/math/: Math general

>>7757787

I always wondered if we could say predict what an alien language sounds like (phonetically) (if they speak one that is). Because our hearing is firstly limited to a certain set of frequencies and I am assuming that all languages do have a collection of similar sounds like the "b" sound.

So I have been wondering what would actually be found if someone was to do something like this. Obviously it would be a lot of work collecting sounds and grouping them according to the phonetic sounds.

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>>7757787
Pic related.

The circumference of a circle is a function of the diameter where
C = dπ

In my diagram, x is a function of the circumference so that

x = kC for some coefficient k,

When d approaches infinite so does the circumference because if C = dπ and d approaches infinity then C = d.

so as C approaches infinity so does x because x = kC and because C approaches infinity
x = C.

So from that we get the notion that a circle with an infinite diameter is a straight line but...

Where does the rest of the circle go? Is it beyond infinity? Does it exist?

>>7757800
>When d approaches infinite so does the circumference because if C = dπ and d approaches infinity then C = d.
you can't say this

>>7757802
Of course I can.

The only way for it to not be that way would be if pi was approaching 0 but that is not the case. Pi has a fixed value that is > 1

For any big d, C is going to be even bigger.

>>7757797
I don't see why you'd think this is a math question, it's rather linguistics and anthropology.
But to answer you anyway, there is some phonetical alphabet, but I'm sure people don't need to use it. There are odd langauges, like that african clicking sound language.

https://www.youtube.com/watch?v=W6WO5XabD-s

https://www.youtube.com/watch?v=lrK-XVCwGnI

>>7757800
x equals the arclength times the radius.

There are formalizations of your concepts, e.g. in
https://en.wikipedia.org/wiki/Projective_plane

and there everything infinite is collapsed to one point far removed point. So the whole circle goes there, if you will.

>>7757800
>so as C approaches infinity so does x because x = kC and because C approaches infinity
x = C.
Two functions having the same limit does not imply that the equations are equivalent in any given scenario. If I take the limit of y=x and y=2x as x approaches infinity, y will approach infinity in both scenarios but x clearly does not equal 2x.

I actually have a question that seemed more appropriate here than in /sqt/ and it's not college advice so bear with me, what exactly do I need to know to be able to learn about tensors? I only just finished multivariable calc and I want to go deeper than that without necessarily delving into analysis since my proof-reading/writing skills are not particularly expert yet.
>So from that we get the notion that a circle

>>7757815

If x is infinity and y = 2x
is y > x?

Well, x is infinite and y is also infinite so does it matter?

>>7757813
>>7757797

Could you do it with markov chains?

>>7757800
Infinity doesn't exist, there's always a maximum.

I wish the infinity meme would die.

>>7757809
Approaching infinity, still [math]C = d \pi[/math], so how are you coming to the conclusion that C=d?

>>7757815
The notion of tensor product of vector spaces is not extremely hard to understand if you are familiar with bilinear/multilinear algebra (determinants, etc), it is simply a convenient (albeit conceptual) way to think about multilinear maps.
Subtleties arise when taking tensor products of modules over a ring but if all you want is what is needed for differential geometry, there is no urgent need to deal with such subtleties.
After that, you need to familiarize yourself with manifolds and vector bundles (roughly speaking, they are "smooth" family of vector spaces indexed by the points of a manifold, an example being the tangent spaces)
It turns out that many constructions that can be made on vector spaces can be extended to vector bundles.
In particular, you can define tensor algebras on manifolds and these allow you to define "infinitesimal volume elements" (the intuition behind this is the fact that the determinant of a family of vectors is the volume of the parallelotope they define), and thus integration on manifolds and all these things we learn about in physics (oriented flux, metrics, etc)

Is he right, /sci/?

>>>/g/52199941

Are negative numbers a meme?

>>7759091
They are a meme.

>>7759110
I mean he has a point though, you can't take something from nothing.

Why are negative numbers even a thing?

>>7757800
>undergrad's first exposure to infinity

>>7759091
How else would you handle debt, opposite charge, etc?

>>7757787

Why can't you prove the chain rule by just dividing the differentials? I've done it the real way, but I'd like to know exactly why dividing isn't rigorous.

>>7759115
Because the system works much better with them than without them. Negative numbers cannot literally be exemplified in nature but the goal of math is not to fully represent nature. It just so happens that most math is used to do so.

>>7759124
whoa what's the real way to derive the chain rule.

