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2

@goblin has already given you a good list of books that you could have a look at. I second this recommendation. Another recommendation is to take a math class on something like abstract algebra or discrete mathematics. In studying, for example, abstract algebra you learn how to think abstractly about concepts. You are given definitions of certain things and ...


3

Note that proof theory isn't the study of how to write an informal proof, its the study of certain logical calculi and the proofs they accept/reject. If I understand correctly, this isn't what you're looking for. You want something more like: How To Prove It How To Think Like A Mathematician Mathematical Proofs: A Transition To Advanced Mathematics Good ...


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Machine learning, if you really want to understand it rather than just believe the algorithms do what they are supposed to do, requires quite a heavy dose of mathematics. That mathematics is written in a language and that language is set theory a la Halmos (i.e., the naive kind). If you wish to properly understand the mathematics involved in ML, then it is a ...


1

The trick I used to memorize them actually stemmed from formal logic (which you may or may not have had any exposure to): The symbol $\land$ is a way to symbolize the binary connective "and". Notice it looks like a "pointy" $\cap$. Similarly $\lor$ (or) looks similar to $\cup$. Now, $$x\in A\cap B$$ can be read as $$x\text{ is in } A \textbf{ and } B$$ and ...


2

Taking $s\neq 0$ $r+s\sqrt t \in \mathbb Q(r+s\sqrt t) \implies s\sqrt t\in \mathbb Q(r+s\sqrt t)\implies \sqrt t\in \mathbb Q(r+s\sqrt t)$ since $s\neq 0$ Now $\mathbb Q(\sqrt t)$ is the smallest field containing $\mathbb Q$ and $\sqrt t$ Thus $\mathbb Q(\sqrt t)\subset \mathbb Q(r+s\sqrt t)$ similarly $r,s,\sqrt t\in \mathbb Q(\sqrt t)\implies ...


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The question is false when $s = 0$ and $t$ is not a square, for obvious reasons. So suppose $s\neq 0$. By definition, $\mathbb{Q}(r+s\sqrt{t})$ is some field (actually, the smallest) containing both $\mathbb{Q}$ and $r+s\sqrt{t}$. But if it contains $\mathbb{Q}$, then in particular it contains $r$ and $s$. Now: as it contains $r+s\sqrt{t}$ and $r$ and ...


1

I suggest you study some Math history to see that people speculated about all sorts of things (they still do), and then they either lived with their speculations or they tried to prove them to be true. Even the most famous 'Mathematicians' have claimed things that were later proved false. I also suggest you ask yourself whether your claim that people ...


5

A typical way to find a formula is to start (and do practical work) with some (more or less) well known formula. Then you come into the unlucky situation that you have to extend it a bit, which often means hard work (brute force, laborious manual work, tricking a computer algebra system to help you). Then you do some guesswork based induction to get a ...


3

You might want to try the book "Birth of a Theorem: A Mathematical Adventure" by C├ędric Villani. There, he describes how he developed one of the greatest theorems ("formulas") of mathematics. In general, you can't generalize it. Nowadays, you can even use computers to find "formulas", such as the classic Four color theorem. You often have a goal you want to ...


22

How do mathematicians find formulas? Short answer: Observation, creativity, and hard work. A formula I personally came up with: As other answers have already addressed the triangle formula, I thought I would share a personal example of observing something, being creative, and then working hard to prove a formula I came up with (I'm sure others have ...


6

I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although ...


4

First note that the equation for a straight line is $$ f(x)= mx+b $$ For simplicity, let $b=0$ and let $mx\geq 0$. Note that now $f(x)$ passes through the origin and we have $$ f(x)=mx $$ Now let's find the area under $f(x)$ from $0$ to $k$, where the distance between $0$ and $k$ represents the base. Also note that $f(k)=mk$ gives us the height of the ...


2

This is what I personally think: They derive it and express them as a function of some variables and constants. How do they derive? By step-by-step calculations/operations from already known axioms, theorem, etc. or may be from the flow of thoughts which strike their minds and then they prove it later using already known axioms, theorems, etc. How was the ...


8

There are many ways in which a mathematician or scientist can create a formula. Sometimes they use proportionality. For a triangle with a fixed base you can observe that the area of the shape is proportional to the height. In other words if you double the height you double the count of unit squares in it. You can follow the same argument by fixing ...


