6} (6(k+1)^2) [/math]
[math] = \frac{1}{6} \left( k(k+1)(2k+1) + 6(k+1)^2 \right) [/math]
[math] = \frac{1}{6}(k+1) \left( k(2k+1) + 6(k+1) \right) [/math]

>>9091325
The second highlighted line might be clearer if the type of brackets change.
[math]\frac{1}{6}(k + 1)\{k(2k + 1) + 6(k + 1)\}[/math]

>>9091347
>>9091348
>>9091356
I think I understand things better now, whereas at the start we have the terms 1/6(k+1)(2k+1) +6/6(k+1)^2 by factorising we are able to change the terms to be 1/6(k+1) and (k(2k+1) + 6(k+1))? Thanks for the help anons, I don't suppose anyone can point me in the direction of an article on khan academy or the like that deals with this sort of factorisation, so that I can deal with it outside of the concept of mathematical induction? I've noticed poor knowledge of factorisation seems to be holding my studies back a lot. My formal mathematics education is seriously lacking, but the mathematics department at a university I applied for agreed to let me into their undegraduate course if I could pass a one month summer school, to prove that I was serious enough about the subject to get up to the level of experience the other undergrads would have in a short space of time. I can't thank you all enough for the help.

>>9090021
Please. This thread is not even in the same league as sc2 generals.

>>9091370
Factoring out a quantity refers to rewriting a formula as a product of that quantity and the remainder quantity.
If [math] A = ab [/math] and [math] B = bc [/math] then [math] A + B = b(a+c) [/math].
It is a consequence of the distributivity of multiplication with respect to addition, except done in reverse.

>>9091381
>>9091370
I have to add though: if factoring an expression is what's tripping you up, you have extremely low chances of making it even as a math undergrad. They teach kids this shit in primary school.

>>9091381
>>9091389
It's not that I find the concept difficult, rather it's just been five years since I'd factorised an expression and had completely forgotten how to do it. I figure I can't be that mathematically challenged given the amount of knowledge I've crammed in over the past month, I'm just deficient in factorisation because I haven't been to school in over five years. Going to grind out all the factorisation modules in Algebra 1 on Khan Academy and then come back to induction. Thank you for your help anon.

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>>9091398
>it's just been five years since I'd factorised an expression and had completely forgotten how to do it

>>9091407
>last factorised an expression age 16
>three years NEET
>two years studying chemistry and biology at community college
>started studying mathematics in july

>>9090891
Yeah I don't expect him to do much nor I don't want him to push him hard. I'll try to make him understand that. Hope that helps.

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Can it be shown without invoking fairy tales that there is a set which is not finite?

>>9091714
according to godel's incompleteness theorems, No.

>>9086149
Hey bros, bionerd here. Is it worth getting in to math if you're 22? It seems like fun

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Is computer science ok in math general? It's technically a subset after all, but I know some people weren't happy about applied math even though it's a subset too.

>>9091742
No. "Computer science" and "applied math" are simply not welcome here. Go create your own thread.

>>9091742
>computer science
We don't like your kind here.
>applied math
We don't like nonexistent garbage here.

>>9091755
>"Computer science" and "applied math" are simply not welcome here.
Which other subsets of math aren't welcome?

>>9091762
>We
>We
Speak for yourself.

>>9091766
"sets" can't really be shown to exist in the first place, so I guess no "subsets" of "math" are welcome here.

>>9091769
I am God and I can speak for my subjects.

>>9091779
>I am God and I can speak for my subjects.
cringe

>>9091742
>Is computer science ok in math general?
No.

>>9086149
Brainlet here, I have a question for you guys.
Does math ever become "transcendent" or is it just a meme? Is it just a really good tool for physics or does it hold more than that? Is mathematical beauty a meme?

Hardest math I ever did was Linear Algebra 101 and got lost at determinants, lol

>>9092040
>meme
>lol
>physics
Try out >>>/r/eddit/

>>9091389
yeah, they teach kids shit in primary school and then you hate math and don't pay attention, then you do shit for a lot of years and then you GROW UP and realise you want different things.
That's a really shitty thing to say, of course you can make it if you apply yourself and ask the questions like he is doing.
You piss me off

What are some good anime to watch in between math study sessions?