>>7757815
Are you interested in calculating things with tensors or the theory behind tensors?
"Tensor Analysis and the Calculus of Moving Surfaces" is a good book on analytic applications of tensors.
There is an accompanying series of lectures by the author on YouTube on his channel, Maththebeautiful.
If you are interested in the THEORY behind tensors, ie k-tensors as multilinear functions from the k times product of a vector space with its self to the reals blah blah, you'd be looking for a more rigorous text aimed at the pure math side of tensors.
Are you into pure math? Check out the rigorous side. Are you into physics, engineering, chem, applied math, etc? Check out the applied side.

>>7759124
Could you outline your proof and then ill try to point out what's wrong?

>>7757815
To continue >>7759143 ,
If you want to use tensors to calculate cool stuff and do physics on complicated surfaces or in curved space time, all you really need is calc III, some linear algebra (matrix multiplication, determinants, maybe some eigen vectors/values) and you can calculate quite a bit.
If you want to get into the rigorous side, you're going to need a lot more in the way of mathematical maturity and definitely a good amount of linear algebra. That would only help you around a bit of an intro to tensors, you can dive deeeeeep into undergrad and grad level algebra with tensors so it depends on the sort of exposure you're looking for.

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>>7759137
>>7759154

How do I respond?

>>>/g/52203443

>>7759183

You don't respond to shitposts.

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I was homeless as a teenager and didn't get to go to high school.

How do I teach myself math? Seriously

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I've been reading a bit on sets after working my way through How to Prove It. Namely Naive Set Theory by Halmos and Bridge to Abstract Mathematics by a few people. I'm finding both books to be very easy and I'm wondering if maybe I should move onto something harder like baby rudin or Spivak (Is that harder?) any advice on that? Also set theory is pretty fun.

>>7759187
at the high school level, a textbook for practice and youtube videos for lectures. find math that you can understand or at least somewhat understand, and then work your way from there. best of luck anon.

>>7759187

Don't waste your money on textbooks, just watch Khan Academy until you get sick of it and realize why people say math is useless.

>>7757787

Has anyone ever appreciated the remarkable fact that

[math] \displaystyle \int\limits_{0}^{ \pi } \sin \; x \; dx = 2 [/math]

?

That the evaluation of the definite integrals over convenient intervals of so many elementary functions (so chosen and described precisely because they make the teaching of calculus itself readily amenable to students) should so frequently give up simple integer answers, or otherwise simple integer multiples of already-familiar irrational numbers (pi and e), was for me deeply surprising once I first got it. The basic trend continued when I became slightly acquainted with some complex analysis, where just an i term would be invoked for this-or-that contour integral, but the order of complexity of "the final answer" had often not changed beyond that place-holder.

>>7759255

Moreover, a thread from a few weeks ago prompted me to look up those weird things /sci/ had mentioned earlier as quasi-meme material, but which I had forgotten - I rediscovered them as Borwein integrals.

https://en.wikipedia.org/wiki/Borwein_integral

When endeavoring to learn the punch line for why these integrals' "early pattern" does not hold good after a few steps (such-and-such series or product associated with the integrals passes a critical point from one step to the next, becoming strictly greater than unity IIRC), I was once again reminded by this more recent example of the basic education around convergence and divergence in a calculus sequence, associated especially with infinite series. For the p-series, in particular (the simple case of the zeta function), when such-and-such is strictly greater than or less than unity, the thing either converges or diverges, and it's pretty easy to appreciate why the greater-than cases converge, geometrically speaking IIRC.

My larger point being that the above "seemingly-pathological-but-not-really" beasties have about themselves the elements of a basic calculus education. I really should evaluate them. Can /sci/ expressly evaluate one of the first few Borwein integrals, showing work?

https://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29#p-series

>>7759255

I feel results like those are always a sign that you were ignoring something obvious or that there is a simple way of viewing the problem. That integral can be reduced to "What is the diameter of the unit circle?"

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Other things I learned last year were how to derive the cubic and quartic, somethings I didn't know before.

I also started a thread regarding a recent discovery of a previously unknown type of pentagon which tiles the plane. I see that the wiki has had major revision to explicate the known types, kek.

IIRC, when I checked the pentagon out I concluded that it has a non-convex "sliding cell" of 24 primitives, to use M.C. Escher's language.

When I made the thread, /sci/ became very angry about the uselessness of mathematics. Why the hell are people being given grant money to do this, robble robble.

It is telling that this new version of a detail picture looks like a stained glass window. Math is an "ivory tower" doing useless nonsense, it would seem!