27

For the particular case of the area of the triangle, here is one way to rationalise it: Decompose the triangle of interest into a purple piece and a red piece. Replicate the red piece, colour that replicate blue, and rotate it and attach it like above. Do the same to produce the green piece. Then, the area of (red piece + purple piece) is $\frac{1}{2}$ ...


5

Mathematicians are interested in combining postulates and axioms to yield new theorems or axioms. For instance, the area of a square is axiomatically the square of its length ( though this can be proven), and if we draw a diagonal across any square we will half the total area into two separate areas. And, these two separate areas are the areas of a right ...


2

As someone who is mostly self-taught and your age, my advice is you must first get comfortable with abstraction and proofs. You should be arguing with your teacher either mentally or verbally. I built up a great relationship with my trigonometry teacher this way because he welcomed my pressing curiosity, although there is no guarantee this can work for you. ...


2

As 5xum mentioned in the comments, union starts with a u and has a symbol $\cup$ that looks very much like a $u$. Then intersection is simply the same symbol flipped $\cap$. As for what they mean, you can think of union $A \cup B$ as a cup (indeed, the LaTeX command for it is \cup) in which you pour all of the elements of both $A$ and $B$, whereas $A \cap ...


1

Honestly, for the most part you can even skip reading most of the book. Problems are millions of times more important. When you get a new book, don't even read it. It doesn't matter. Go directly to the problems, skipping everything in the book. Don't even glance at chapter titles. Go directly to the first problem. Obviously the problem probably won't make ...


0

A strong base in Precalculus course (US curriculum) is highly recommended. It'd be nice to learn the basics of differential and integral calculus as well, but your university will certainly cover that. Focus on learning precalc thoroughly, and if you have time, begin study of calculus. However, in college, as Joel commented, retake calculus from the ...


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For starting your mathematics journey you have to work on three areas which are considered as foundations for reading advanced math. Linear Algebra, Calculus, Probability - These topics are essential for your future goals in your mathematical investigations. For linear algebra, MIT courses are really good. There are bunch of books out there for linear ...


0

First one should note that if $p=1$, the $I = \ln(x)|^b_a$, and I guess you mean $p\ge 0$ is a non-negative number. In this case, if $0<a,b<\infty$, then the integral will converge. The problem comes in if either one of $a,b=0$.


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If $-p>0$ there isn't problems. If $-p<0$, then the integrand function is $$\frac{1}{|x|^p}$$ this integral doesn't diverge if $|x|\neq 0$ in $[a,b]$.


3

You didn't say your geographical area. In the United States, departments are typically not only willing to consider applicants without an MS--that most admitted PhD candidate have not had master's training is sort of the norm. As a consequence, the US degree process usually involve some number of years of course work (1 or 2 typically) with a ...


2

Just realized that this question is old; oops, I've already written an answer. Guess I might as well post it. I completely agree with Paul Garret when he says: My advice would be to not feel obliged to comply/obey the implied commands in textbooks and other school-math. Instead, try at every moment to move toward a thing that interests you, that ...


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If you are truly interested in Cryptography and Computer Science you can complete a CS course in college, with a focus on mathematics.Thus combining two things you love. It's entirely possible and quite a few people follow this path. Most of the courses, focus on possibilities, algorithms,advanced math and so on. Later you can pursue a career as a ...


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Many French dissertations can be found on "HAL archives ouvertes". It is a law requirement in Germany to publish ones PhD dissertation. I believe all of them are visible by Google SCholar. I'm not sure whether you can download them all thought. You may have to require the hardcopy.


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Partial answer, to find at least the titles, see The Mathematics Genealogy Project. From this you might google the title...


2

To gain mathematical maturity, I think you should work on both new stuff (that you might find quite challenging) and also old stuff (which is usually easier). Good ideas feed off each other given a half a chance. When learning new stuff, it is often good to seek out ideas that unify and/or clarify what you already know. The following subjects are ...


0

Testimonials will not be a good way of deciding what type of medium you learn from. The way one person learns best may not be the way another person does. I suggest whenever starting a new subject, try out a few different materials and see what works best for you. If you like the video lectures you find for a subject -- use them. If you find a textbook ...



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