>>9091742

fukken saved

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>>9092105
cromartie high school

>>9086149
Lads, can someone help me to find the connection between dot product and an angle between vectors.

>>9092062
but it's filled with xkcd comics and smug pseudointellectuals

>>9092105
>not having a octocore brain that can handle simultaneous mathematics and anime
must be tough being a brainlet

Was my decision to learn maths and stats in any way good?

>>9092175
I see, so you are indeed a r*dditor. Don't shit up this thread with your nonsense, please.

A question I've been pondering since this morning that I'm not sure is [analytically] solvable:

Which is bigger, 2^333 or 3^222 (without explicitly calculating)? How do you know?

>>9092240
(2^3)^111=8^111 < 9^111 = (3^2)^111

>>9092240
It isn't clear if 2^333 or 3^222 even exist.

>>9092240
222 log 3 > 333 log 2, hence 3^222 is larger.

>>9090727
>if i say that a tree falls in the forest than it actually does.
Assuming something exists has no bearing on whether it exists or not.

>>9090727
If you knew anything about reality at all you would realize this notion comes from how this universe actually works, not how we want it to or think it should.

>>9086149

The celebrated Barnett identity finds perhaps its most natural expression by means of the LHS of the below equation 1)

[eqn] 1) \;\;\;\;\;\;\;\;\; \Bigg( 1 + \sum_{k=1}^{ \infty } \frac{1}{ \prod\limits_{j = 1}^{k} j} \Bigg) ^{ 2 i \int\limits_{-1}^{1} \sqrt{1-x^2} dx } + 1 = 0 [/eqn]

Which beautifully relates sum, product, and integration.

>>9092272
>without explicitly calculating

>>9092314
can i masturbate to this ?

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>>9092175
That's actually more than /sci/ has to offer, as we don't have that much xkcd comics.

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>>9092314
>[math]\infty[/math]
No such thing.

>>9091212
[math]\mathbb{N}[/math]

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Currently I'm thinking of negating the axiom of "power set" and the axiom of "infinity". Anything else I should add to my list? The negation of "AC" will of course be provable from the rest of the axioms I'm going to add.

>>9092451
throw those 3 out and then add "there are no more sets than the ones that can be built by these"???

Isn't space infinite?

>>9092467
>space
Sounds like some type of "physics" garbage. Please keep it contained to some other thread.
>infinite
No such thing.

>>9091212

I actually made this one up myself:

Most matrices live on campus. In general, they don't commute.

>tfw you will never be a professor and force your students to endure your :^) - tier jokes at most once per month to give a bit of cognitive flavor and induce groans in the class but not be overbearing about it and just be mostly straight business the rest of the time

>tfw you will never pull out that old saw exactly once a year as the teaching schedule permits

>>9092478
>reddit spacing
Your kind isn't welcome in this thread.
>>>/r/eddit/

>>9092481

What would your reaction be if you were actually made to understand that:

I've used this site regularly since early 2004,
I've spent something like less than two days, lifetime, actually browsing reddit (I didn't care for it but the odd search links there)
and I regularly format my posts in this way?

If you would reply "cuck faggot" etc, or otherwise object, then you would be wrong. I am regularly identified in this wise on /tv/, who also falsely believe themselves to have pegged me.

4chan's userbase is objectively wrong about what it describes as "reddit spacing". It looks better and helps to bracket thoughts in a line-based environment like this which doesn't lend itself to long-form prose If this is how reddit actually formats its stuff regularly, then yes, reddit is right,and 4chan is wrong, and I shall go on being right, and there is absolutely nothing that you can do about it, cuck faggot.

>>9092492
>writing out an entire rant in response to a bot

>>9092492
Didn't even read your post.
see >>9092481

>>9092492
>admitting to being a r*dditor
R*dditors aren't welcome here.
>>>/r/eddit/

>>9092243
Oooh, that's a pretty way to solve it. Thanks!

>>9092245
Wildburger go to bed

>>9088987
See I wouldn't do that when thinking about charts on manifolds, but I would do exactly that when thinking about the equivalent on schemes. Idk why.