Passed Differential Equations with an A and after going from remedial to now, I can now say that Math truly is completely fucking useless.

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>>7759280

Right. Of course the deep punch line in calculus is that finding rates of change and areas under geometric curves turn out to be related.

But how do you intuitively graphically relate the two (if in fact they can be legitimately related? Just because the two numbers here happen to be equal is no confirmation.

I set an exercise: if feasible describe an animation a la pic related to relate the area of a sine function's "sector" to "2".

>>7759280
>That integral can be reduced to "What is the diameter of the unit circle?"
This.

>>7759297
>But how do you intuitively graphically relate the two

It's a great question really. I believe the second fundamental theorem of calculus makes intuitive sense and perfectly links the concepts of the derivative and the integral. If you are accumulating the area beneath a curve with a Riemann integral, what is the rate of change of the accumulation? Because [math]\Delta x[/math] is (an infinitesimal) constant, the only factor it can depend on is the value of the function itself. Thus, [math]\fract{d}{dx}\int f(x)dx = f(x).[/math]

>>7759340
* [math]\frac{d}{dx}[/math]

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Prove me wrong /sci/.

So basically I'm retarded and finished a semester of university as a maths student and realized I want to do engineering since I can't see myself doing proofs for the rest of my life. My dilemma is that I need to do high school chemistry and physics in order to transfer to a closer university.

I'm not going to be doing any maths for essentially 9 months since I start next fall, and I'm going to be transferring my calc 1 credit. How, and where do I go to prepare myself for calc 2?

>>7759366
Er, it's kind of impossible with the existential quantifier there. Ex f(x) = 1, def. integral of the derivative from 0 to x is 0, and inequality is obvious.

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>>7759366

>>7759397
Now we're talking

How do you guys take in large amounts of math quickly?

People always stress not memorizing things but at this point it just seems like everything is all memorization because there's no way I'm going to remember massive proofs ("just remember the basic idea, the rest should come naturally") if I'm trying to review math textbooks in just a few days. Likewise now that I'm trying to learn statistics, it seems like I just need to memorize loads of distribution formulas and such.

>>7759202
Try some abstract algeba (Pinter's book is cheap) or Rudin for analysis.

>>7759404
Nobody ever said it would be easy. Sure you can try and memorize it, but you'll just forget it in a month anyway.

>>7759202
Also Rudin will be harder as it's a legit analysis book. If you have never studied calculus then try spivak.

What are some objective applications from the math field that the average joe benefits from. Aside from making movies with Matt Damon about it.

>>7759415
>What is engineering
Like..a bridge for example?
Every single electronic you've ever used?
Everything we've ever had since before the Greeks, even music?

>>7759288
Differential equations is still pretty elementary lol

>>7759424
Your proving my unspoken point that math is a meme anon...

>>7759288
Are you retarded? Diff. Eqs are incredibly useful.

I'm taking a course in Algebraic Topology. THAT shit feels completely fucking useless.

>>7759420
Those are all products of pure math?

>>7759427
All Fourier series and transforms basically. Yeah.

>>7759160
That's not a legit proof. You'll often see it as a justification in non rigorous courses.

>>7759429
This should be taught in elementary school. That math I'd the foundation on which our world is built. Maybe they do and I went to school in the 60411 zip code. Our system was bad, I wouldve appreciated being tough just how vital it is.

>>7759427
At some point all math was pure math. It's a progression. Cryptography is taken extremely seriously by governments and it depends on number theory, a subject even a century ago that was considered completely useless

>>7759440

Why isn't it rigorous?

>>7759202
Those books are meant to be easy. The reason they're easy is because they're fundamental.

In other words, you gotta learn how to walk before you can run and those books are teaching you to stand up. Rudin would be like skipping.

I have a bachelors in math and want to learn some physics now that I am graduated. Clearly I am good on the math obviously but is there any tips on grasping the physical constraints of everything or getting intuition behind it.
The book I am armed with is Theoretical Physics by Georg Joos.

>>7759450
At what point would you say it's appropriate to run then? I'm somewhat expecting Bridge to Abstract Algebra to get somewhat harder as I hit the number system constructions, but overall it's a leisurely read with some interesting exercises here and there.

Also, pretty sure these are distinct but I'd like a second opinion:
[math] \emptyset {\emptyset} {{\emptyset}} {{{\emptyset}}} [\math]

They are, correct?

>>7759467
Rudin is not really that tough, it's mostly just artificially difficult in that he cuts most of the fat out of his explanations and proofs, it's still just basic Real Analysis at the end of the day. Don't put this crap on a pedestal. Your objective is to learn Analysis at the end of the day, so if you can't seem to get a grip on what he is saying check out another resource to help get it to click.

