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Real world uses of hyperbolic trigonometric functions
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I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses outside academia?
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On a map using the Mercator projection, the relationship between the latitude L of a point and its y coordinate on the map is given by $y = \operatorname{arctanh}(\sin(L))$, where $\operatorname{arctanh}$ is the inverse of the hyperbolic tangent function.
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|soft-question|big-list|applications|hyperbolic-functions|
| 0
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Do complex numbers really exist?
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Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are? This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people actually ask this question.
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To the degree that anything actually "exists" in math, yes complex number exist. Once you accept that groups, rings and fields exist, and that isomorphism of rings makes sense, complex numbers can be recognized as (isomorphic to) the subring (which happens to be a field) of the ring of $2 \times 2$ real matrices. Generators of this subring are the following matrices $ \begin{vmatrix} 1 & 0 \\ 0 & 1 \\ \end{vmatrix} \;$ and $\; \begin{vmatrix} 0 & -1 \\ 1 & 0 \\ \end{vmatrix} $ which correspond to $1$ and $i$ in the normal notation of complex numbers. As people tend to accept that matrices exist, this may be a convincing argument.
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|soft-question|complex-numbers|education|philosophy|
| 0
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How do you determine if a point sits inside a polygon?
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Given the coordinates of a point $(x, y)$, what is a procedure for determining if it lies within a polygon whose vertices are $(x_1, y_1), (x_2, y_2), \ldots , (x_n,y_n)$?
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I will assume the polygon has no intersections between the edges except at corners. Call the point $(x_0, y_0)$. First we determine whether we are on a line - this is simple using substitution and range checking. For the range checking, suppose we have a segment $(x_1, y_1)$, $(x_2, y_2)$. We check that $x_1\leq x_0\leq x_2$ or $x_2\leq x_0\leq x_1$ and do the same for $y$. Now, if we aren't on a line, we draw a ray from the point and count how many lines you intersect. If the line doesn't intersect a corner or contain a non-degenerate segment of an edge, then an odd number of intersections should mean you are inside and an even number mean outside. If we intersect with a corner, it is more difficult. Clearly, the ray could leave with it only intersecting a corner, but it could also pass through the corner without entering the polygon. One solution is to pick the ray so that it doesn't intersect a corner or go along a line - this should always be possible. We can also use cross product
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|geometry|algorithms|
| 1
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Are the "proofs by contradiction" weaker than other proofs?
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I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons. For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice. Is there a similar case for proofs by contradiction?
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Nearly always the direct proof is easier to understand, shorter, and more helpful!
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|logic|proof-theory|
| 0
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Are the "proofs by contradiction" weaker than other proofs?
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I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons. For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice. Is there a similar case for proofs by contradiction?
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Sometimes you might want to know not just that there exists something, you might want to know how to actually go about finding it (and related questions like how quickly you can find it). Proofs by contradiction are non-constructive, while direct proofs are typically constructive in the sense that they actually construct an answer. For example, the proof that there are infinitely many primes usually proceeds by contradiction. However, you can make it a direct proof which gives the stronger result that the nth prime is less than e^{e^n}. (This is a good exercise to work out for yourself, but you can also find it as Prop 1.1.3 in my senior thesis and probably many other places as well.)
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|logic|proof-theory|
| 0
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Are the "proofs by contradiction" weaker than other proofs?
|
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons. For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice. Is there a similar case for proofs by contradiction?
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Most logicians consider proofs by contradiction to be equally valid, however some people are constructivists/intuitionists and don't consider them valid. ( Edit: This is not strictly true, as explained in comments. Only certain proofs by contradiction are problematic from the constructivist point of view, namely those that prove "A" by assuming "not A" and getting a contradiction. In my experience, this is usually exactly the situation that people have in mind when saying "proof by contradiction.") One possible reason that the constructivist point of view makes a certain amount of sense is that statements like the continuum hypothesis are independent of the axioms, so it's a bit weird to claim that it's either true or false, in a certain sense it's neither. Nonetheless constructivism is a relatively uncommon position among mathematicians/logicians. However, it's not considered totally nutty or beyond the pale. Fortunately, in practice most proofs by contradiction can be translated into
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|logic|proof-theory|
| 0
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Are the "proofs by contradiction" weaker than other proofs?
|
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons. For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice. Is there a similar case for proofs by contradiction?