Can someone give me a quick rundown on K-theory? Assuming I've seen the definition of the Grothendieck group of a Scheme/Manifold in the form of an exercise, but thats it.

>>9091766
All subsets of mathematics are welcome, but applied mathematics is not a subset of mathematics.

>>9092096
Don't know what shit education you had, but I was thought how to factor algebraic expressions in 3rd grade. Learned about parentheses and distributivity in 2nd grade.

>>9092284
>>9092295
If you knew anything about mathematics you'd realise how retarded it is to say that distances smaller than the Planck length are undefined.
>every step of the way to the derivation of the quantity known as the Planck length uses (and hence implicitly assumes the validity of) calculus with real numbers
>the transcendental irrational [math] \pi [/math] appears several times in the process
>lengths smaller than Planck's are undefined bro look at these equations you can't measure anything shorter than that
Physishits should be held down on the ground and pommeled with a straight edge over the head until their brain is mincemeat, because theirs' are as good as mincemeat anyway.

>>9092731
The empty set is a subset of mathematics.

>>9092747
>Physishits should be held down on the ground and pommeled with a straight edge over the head until their brain is mincemeat
I wholeheartedly agree. The world would truly be a better place.

>>9092470
Kill yourself you fucking physishit false flagger. Neo-Pythagoreans aren't welcome here.

>>9092492
I say that you're a lying sack of reddit shit.

>>9092750
Applied math is not an empty set.

>>9092754
>physishit
I despise physishits more than anything on this shitty planet.
>Neo-Pythagoreans aren't welcome here.
Please don't say that.

>>9092750
>>9092762
So-called "applied math" can't be shown to exist. It's about as real as the "real" numbers.

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>>9086149
I don't get the third equation, i see this meme everywhere but can someone explain this?

>>9092767
>i see this meme everywhere
Which "meme"?

>>9092767
It's a notational quirk. Look up Ramanujan summation.

What is a good system in which the negation of AC is provable without any additional axioms?

>>9092811
If you were able to prove anything you wouldn't need to be asking such questions, brainlet.
Kill yourself.

>>9092817
An interesting assumption. Such a worthless post though.

So modern mathemathical breakthroughs have been essentially trying to find more officient ways to do computation, right?

My brain has been configured to immediately disregard any math that can't be derived from geometric analisis. The math side of /sci/ feels cray-cray to me

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>>9092829
>So modern mathemathical breakthroughs have been essentially trying to find more officient ways to do computation, right?

>>9092829
Geometric Analysis is not very computational.

Can integers truly be shown to exist or are they yet another fabrication? I'm genuinely interested in some answers.

>>9092899
Surely we must be applying modern mathematics to something, right?

>>9092900
that's me stating why I don't get all this set and -1/12 mumbo jumbo because I can't see a clear purpose in it

>>9092935
>we
Who are you talking about?
>"applying"
"applied math" is nonexistent. Nonexistent things aren't welcome here.

The situation with finite and infinite sets is clear. But what about sets which are neither finite, nor infinite?

Quantum mechanics decisively disproves finitism by demonstrating that our universe needs infinities to function. So it would be appropriate to say that the finitist's beliefs are neither based on empirical evidence nor on aesthetic considerations. Indeed at this point the only hope for the finitist would be to kill himself.

Is there a way to improve my mental math skills /mg/?

>>9093106
>uses physics to disprove mathematics

>>9093106
>>9093129
>physics
Such garbage isn't welcome in this thread.

>>9093106
>Quantum mechanics
Which is invalid.
>phys*cs disproves
It simply isn't capable of doing so.
>finitism
What is "finitism"?
>infinities
What are these "infinities"? Is this the usual level of rigor in phys*cs?

>>9093129
>>9093133
Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.

>>9093106
Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.

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>>9093106
>>9093139
>>9093142
Physishits should be held down on the ground and pommeled with a straight edge over the head until their brain is mincemeat, because theirs' are as good as mincemeat anyway.

>>9093137
Assuming anything without proof can only be used to prove that it is wrong.

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Ladies (male) and gentlemen, the Tate cohomology is really nice! Very clever stuff.