>>7759091
You sound like a middle-schooler here to only rant on about how useless the math you are learning is. Go back s4s.

>>7759426
Not if you want to prove the Riemann hypothesis over finite fields.

>>7759471
I am reading baby Rudin again right now because I did very poorly in a class in it and now have to take the next class which builds on it.

Advice?

Is not wanting to do math exercises extensively in my free time a sign that I don't have the intrinsic motivation to succeed in math? When I'm assigned problems for class I like to get a head start on them and I often read ahead to material we will be covering soon. In addition I go over the textbook carefully to make sure I understand the concepts. In my free time I often like to read about math, things like expository articles, wikipedia, documents I find on the web, the history of various concepts, etc. But when it comes to sitting down with a book and just going through a bunch of exercises on my own, the idea just feels way too exhausting.

>>7759366
function butthole does not equal integral of circle to x dick function butthole divided by sideways tinier dick x of the bigger dick x

Answer is I don't know.

>>7759512
crashing this discussion

with no survivors

>>7759366
Pick any continuous function that does not vanish at 0

>>7759610
Like cosine?

>>7759366
For integrable functions, it's equivalent to f(x) != f(x) - f(0). There's a shitload of such functions.

A math question from a different thread that I found interesting:

>>7759439

>>7759479
Use any resource necessary to learn the material. If you are reading Rudin and it is not making sense, get another resource that can provide another explanation, merge the two to get a better idea or maybe the other explanation will suffice. If I am stuck I like developing some small intuition and going from there and that usually involves looking at the simplest cases and building up. In terms of Rudin itself, I did not like his Topology discussion but I do like Munkres and Gamelin/Greene better since those are topology books, so I took from that what I needed. Afterwards I understood the material and that's all that matters really as opposed to wanking off to some coveted book.

>>7757787
>Post your mathematical thoughts, questions, problems and have people discuss them with you.
why doesn't /sci/ have a sticky for people who want to get good at math?

>>7757834

This, this is how kikes ruined physics.

>>7760402
It used to a few years ago. I remember that at some point there was an olympiad sticky but it disappeared, for some reason.

>>7760446
Apparently that was a guy who hacked a mods account. He would sticky a new Olympiad question every day. Stopped once he got caught.

>>7760446
that's too bad, it would be quite useful

>>7760470
A lot of sci is composed of university students. Most of the people coming in from other boards wanting learn math are often talking about math at a highschool level. It would pretty much be a sticky for everyone but /sci/fags.

Not saying it wouldn't be useful, but I doubt any mathfag is going to make it. It's worth mentioning that some autistic guy who would spam a book list regularly decided to make a wiki around it. It has books for intro math. You may want to look for that.

>>7759448
Not the guy you're replying to. The proof you posted is valid as far as I can see, other than assuming the approximation is good enough to equate the limits. This isn't true for all limits, but it holds here. Wikipedia has a similar proof which covers this detail.

The pitfall that people refer to when they talk about 'proofs by cancellation' is where you have [math]\frac{\Delta f}{\Delta x}=\frac{\Delta f}{\Delta g}\cdot\frac{\Delta g}{\Delta x}[/math], and fail to account for when [math]\Delta g(x) = 0[/math].

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Why the fuck did I chose to study math again

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>tfw reading about Green-Tao theorem

>>7759273
I haven't fully understood your question, so this reply might not be relevant to what you were looking for.

For the Borwein integrals, the problem becomes more visual when we consider the Fourier transforms of the functions, and find that the integrals represent the convolution of functions (whose graphs are rectangles) evaluated at 0. If you're not familiar with Fourier transforms or convolution, wikipedia or somewhere else will provide more helpful introductions than I could. The significance of [math]\frac{1}{15}[/math] term is that these rectangles fail to overlap properly starting here (when the sum of their base lengths becomes too large), and the convolution of these rectangles fails to attain the value [math]\frac{\pi}{2}[/math].

I don't expect that last paragraph to be incredibly enlightening, but it's a brief overview of what's discussed in this paper (which includes the evaluations you were looking for):

http://www.schmid-werren.ch/hanspeter/publications/2014elemath.pdf

It should be an interesting and digestible read, and it emphasises the geometric interpretation of these integrals, which I hope is what you were asking about.

>>7759397
It's late so I might be wrong, but the result should be f(x)-f(0), not f(x). So any function with f(0)=/=0 is solution.