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At first this seems like a silly question - after all isn't the point of a mathematical proof to be a proof and hence to be beyond question. But of course, to prove anything we need assumptions and some people do disagree with the axioms commonly used by mathematicians. I don't have much knowledge of this view, but I am sure they have theorems that show that contradiction like proofs are valid (given their axioms) within certain conditions. I would recommend going along with what everyone else does and treating "proofs by contradiction" as equally valid, unless you have investigated the Constructivism view and you decide that they are correct. As to whether they are clearer, that will depend on the actual proof. Sometimes the clearest way to make a proof is to start from the assumptions and see what they are really saying and why that is going to lead to a contradiction. The most illustrative proof depends on the circumstances.
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|logic|proof-theory|
| 0
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Are there any functions that are (always) continuous yet not differentiable? Or vice-versa?
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It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable? The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But are there any that do not satisfy this?
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Actually, in some sense, almost all of the continuous functions are nowhere differentiable: http://en.wikipedia.org/wiki/Weierstrass_function#Density_of_nowhere-differentiable_functions
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|real-analysis|continuity|
| 0
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What exactly does it mean for a function to be "well-behaved"?
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Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case). Can someone explain it to me, and if there is an explanation of the why as well, I would be grateful. Thanks! EDIT : To give one example where the term appears, see this Wikipedia entry for utility functions, which says at one point: In order to simplify calculations, various assumptions have been made of utility functions. CES (constant elasticity of substitution, or isoelastic) utility Exponential utility Quasilinear utility Homothetic preferences Most utility functions used in modeling or theory are well-behaved . They are usually monotonic, quasi-concave, continuous and globally non-satiated. I might be wrong, but I don't think "well-behaved" means monotonic, quasi-concave, continuous and gl
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In the sciences (as opposed to in mathematics) people are often a bit vague about exactly what assumptions they are making about how "well-behaved" things are. The reason for this is that ultimately these theories are made to be put to the test, so why bother worrying about exactly which properties you're assuming when what you care about is functions coming up in real life which are probably going to satisfy all of your assumptions. This is particularly ubiquitous in physics where it is extremely common to make heuristic assumptions about well-behavior. Even in mathematics we do this sometimes. When people say something is true for n sufficiently large, they often won't bother writing down exactly how large is sufficiently large as long as it's clear from context how to work it out. Similarly, in an economics paper you could read through the argument and figure out exactly what assumptions they need, but it makes it easier to read to just say "well-behaved."
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|terminology|analysis|economics|
| 1
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
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The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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Now that Wiles has done the job, I think that Fermat's Last Theorem may suffice. I find it a bit surprising still.
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|soft-question|big-list|examples-counterexamples|
| 0
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
|
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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An infinite amount of coaches, each containing an infinite amount of people can be accommodated at Hilbert's Grand Hotel . Visual demonstration here .
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|soft-question|big-list|examples-counterexamples|
| 0
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
|
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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All of three dimensional space can be filled up with an infinite curve .
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|soft-question|big-list|examples-counterexamples|
| 0
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
|
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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My first thought is the ham sandwich theorem --given a sandwich formed by two pieces of bread and one piece of ham (these pieces can be of any reasonable/well-behaved shape) in any positions you choose, it is possible to cut this "sandwich" exactly in half, that is divide each of the three objects exactly in half by volume, with a single "cut" (meaning a single plane).
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|soft-question|big-list|examples-counterexamples|
| 0
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What are some classic fallacious proofs?
