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1

Applications to proving linear-algebraic results: Doron Zeilberger, A combinatorial approach to matrix algebra. The same, but scanned better. Richard Swan's combinatorial proof of the Amitsur-Levitzki theorem and a correction. Dodgson's theorem proved using Zeilberger's famed talent for matchmaking. Donald Knuth, Overlapping Pfaffians. Jiang Zeng, A ...


3

The classical example is Straubing's combinatorial proof of the Cayley–Hamilton theorem.


0

I believe that one important aspect of math teaching is linking the formal and the intuitive. Thus, epsilon-delta or epsilon-M proofs should be taught only if it is somehow linked to students' intuitive idea of a limit. I think this is possible by: motivating the definition, and perhaps playing a "game" with students, where you give them a function, say ...


0

As someone mentioned above, your question is really difficult to answer, especially the modern topics in "pure" linear algebra. It is hard to say whether some topics are "pure". Anyway, I think Brian C. Hall's book (maybe GTM222) about matrix lie group, somehow may help you.


0

This problem that has a bounty on it, contains a reference to a solution using probability theory. I think the probability solution is quite elegant. It works though by showing an even stronger result than the one asked for. So, I suspect another proof must exist, but that proof might be more complex.


3

You can reduce the possibilities quite quickly. Clearly the final digit has to be $0$, and then divisibility by $10$ is assured as is divisibility by $9$. You can only have divisibility by $5$ if $5$ is in the fifth place, and divisibility by $2$ only works if the digits in even places are even. Divisibility by $8$ means the digits in the $7, 8$ places are ...


2

Try to do putnam competition tests. You don't really learn a lot of theory but it's a good exercise in coming up with proofs.


0

Assembling a 100+ piece jigsaw puzzle. This will improve your ability to visualize shapes.


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Project Euler: is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems. Solving Project Euler problems is an ...


-1

How about calculating the value of 2^52 without a calculator


1

Carry a book about an area of maths, and read bits of it? Do smallish mathematical puzzles? Practice mental arithmetic?


2

Learn a little coding (for example python) and practise with old problems from the google coding championships.


2

How about inventing games with dice or coins, and then analyzing them?


0

Have you studied abstract algebra? I believe it is a natural step after learning linear algebra, as a vector space is often one of the (if not the) first abstract algebraic structure a student encounters. Perhaps take a look at Dummit and Foote's text. It also covers much of what an "advanced" linear algebra text might, and much more.


0

You really don't need sets at all. In 'Modern Algebra I-II' of van der Waerden there is not one single notation from set theory. You could handle the property of being a natural number instead of the set of natural numbers etc and in the reality properties often works better than sets. I mean, the property of being an electron is more adequate than the set ...


0

No $3$ consecutive odd integers are prime except $3,5,7$.


1

You should read Martin Gardner's books more than anything I think. They'll motivate you to learn mathematics far more than a standard text in some subject like trigonometry or basic calculus. Check out ...


0

I think it totally depends on yourself. You like mathematica, then you use it. You wish to learn Latex, then you learn it. And all the communication problems seems unreasonable, if you wish, you can even write thing down and take picture of it. There is a lot of ways to learning, discussing math. No big deal which way you prefer.


3

I think that number theory and group theory are two subjects that are easy enough to understand with no prior knowledge and most importantly are very good for learning styles in proof writing. Two good books. Friendly Intro To Number Theory Silverman ISBN 9780321816191 Edition 4 Book Of Abstract Algebra Pinter ISBN 9780486474175 Edition 2 I hope this ...


2

One very important subject of mathematics you must learn is Linear Algebra. Calculus is a MUST because it is the foundation for computation. I recommend this book because a provides you a structured foundation for proof strategy. It is one that all higher mathematicians (even me) have read. So, the path that I recommend to you is finish trigonometry, then ...