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>>7760693
best comment section on the internet is to the answer here:

http://mathoverflow.net/questions/25402/is-the-green-tao-theorem-true-for-primes-within-a-given-arithmetic-progression

>>7759255
My attempt for an explanation why many such integrals come out simple is that you look at integrands defined by series with small Kolmogorov complexity.

E.g. setting up a top-down theory of formal power series can be done concisely. Here the sine is something more compactly written down than some fourth-order polynomial, even. It's just a sequence [mat
h] a: {\mathbb N} \to {\mathbb Q} [/math] with [math] a_n := (-1) \frac {1} {(2n+1)!} [/math] that you eventually associate with the uneven monomials. You can set up algebraic differentiation here, mapping sequences with coefficients [math] (a_n) [/math] to [math] (a_n)' := (n·a_{n+1}) [/math] (as [math] \frac {d} {dX} X^n = n·X^{n-1} [/math]).
Now you think of [math] \pi [/math] not as a number in the first place, but a an equivalence class of certain very basic sums, which is a primitive (primitive recursive, lel) operation on such infinite lists again. E.g.
[math] \sum_{k=0}^\infty \frac {2k!} {(2k+1)!!} = \pi [/math]
It’s a simple ingredient of the theory, and if you stick to simply representable series, you’ll end up with simply representable results like that.

My point here being really that if you look at results in a refined area, then you can axiomize concisely, you don’t leave it soon.

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>>7760776
(cont.)

And maybe you view and understand
[math] \int_{-\infty}^\infty \frac {1} {1+x^2} dx = \pi [/math]
in a geometric context, where it derives easily.
Then you look at
[math] \int_{-\infty}^{\infty} e^{-x^2} dx = \pi^{1/2} [/math]
and think it’s funny two simple integrals end up at pi again.

But then you set up a concise probability theory on the reals and when searching for the normalization of the student-t distribution you’re lead to
[math] f(t) = \frac{ \Gamma ( \frac {\nu+1} {2} ) } { \sqrt { \nu \pi } \Gamma ( \frac {\nu} {2} ) } \left( 1 + \frac {t^2} {\nu} \right)^{- \frac { \nu + 1 } { 2 } } [/math]
and now in this theory the two [math] \pi [/math]’s above are the same pi. And I don’t mean they are the same number (which they are, in the theory of infinite series, of course), but I mean they are the same post in their individual theories.
(it was funny to me to see how Mochizuki does heuristic explainations using that second basic integral in his context btw., pic related - he's giving new "why"'s)

For those examples in particular, it’s important to note the impredicative character of limits. The successor function in Peano arithmetic
[math] s : {\mathbb N} \to {\mathbb N} [/math]
[math] s(n) := n+1 [/math]
is constructive, while saying
[math] \exp(x) : {\mathbb R} \to {\mathbb R} [/math]
[math] \exp(x) := \lim_{m \to \infty} \sum_{n=0}^m \frac {1} {n!} x^n [/math]
is, for starters, a request that this even defines a function. What I mean is that := must be put in quotes in the second definition.

>>7760780
*It's important because an impredicative definition of some shit might evaluate to whatever. Existence of "stuff" in a set depends on the ambient theory.
A limit on R requires a metric [math] \pi [/math] per definition arises as 2· the circumference of a unit circle in the 2-norm. Here we'd get to more why question if we hunt down all the magic numbers. I suppose unit 1 should be natural because of the ring-structure and the role of 1 in the reals and the 2-norm because ... something with bilinar metric forms in Riemann geometry making for a rich theory?

Consider the function [math]\cos(\pi n)[/math] where n is some natural number. This can be re-written as [math](-1)^n [/math]. Now consider the function [math]\cos( 3 \pi n)[/math]. Can this be re-written as [math](-1)^{n/3} [/math] even though this function is not entirely real for all n? Are they equivalent?

>>7760839
[math] \cos( \pi z) = \frac {1} {2} \left( e^{i\pi z} + e^{-i\pi z} \right) [/math]

and depending on how you choose the branch of the log (here [math] \log \left( (-1) \right) = i \pi [/math] )

[math] (-1)^z = e^{ \log \left( (-1)^z \right) } = e^{ z·\log \left( (-1) \right) } [/math]
[math] = e^{i\pi z} [/math]

>>7760865
(cont.)
or maybe you're already helped by pointing out 3n != n/3

>>7760839
Na man, cos(3n*pi)=cos(n*pi), if n is a natural number. So cos(3n*pi) cant be equal to (-1)^(n/3), since as you said cos(n*pi)=(-1)^n

>>7759280
That integral can be reduced to "What is the diameter of the unit circle?"