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If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already: Let $a = b$. Then $a^2 = ab$ $a^2 - b^2 = ab - b^2$ Factor to $(a-b)(a+b) = b(a-b)$ Then divide out $(a-b)$ to get $a+b = b$ Since $a = b$, then $b+b = b$ Therefore $2b = b$ Reduce to $2 = 1$ As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.
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The two envelopes problem is a good one. See also: Card doubling paradox and: https://mathoverflow.net/questions/9037
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|soft-question|big-list|fake-proofs|
| 0
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
|
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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The Monty Hall problem fits the bill pretty well. Almost everyone, including most mathematicians, answered it wrong on their first try, and some took a lot of convincing before they agreed with the correct answer. It's also very easy to explain it to people.
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|soft-question|big-list|examples-counterexamples|
| 0
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Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
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Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.
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Let's group the terms as follows: Group $1$ : $\displaystyle\frac11\qquad$ ($1$ term) Group $2$ : $\displaystyle\frac12+\frac13\qquad$($2$ terms) Group $3$ : $\displaystyle\frac14+\frac15+\frac16+\frac17\qquad$($4$ terms) Group $4$ : $\displaystyle\frac18+\frac19+\cdots+\frac1{15}\qquad$ ($8$ terms) $\quad\vdots$ In general, group $n$ contains $2^{n-1}$ terms. But also, notice that the smallest element in group $n$ is larger than $\dfrac1{2^n}$. For example all elements in group $2$ are larger than $\dfrac1{2^2}$. So the sum of the terms in each group is larger than $2^{n-1} \cdot \dfrac1{2^n} = \dfrac1{2}$. Since there are infinitely many groups, and the sum in each group is larger than $\dfrac1{2}$, it follows that the total sum is infinite. This proof is often attributed to Nicole Oresme.
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|calculus|sequences-and-series|harmonic-numbers|
| 1
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
|
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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There are true statements in arithmetic which are unprovable. Even more remarkably there are explicit polynomial equations where it's unprovable whether or not they have integer solutions with ZFC ! (We need ZFC + consistency of ZFC)
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|soft-question|big-list|examples-counterexamples|
| 0
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Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
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Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.
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This is not as good an answer as AgCl's, nonetheless people may find it interesting. If you're used to calculus then you might notice that the sum $$ 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$$ is very close to the integral from $1$ to $n$ of $\frac{1}{x}$. This definite integral is ln(n), so you should expect $1+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{n}$ to grow like $\ln(n)$. Although this argument can be made rigorous, it's still unsatisfying because it depends on the fact that the derivative of $\ln(x)$ is $\frac{1}{x}$, which is probably harder than the original question. Nonetheless it does illustrate a good general heuristic for quickly determining how sums behave if you already know calculus.
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|calculus|sequences-and-series|harmonic-numbers|
| 0
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What is the single most influential book every mathematician should read?
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If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
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William Dunham's "Journey through Genius." Well, rather that is the book I read that made me want to be a mathematician.
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|soft-question|big-list|reference-request|
| 0
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What is the single most influential book every mathematician should read?
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If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
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Following Noah's lead I will mention; "The Man Who Loved Only Numbers" and "How to Read and Do Proofs"
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|soft-question|big-list|reference-request|
| 0
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What is the single most influential book every mathematician should read?
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If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
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Probability Theory: the Logic of Science. Or anything by Edwin T Jaynes .
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|soft-question|big-list|reference-request|
| 0
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What is the single most influential book every mathematician should read?
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If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
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This is an extremely broad question, especially given the wide variety of mathy people here, but I'll bite. HSM Coxeter's Introduction to Geometry is a book that was very important to the development of my interest in mathematics and inclination towards its geometric aspects.
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|soft-question|big-list|reference-request|
| 0
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What is the single most influential book every mathematician should read?
|
If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
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Nicolas Bourbaki's Éléments de mathématique (specifically Topologie Générale and Algèbre).
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|soft-question|big-list|reference-request|
| 0
|
Is there a real number lookup algorithm or service?