2

You need to learn trigonometry to understand calculus. You need to learn calculus to understand the set $\mathbb{R^n}$ and its properties. You need to learn those properties to prove things. Pick up a Stewart's calculus book and begin there. Oh yeah, and don't get frustrated if it takes you over three hours to understand a concept. That's uncommon, but ...


1

I'm partial to Ullman, Hopcroft's text. Jeff Ullman runs a Coursera course on Automata theory, and I have heard very good things from someone I know who is a linguist. The text gives a very comprehensive overview of formal languages and automata, as well as issues of computability and complexity. I think it is very thorough and well done, plus having a ...


0

I have found that this pdf http://www.staff.science.uu.nl/~ooste110/syllabi/compthmoeder.pdf helps give a mature & brief theoretical introduction. Oosten also includes detailed proofs of the main theorems in recursive function theory that are necessary for carrying out rigorous details in every domain of mathematical logic related to the use of recursive ...


1

There are Graeco-Latin squares of all orders greater than two except of order six.


1

Classics: Introduction to Metamathematics by S.C. Kleene. One of the first books about computation theory. General introduction to the mathematical logic. Includes very basic set theory, first order logic, formal number theory (including Gödel), recursive functions and Turing machines. Centered around the logic. See Teach Yourself Logic, #8. The Big Books — ...


0

I really like Introduction to Matrix Analysis by Richard Bellman.


3

I would go for some later chapters of Advanced Linear Algebra, from Steven Roman, and Linear Algebra and Geometry, from Kostrikin. About open problems in Linear Algebra, you can take a look at the comments in this question: Are there open problems in Linear Algebra?. Particularly, I find it difficult to find open problems in linear algebra, since (in my ...


1

For all $n$, the braid group on $n$-strings $B_n(M)$ of a closed orientable surface $M$ is torsion free unless $M=S^2$, the $2$-sphere. The reason the same proof for other surfaces can't be used for the sphere is that the sphere is the only closed surface with non-trivial higher homotopy (because all other surfaces have a contractible universal cover, ...


4

Here is an example. The question is how to show that $$\binom{n}{k}^{-1}=(n+1)\int_0^1 x^k (1-x)^{n-k} \, dx. $$ To make this self-contained, I'll paste this answer below: Let's do it somewhat like the way the Rev. Thomas Bayes did it in the 18th century (but I'll phrase it in modern probabilistic terminology). Suppose $n+1$ independent random variables ...


2

Thomas Bayes showed that $${n \choose k}\int_0^1 x^k (1-x)^{n-k}\mathrm dx = \frac{1}{n+1}$$ by pure thought, without using calculus (for all integers $k,n$ with $0 \leq k \leq n$). His argument, known as the Bayes' billiards argument, uses two equivalent probabilistic stories about picking random points on a number line from $0$ to $1$. That's just one ...


1

All compact orientable two dimensional surfaces admit a metric with a constant non-positive curvature, except the sphere $S^2$.


22

Why study set theory?      We like to think that mathematics developed from the need of our ancients to count things. I have four sheep, you have sixteen camels, my tribe has ten dozens of men, you have six hundred wives... etc. etc. But if you look closely, counting how many things you have of a certain type, first required you have be ...


1

There are many books which will generate enthusiasm for mathematics in a layman reader. Some of them are listed below: 1) One, Two, Three, Infinity : George Gamow 2) Fractals, Music, Hypercards and More ..., Mathematical Recreations from SCIENTIFIC AMERICAN Magazine : Martin Gardner 3)Time Travel and Other Mathematical Bewilderments : Martin Gardner ...


3

To expand on Ittay Weiss's post, and maybe this is not exactly what you are looking for, but set theory deals with the mathematical universe. More concrete, set theory has the ability to describe independence results. This, in my opinion, is the real importance in set theory. As Ittay alluded to above, consider $\mathbb{R}$. Does there exist a set $A$ such ...