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It depends I suppose on what you mean by infinite object. Is there a precise definition of this term?

>>9093158
>Is there a precise definition of this term?
something like [math] \pi [/math] which never ends

>>9093159
It depends I suppose on what you mean by "something which never ends". Is there a precise definition of this term?

>>9093166
Something with infinite Kolmogorov complexity

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>>9093169
It depends I suppose on what you mean by "infinite". Is there a precise definition of this term?

>>9093173
>Is there a precise definition of this term?
'not finite'

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>>9093176
It is not clear that there are things which are not finite.

>>9093185
Space is infinite even though the mass on it probably isn't

>>9093216
>Space is infinite
Formally prove it without using dubious concepts such as the axiom of "infinity".

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>>9093216
Your """physical intuitions""" aren't a proof. Please do not spread misinformation.

>>9093239
if space is a vacuum without matter or energy, then let's use a proof by assuming it's finite. If anything exists beyond space, it would have to be the opposite of this vacuum (something that fills space: a planet, radiation, matter) and thus, it would be a continuation of our universe. If emptiness exists beyond space, then that emptiness is part of space's vacuum and hence still part of our universe. The only way space could not continue indefinitely was if it was surrounded by an encompassing and continuous black hole that wrapped around it, which due to physical laws, is implausible.

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>>9093266
Is this the phys*shit's idea of a formal proof? Such miserable vermin shouldn't even be participating in this thread in the first place.

First qual tomorrow boys

I wanna die

>>9093264
Well what about time then? Isn't it infinite?

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>>9093349
It depends I suppose on what you mean by "infinite time". Is there a precise definition of this term?

Post a real number which is also not a rational number. But with the constraint that you can't say that it is the result of some process, including the ill-defined "..." function.

>>9093185
It is actually very clear. For instance the position and momentum operators in QM are infinite dimensional.

https://physics.stackexchange.com/questions/10230/trace-of-a-commutator-is-zero-but-what-about-the-commutator-of-x-and-p

>>9093458
>ph*sics
Take your garbage to some other thread.

>>9093453
>Post a real number
So you mean either natural numbers, integers or rationals?

>>9093461
>asks for physical evidence of infinity
>denies physics
Wew lad

>>9093453
Problem is that you are asking us to right down a number with an infinite amount of non repeating decimals numbers. The only way to write an irrational number as a finite expression is by defining it by an expression.

What you can do is take random number generator from 0-9, start with 1.1 and fill the rest of the numbers by using the rng for the rest of time, and even then, the number you wrote down will still be rational, but it will aproach an irrational number.

>>9093464
Irrational numbers are Real numbers too.

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>>9093458
The formal theory of "QM" relies on dubious notions such as "infinite" sets which simply can't be shown to exist.
>>9093465
>physical
A theory which extensively uses "real" numbers isn't and can never be "physical". Perhaps you might have a point when you and your retarded brethren develop a more sensible version of it. Although I doubt that will ever happen.
>evidence
Ph*sics is a complete fucking joke. It can't provide "evidence" for anything.

>>9093477
Yes, nonexistent numbers are nonexistent numbers too. Your point?

>>9086149
Why do anti-infinity fags post so often? Every time someone says real or infinite they respond. We know you are larping so how come you have so much free time?

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>>9093487
It's only one obsessed poster. He has psychological problems and should try to get help with those when his NEEThood ends (I think that should happen within a month).

>>9093479
>which simply can't be shown to exist
Haha, you're retarded.
>what is the axiom of infinity?

>A theory which extensively uses "real" numbers isn't and can never be "physical".
But it actually is so your opinion doesn't mean shit.

>>9093499
>the axiom of "infinity"
You can assume something exists all you want, that doesn't make it less retarded.
>But it actually is
The "real" numbers can't be shown to exist. So any theory which makes heavy use of them isn't "physical". Take your retarded garbage to the ph*sics general. We don't want your kind here.

>>9093498
>He has psychological problems
What sort of problems?
>I think that should happen within a month
What makes you think so?

>>9092747
I never said that you fucking moron. The mathematical "space" is different from spacetime. Spacetime has properties, you can't just wildly apply mathematical notions if it isn't a good model. Nobody said anything about being pi meters away from something.