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Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write a program to calculate an approximation to $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$, look up the answer ("looks close to $\displaystyle\frac{\pi^2}{6}$") and then use the likely answer to help find a proof that the sum really is $\displaystyle \frac{\pi^2}{6}$. Does such a thing exist?
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Try Wolfram Alpha . It actually does sequences as well.
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|math-software|experimental-mathematics|
| 1
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Can you recommend a decent online or software calculator?
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I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any?
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For desktop software, I use SpeedCrunch . Has a history, lots of mathematical functions, supports variable, etc.
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|soft-question|big-list|math-software|computer-algebra-systems|
| 0
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Best Maths Books for Non-Mathematicians
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I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is " Journey Through Genius ", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking abo
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I've been successful in using Courant and Robbins' What Is Mathematics? An Elementary Approach to Ideas and Methods for adults who have not had a math class for a few decades, but are open to the idea of learning more about mathematics. Some sections are too advanced for someone with only high school mathematics, and many more will appear that way to the person at first, but do not actually rely on anything beyond high school mathematics.
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|reference-request|soft-question|big-list|book-recommendation|
| 0
|
Is there a real number lookup algorithm or service?
|
Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write a program to calculate an approximation to $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$, look up the answer ("looks close to $\displaystyle\frac{\pi^2}{6}$") and then use the likely answer to help find a proof that the sum really is $\displaystyle \frac{\pi^2}{6}$. Does such a thing exist?
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Sometimes the decimal digits of numbers will appear in Sloane's On-Line Encyclopedia of Integer Sequences OIES . E.g. here is the decimal expansion of pi.
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|math-software|experimental-mathematics|
| 0
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Online resources for learning Mathematics
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Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, most resources I know of tend to either assume a working knowledge of maths beyond secondary school level, or only provide a brief summary of the topic at hand. I'll start off by posting MIT Open Courseware , which is a large collection of lecture notes, assignments and multimedia for the MIT mathematics courses, although in many places it's quite incomplete.
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Purplemath has a list for math lessons and tutoring. It's a list with various links and short reviews referring to tutoring and instructional resources.
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|reference-request|online-resources|
| 0
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List of Interesting Math Blogs
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I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog
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It isn't quite a blog, but Steven Strogatz's 15 part series for the New York Times was excellent.
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|soft-question|big-list|online-resources|
| 0
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How come $32.5 = 31.5$? (The "Missing Square" puzzle.)
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Below is a visual proof (!) that $32.5 = 31.5$ . How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle .)
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The red and blue triangles are not similar (the ratios of the sides are 3/8 = 0.375 and 2/5 = 0.4 respectively), so the "hypotenuse" of your big triangle is not a straight line.
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|geometry|recreational-mathematics|fake-proofs|
| 0
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How come $32.5 = 31.5$? (The "Missing Square" puzzle.)
|
Below is a visual proof (!) that $32.5 = 31.5$ . How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle .)
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It's an optical illusion - neither the first nor second set of blocks actually describes a triangle. The diagonal edge of the first is slightly concave and that of the second is slightly convex. To see clearly, look at the gradients of the hypotenuses of the red and blue triangles - they're not 'similar'. gradient of blue triangle hypotenuse = 2/5 gradient of red triangle hypotenuse = 3/8 Since these gradients are different, combining them in the ways shown in the diagram does not produce an overall straight (diagonal) line.
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|geometry|recreational-mathematics|fake-proofs|
| 1
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List of Interesting Math Blogs
|
I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog
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John D Cook writes The Endeavor One of the MathWorks blogs: Loren on the Art of Matlab ... a few more: eon Peter Cameron's Blog Walking Randomly Todd and Vishal's Blog (Check their blogrolls for more)
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|soft-question|big-list|online-resources|
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Is $0$ a natural number?
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Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.
