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Conspicuous by its absence among the above answers is any celebration of the usual notation for multiplication.


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$(\mathbb{R^{+}},\bullet)$, the multiplicative group of positive real numbers, is isomorphic to $(\mathbb{R},+)$, the additive group of real numbers. Consider logarithmic function $x \mapsto ln(x)$. Since $ln(ab)=ln(a)+ln(b)$ and it is totally defined on $\mathbb{R^{+}}$ and surjective, it is a homomorphism from $(\mathbb{R^{+}},\bullet)$ onto ...


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Time. You can't turn it back (no inverse) and you can't stop it (no identity).


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I like french invention ∃! to designe unicity. It is nicely useful in mathematics but is not universally used. There are a lot of illustrations of this ("∃x such that f(x) is something and this x is unique" changes by "∃!x such that f(x) is something" or "Prove that ∃!x such that something". A professor of University of Toledo (Ohio) adopted this notation ...


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The switch from Roman to Arabic numerals has to be a fundamental improvement. Try to calculate IL divided by VII !!


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Coxeter co-joined Stott's operator and Schläfli's description of regular polytopes into a combined notation that is used to this day. Stott wrote the truncated icosahedron as e_1 I. Wythoff wrote it as t_0,1 I. Coxeter writes it as t_0,1 {3,5}. Krieger writes it as x3x5o. If one wants to write the figure where the inter-hexagon edge is phi, then only ...


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Great question! I would like to suggest you to have a look to some Python libraries specialized on Mathematics, like sympy or gmpy. I use them often to study and make my tests, and I miss always some extra explanations or samples in the online documentation. They are really great, and some extra samples and theory-related explanations would be imho a good ...


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One place to look at is Topospaces (The Topology Wiki), which was suggested by a now-deleted user in a now-deleted answer. The description says This is a pre-alpha stage topology wiki primarily managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago. We have over 400 articles including some material in basic point-set topology. As ...


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A famous example is the law of quadratic reciprocity. Wikipedia says that several hundred proofs of the law of quadratic reciprocity have been found. Many details concerning proofs by different methods can be found at the question at MO.


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Almost all Gelfand's books (like Discriminants, Resultants, and Multidimensional Determinants,Calculus of Variations ) seem to fit the bill.


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I feel this way about Quantum Computing Since Democritus by Scott Aaronson, and also Gilbert Strang's books on applied math.


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In advanced mathematics, it is pretty much impossible to avoid "technical jargon". Without the technical terms, and the precisely defined concepts they stand for, you can't do very much.


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This is an answer covering a somewhat more exotic proof. The claim is one I used in my answer at prove or disprove $H$ is a subgroup but I will expand on the details here and write it as nicely as possible. For more about when induction can be done on these more exotic examples, see my blog post ...


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Initial comments: This is an excellent question in my opinion and is just what the proof-writing tag is for. Unfortunately, there are often many problems plaguing beginners when it comes to induction proofs: Why induction is a valid proof technique should be understood at the outset, and this is rarely the case. Less relevant in high school or undergrad, ...


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Here is my template for case #$1$: As an example, let's prove by induction that $\sum\limits_{k=0}^{n-1}2\cdot3^k=3^n-1$. First, show that this is true for $n=1$: $\sum\limits_{k=0}^{1-1}2\cdot3^k=3^1-1$ Second, assume that this is true for $n$: $\sum\limits_{k=0}^{n-1}2\cdot3^k=3^n-1$ Third, prove that this is true for $n+1$: ...


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A (canonical, if not exciting) proposal for case 2 (strong induction): Let's prove that every integer $\ge 2$ can be expressed as a product of one or more prime numbers. For $n \ge 2$, we define the statement $P_n \;:\; \forall k \in \{2,\ldots,n\}$, there exists a (finite) sequence of prime numbers $p_{n_1}, \ldots, p_{n_k}$ such as $k = p_{n_1} \ldots ...


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A proposal for case 1 (simple induction): Here are some generic recommendations for beginners I have given in previous answers (here or here): Write down in full length the statement $P_n$ to be proven at rank $n$, and the range of values $n$ over which $P_n$ should stand Clearly mark the anchors of the induction proof: base case, inductive step, ...


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For a general audience I would like to talk about Origin and Convergence of infinite series. I would like to use the following example and explain why that happens. Let $$S=\sum_{n=1}^{\infty}(-1)^n$$ $$S=-1+1-1+1-1+\cdots=-1-S\implies S=-\dfrac{1}{2}$$ $$S=(-1+1)+(-1+1)+(-1+1)\cdots=0$$ $$S=-1+(1-1)+(1-1)+(1-1)+\cdots=-1$$ Therfore ...


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$$\int_{0}^{1} \frac{\ln(x^2+1)}{x^2+1}$$ I really like this. It seems to be posted again and again. Solutions can be found here: Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$


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I have found the identity below from which can be deduced infinitely many others of increasing degree by the elementary theory of elliptic curves. The 6-tuples indicate the coefficients of, respectively $n^5$, $n^4$, $n^3$, $n^2$, n and 1 A = (1, 10, -8, 16, 64, -32), B = (1, -10, -8, -16, 64, 32), C = (-1, 8, 8, -16, 80, 32), D = (-1, -8, 8, 16, 80, ...


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Some inductions are nice because they use nice symmetries. This question has some nice inductions.


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The Jacobson radical. Take a noncommutative ring $R$ with 1. Any left ideal is either contained in another or is maximal. The elements common to all maximal left ideals, i.e. $$ J = \bigcap_i M_i, $$ is a group in two ways: It is an abelian group because it is an ideal (inherits group additivity from $R$, pretty obvious). It is group under circle ...


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I used to be exactly the same way. It would take me several seconds to multiply even single-digit numbers. I'm getting better now using 2 things: Practice — just find the kind of problems you want to get better at solving, and solve them. The Trachtenberg system — provides some great techniques for doing multiplication and division more efficiently. The ...


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I assume you're going for a big list? Here are several: Many strongly minimal theories (e.g. $\mathbb Q$-vector spaces, algebraically closed fields of a fixed characteristic) More generally, any $\aleph_1$-categorical theory which is not also $\aleph_0$-categorical. Shelah has an $\omega$-stable example with $\textrm{ENI}$-depth two (so is not ...


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Graph Theory: the complete graphs; the complete bipartite graphs; trees; the Petersen graph.


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Finite groups: The cyclic groups; the dihedral groups; the symmetric groups; the alternating groups; the quaternion group.


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When I was in kinder-garden I was sick one day so I missed class and the next day the teacher gave us a quiz on basic multiplication. But I didn't know what multiplication was so when I got the test paper I thought that she wrote the + signs really badly. I didn't want to make her feel bad by pointing it out (not that my writing was any good either). So I ...



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