I'm confused by this statement because half the area of the unit circle does not equal 2. Yet in the integral the area under the curve from 0 to pi is 2. Wouldn't the area under the curve of the integral match the area of half the unit circle?

6/2(1+2)

>>7759481
consider amphetamines

>>7759481
Until you get through the basic courses most exercises will be extremely boring/mechanical

>>7759160
The proof is basically just cancellation but you need to deal with the denominator being 0. I don't recall the details to be honest friend.

>>7759160
>>7762064
I.e. delta g can be identical 0 which is a problem. Need an auxiliary function to deal with this.

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>>7759448
The ideas are there and only a little adjustment would yield a perfectly rigorous proof yield a perfectly good proof using little-oh notation, however you do not explain how good an approximation f(x)+hf'(x) is to f(x+h), what the approx sign means and how small the error is so is not really a proof.
Here is how to do the proof in a completely elementary and rigorous way.

Observations regarding primes?

>>7762215
All twin primes are at least two numbers away from another prime
All primes except for the number 2 are odd
If you string together every prime number in order in a decimal, the number you get is a huge waste of time, as proved by Copeland and Erdos
No composite number is prime
All primes greater than 2 are at least two apart from another

>>7762246
>>Observations regarding primes?
So....no?

>>7762215
You can find arbitrarily large gaps between primes
If you arrange the primes in increasing order, then [math] p_{k+1} \le p_1 \dots p_k +1[/math]
For each n greater than one, there is a prime p such that n < p < 2n

>>7762246
3 is one apart from 2 :^)

>>7760339
I happen to have Munkres book right next to Rudin. I'll take your advice and do that, thank you.

>>7762361
It's also still 2 apart from another prime :^)

>>7759137
http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx#Extras_DerPf_ChainRule
It builds on previous proven stuff. In the end it all boils down to epsilon delta proofs:
http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx

I'm working through Eisenbud's commutative algebra book. I've seen primary decomposition of ideals before, and that's all good and well, but the business with primary decomposition of modules is bothering me. Is there some intuition for the definition of, say, an associated prime of a module? What does primary decomposition of general modules buy you in algebraic geometry, aside from just isolated/embedded components (which doesn't seem that useful).

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Hello /sci,
In highschool I used to play a game called 24, the objective is to use basic operations to reach 24.
Pic related, for instance
24 = 24 + 8 - 6 - 2
What I've noticed is you are allowed to use any operation not having other digits, so for instance sqrt(), n!.
The interesting thing I discovered is you can make 24 out of any number using the operation (4!=24)
For instance: sqrt(sqrt( ... sqrt(n))...) tends to 1, and once you have four ones you have (1+1+1+1)! =24
This ultimatly destroys the game, but hee what you gonna do.
Also can anyone find the solution for this: 8 1 9 2.

>>7759479
(Somewhat) unpopular opinion, but this is a math ramble thread so I don't feel too bad.

Go find another analysis book.
Rudin is a terrible text that should never be used to teach anybody and it's a travesty that it's as popular as it is.

It's a difficult book not by nature of teaching hard material (introductory real analysis is not that difficult) but by intentionally obscured and incomplete presentation.

Yes, you do need to learn how to fill in gaps and examples, but throwing a shitty book at you and telling you to work it out yourself because it's not the book's job to be clear isn't a good way to learn that skill.

>>7763072
(sqrt(9) + cbrt(8) - 2 + 1)!

>>7763072
[-(8-1)+9+2]!

>>7763083
That is an invalid operation since the writen form for cbrt has a 3.
I know I know, seems picky but I don't make the rules (or do I??).
8*(9/(2+1)) is a solution.

I have a question, is there a way to discover the number of impossible tiles by brute forcing it?
I'm guessing ruling out symmetries and using only (+; - ; * ; / ; !) and integers 0 to 10, something like 20% of tiles are impossible.

To clarify starting with tile (0, 0, 0, 0) you get (0!+0!+0!+0!)!=24 so it is possible
then (0, 0, 0, 1) also possible (0!+0!+0!+1)! =24.
and remember to skip (1,0,0,0); (0,1,0,0); (0,0, 1,0) for they are permutations of the forementioned.
There are 104 different tiles, and remember that the order of operation counts.

For you yongsters out there, how bout this tile (9, 4, 2, 3)?

>>7763489
9*4*2/3