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Simple answer: sometimes yes, sometimes no, it's usually stated (or implied by notation). From the Wikipedia article : In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers $\{1, 2, 3, \dots\}$ according to the traditional definition; or the set of non-negative integers $\{0, 1, 2,\dots\}$ according to a definition first appearing in the nineteenth century. Saying that, more often than not I've seen the natural numbers only representing the 'counting numbers' (i.e. excluding zero). This was the traditional historical definition, and makes more sense to me. Zero is in many ways the 'odd one out' - indeed, historically it was not discovered (described?) until some time after the natural numbers.
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|terminology|natural-numbers|
| 1
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Is $0$ a natural number?
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Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.
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There is no "official rule", it depends from what you want to do with natural numbers. Originally they started from $1$ because $0$ was not given the status of number. Nowadays if you see $\mathbb{N}^+$ you may be assured we are talking about numbers from $1$ above; $\mathbb{N}$ is usually for numbers from $0$ above. [EDIT: the original definitions of Peano axioms, as found in Arithmetices principia: nova methodo , may be found at https://archive.org/details/arithmeticespri00peangoog : look at it. ]
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|terminology|natural-numbers|
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Why is $1$ not a prime number?
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Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
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actually 1 was considered a prime number until the beginning of 20th century. Unique factorization was a driving force beneath its changing of status, since it's formulation is quickier if 1 is not considered a prime; but I think that group theory was the other force. Indeed I prefer to describe numbers as primes, composites and unities , that is numbers whose inverse exists (so if we take the set of integer numbers Z, we have that 1 and -1 are unities and we still have unique factorization up to unities). We can always amend the defition of a prime number and say it is a number with exactly two divisors: in this way 1 is not a prime by definition :-)
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|abstract-algebra|elementary-number-theory|ring-theory|prime-numbers|terminology|
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Aren't constructive math proofs more "sound"?
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Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction?
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when dealing with infinite sets, many propositions cannot be proved mithout using non-constructive proofs. Axiom of Choice (AC, in brief) or an equivalent proposition is required. Russell's paradox is not a problem per se, you just rule out certain collections of things as sets; Banach-Tarski paradox (you may take a ball, "divide" it in a finite number of parts, translate and rotate them and obtain two balls equal to the first) may be worse indeed. But few mathematicians would prefer not to do a lot of maths because AC is not allowed!
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|proof-theory|constructive-mathematics|
| 0
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Is $0$ a natural number?
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Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.
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I remember all of my courses at University using only positive integers (not including $0$) for the Natural Numbers. It's possible that they had come to an agreement amongst the Maths Faculty, but during at least two courses we generated the set of natural numbers in ways that wouldn't make sense if $0$ was included. One involved the cardinality of Sets of Sets, the other defined the natural numbers in terms of the number $1$ and addition only ($0$ and Negative Integers come into the picture later when you define an inverse to addition). As a result when teaching the difference between Integers and Natural Numbers I always define $0$ as an integer that isn't a Natural Number.
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|terminology|natural-numbers|
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Online resources for learning Mathematics
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Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, most resources I know of tend to either assume a working knowledge of maths beyond secondary school level, or only provide a brief summary of the topic at hand. I'll start off by posting MIT Open Courseware , which is a large collection of lecture notes, assignments and multimedia for the MIT mathematics courses, although in many places it's quite incomplete.
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For basic mathematics mathpage has a long course, including the list below. Don't be put off by the elementary school style lesson names, it does go into some depth with each one. Lesson 1 Reading and Writing Whole Numbers Lesson 2 The Meaning of Decimals Lesson 3 Multiplying and Dividing Lesson 4 More Elementary Addition Lesson 5 Adding Whole Numbers and Decimals Lesson 6 The Meaning of Subtraction Lesson 7 Subtracting Whole Numbers and Lesson 8 The Meaning of Multiplication Lesson 9 Multiplying Whole Numbers Lesson 10 The Meaning of Division Lesson 11 Short Division Lesson 12 Dividing Decimals Lesson 13 Percent with a Calculator Lesson 14 Parts of Natural Numbers 1 Lesson 15 Parts of Natural Numbers 2 Lesson 16 Ratio and Proportion 1 Lesson 17 Ratio and Proportion 2 Lesson 18 Proportionality Lesson 19 Proper Fractions, Mixed Numbers Lesson 20 Unit Fractions Lesson 21 Equivalent Fractions Lesson 22 Lowest Common Multiple Lesson 23 Fractions into Decimals Lesson 24 Adding and Subtracti
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|reference-request|online-resources|
| 0
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How does the wheel paradox work?