1

I've not ever heard of either of those books, but I've taught linear algebra before and my advice is to buy (or borrow or rent or whatever) the book your instructor assigned. My reasons: Homework is often assigned out of the book so you need access to the class text anyway. Most linear algebra books cover pretty much the exact same stuff so the advantage ...


0

Here is one I found. The sum of two complex sinusoids with the same frequency is another sinusoid of that frequency, with 1 exception: For amplitudes $A_1,A_2 \in \mathbb C$ and phases $\varphi_1, \varphi_2 \in \mathbb R$ and frequency $f \in \mathbb R$: If $f_1(x) = A_1\cos(2\pi f~x + \phi_1)$ and $f_2(x) = A_2 \cos(2\pi f ~x + \phi_2)$ ...


11

Naive set theory: Set theory is the common language to speak about mathematics, so learning set theory means learning the common language. Another aspect is that of counting. Cardinality of sets is a very fundamental notion which be treated naively quite efficiently. Cardinality means counting, so learning set theory means learning to count (beyond the ...


0

Mersenne prime numbers $M_{n}=2^n-1$ have $n$ odd except $n=2$.


0

To me*, restriction estimate means a bound on the norm of an operator $T:X\to Y$ where $X$ is some space of functions in a domain $\Omega$ (often the entire Euclidean space) $Y$ is some space of functions on a subset $E\subset \Omega$ of strictly smaller dimension than $\Omega$ itself For smooth functions, the operator agrees with the restriction ...


0

Fermat's last theorem: There are no solutions in $\mathbb{Z}$ in the equation $a^n+b^n=c^n$, except for $n=2$ $0$ properties: The only number that does not have a multiplicative inverse is $0$. The only number that satisfies $-x=x$ is $0$. $\frac xx=1, x \not \in \{0\}$ $\frac 0x=0, x \not \in \{0\}$ $x^x\in \mathbb{C}, x \not \in \{0\}$


7

$x^a-y^b=1$ has no solution in prime numbers, except $3^2-2^3$.


3

For any prime $n$ except $2$, $x^n+y^n=z^n$ has no solution in nonzero integers.


1

Just an observation from $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$: For any prime $p$ such that $p\neq 2$, $$\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}}<p$$ and $$\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2.$$


1

Zsigmondy's theorem. If $a,b\in\mathbb Z, n\in\mathbb N_{\ge 2}, (a,b)=1, a>b$, then $$\exists p\in\mathbb P( \forall k\in\mathbb N\cap[1;n)(p\mid a^n+b^n\wedge p\not\mid a^k+b^k))$$ The only exception to this is $(a,b,n)=(2,1,3)$. The following variation of it If $a,b\in\mathbb Z, n\in\mathbb N_{\ge 2}, (a,b)=1, a>b$, then $$\exists p\in\mathbb ...


2

For the "foundational" debate of the '30s, see : Marcus Giaquinto, The Search for Certainty : A Philosophical Account of Foundations of Mathematics (2002). For some impact on mathematics of one of the above philosophical "schools" (e.g. Intuitionism and related : Intuitionistic Logic, The Development of Intuitionistic Logic, Luitzen Egbertus Jan Brouwer, ...


1

I want to highlight the Aristotelian logic. In human language there was discovered the most general mathematical system which even today forms the basis of science and especially mathematics. From tautologies, combinations of sentences which independently of the context seems to be true, truly non-trivial systems of logical deductions was evolved.


3

Here is an extremely good source I have recently found: Panorama And in general, all the page Mathigon is outstanding well done.


6

$$\forall x,y \in \mathbb{Z_+}$$ $$|y^3-x^2| \ne 2 $$ except for $x = 5$ and $y=3$ Ie 26 is the only number between a square and a cube.


4

The word I've most commonly seen used for a point at which a function is continuous but not differentiable is "kink", as in: "The function $f(x) = |x|$ has a kink at the origin."


5

All spheres $S^n$ are simply-connected except $S^1$.



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