I also didn't say the distance was undefined, I said that a hypothetical thing's existence would be unverifiable.

>>9093508
>spacetime
>reddit spacing
Physishits should be held down on the ground and pommeled with a straight edge over the head until their brain is mincemeat, because theirs' are as good as mincemeat anyway.

>>9093507
>what sort of problems
Let's start with this https://en.wikipedia.org/wiki/Obsessive%E2%80%93compulsive_disorder
>what makes you think so
Assuming you are the Kyoani poster, and he didn't lie about becoming a researcher, your work should begin this autumn.

>>9093510
>I don't understand reality so I make my own

@9093504
This """"argument""" clearly isn't going anywhere (like your career) so I'll just ignore your posts from now on.

>>9093512
>"OCD"
see >>9089717
>Assuming you are the Kyoani poster
Your assumption is false. I'm not subhuman so I wouldn't speak about such things on this website.

>>9093522
well then you are even worse. At least he is potentially becoming a researcher and you are nothing.

>>9093520
That sounds a lot like the axiom of "infinity". If something can't be shown to even make sense, just make up your own fairy-tale "reality".
>>9093521
A good move. It's the only thing a defender of fairy-tales can do at this point.

>>9093525
>you are nothing
Am I now? You can't possibly know this. Or is it yet another one of your axioms of "infinity"?

>>9093529
Your behaviour makes it very obvious. Kill yourself before your finitism does.

>>9093531
>Your behavior* makes it very obvious
I see. Is it one of your """physical intuitions""" then?
>Kill yourself before your finitism does.
What makes you think I'm a "finitist"?

>>9093534
Indeed it is an empirical fact that you are an obsessive loser who has nothing better to do but to spam these threads full of your stuff.
>What makes you think I'm a "finitist"?
Left as an exercise to the reader.

>>9093540
>"empirical" "fact"
Physics doesn't deal with "empirical" "facts".
It sounds more like one of your retarded axioms.
>Left as an exercise to the reader.
Please answer the question.

>textbook offers an example of a real number
>it's a rational or relies on "and so on" pig-disgusting ellipses or a function which promises to output an irrational
Convenient

>>9093545
>Physics doesn't deal with "empirical" "facts".
What physicists do is completely irrelevant. Neither of us is a physicist, I hope.
>It sounds more like one of your retarded axioms.
Yeah and I could also be an NSA agent trying to see if your superior intellect is a potential risk, if you want more ways to refute my claims and keep your psychological defences up. Your mind is crumbling because you aren't capable of doing modern math?
>Please answer the question.
Read your own posts and it should be obvious. You are not worth proper answers, only insults.

>>9093554
>Neither of us is a physicist
I'm definitely human, I hope you are too. I doubted it since you relied on your ""physical intuitions"" like those animals do.
>Read your own posts and it should be obvious.
It's not obvious. Why am I a "finitist"?
>You are not worth proper answers, only insults.
Please don't say such things.

>>9093554
>Your mind is crumbling because you aren't capable of doing modern math?
I'm currently working on creating a non-retarded version of a modern field of math which is very dear to me.

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>>9093558
Notice how you have spent yet another 30 minutes replying to a poster who has no respect for you or your posts? You are not having shitpost breaks between working/studying, but you are shitposting all day long. I'd say I have shown my point of your uselessness and shall now continue doing something more productive.

>>9093560
What you are currently doing is wasting precious hours. Good luck with your reformulations. Maybe you will get a dozen readers if you ever get it done. See ya later.

>>9093560
Are you... the wild one?

>>9093568
>Notice how you have spent yet another 30 minutes replying to a poster who has no respect for you or your posts?
You know that only makes it better, right?
>You are not having shitpost breaks between working/studying, but you are shitposting all day long
It this yet another one of those "physical intuitions"?

NEW THREAD
>>9093634

>research in autumn
Oh so now I am a kyoaniposter? Just FYI I take the maximal position on infinities: inaccessible cardinals exist. Because Grothendieck-Tarski. I've said as much before.
My advice is to shoot finitists on sight (report the loon and ignore his posts ffs).