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I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel?
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the smaller wheel does not just rotate, but also slides. If you had cogwheels instead of smooth wheels, you'd notice that movement is not possible.
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|geometry|
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Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
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How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ is prime? Thanks.
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$$n\text{ is prime if }(n-1)! \equiv -1 \pmod n$$ This direction is easy. If $n$ is composite, then there exists $k|n$ and $k\lt n$ . So $k|(n-1)!$ and $k \equiv 1 \pmod n$ . This means $k$ needs to divide $1$ . So $n$ must be prime (or $1$ , but we can eliminate this by substitution). $$(n-1)! \equiv -1\text{ if }n\text{ is prime}$$ Wikipedia contains two proofs of this result known as Wilson's theorem. The first proof only uses basic abstract algebra and so should be understandable with a good knowledge of modular arithmetic. Just in case, I prove below that each element $1, 2, ... n-1$ has a unique inverse $\mod n$ . They use the fact that integers $\mod p$ form a group and hence that each element $x$ not congruent $0$ has a multiplicative inverse (a number $y$ such that $xy \equiv 1 \mod n$ . We show this as follows. Suppose $n \nmid x$ , for $n$ prime. From the uniqueness of prime factorisations, $xn$ is the first product of $x$ , after $0x$ , divisible by $n$ (use prime factorisa
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|group-theory|elementary-number-theory|modular-arithmetic|
| 1
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What are all the homomorphisms between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$?
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Any homomorphism $φ$ between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$ is completely defined by $φ(1)$. So from $$0 = φ(0) = φ(18) = φ(18 \cdot 1) = 18 \cdot φ(1) = 15 \cdot φ(1) + 3 \cdot φ(1) = 3 \cdot φ(1)$$ we get that $φ(1)$ is either $5$ or $10$. But how can I prove or disprove that these two are valid homomorphisms?
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If one has a homomorphism of two rings $R, S$ , and $R~$ has an identity, then the identity must be mapped to an idempotent element of $S$ , because the equation $x^2=x$ is preserved under homomorphisms. Now $5$ is not an idempotent element in $\Bbb Z_{15}$ , so the map generated by $1 \to 5$ is not a homomorphism. However, $10$ is an idempotent element of $\Bbb Z_{15}$ . In particular, the subring $T \subset \Bbb Z_{15}$ generated by $10$ has unit $10$ . Since it is annihilated by $3$ , and consequently by $18$ , there is a unital homomorphism $\Bbb Z_{18} \to T$ (i.e., mapping $1$ to $10$ ). So your second map is a legitimate homomorphism of rings (composing with the injection $T \to \Bbb Z_{15}$ ). Basically, the point of this answer is to check that one of your maps preserves the relations of the two rings, while the other doesn't.
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|ring-theory|abstract-algebra|
| 1
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Why are $\Delta_1$ sentences of arithmetic called recursive?
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The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ if it is provably equivalent to a prenex normal form with only existential quantifiers. A formula is $\Delta_1$ if it is both $\Pi_1$ and $\Sigma_1.$ These formulae are often called recursive: why?
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If a formula $\phi(x_1, ... x_n)$ is in both $\Sigma_1, \Pi_1$, then one can define a Turing machine to determine whether it is true or false. Namely, in parallel, search for a collection of parameters that makes true the existential formula, and search for a collection of formulas that makes false the universal formula. If the first happens, return true; if the second happens, return false. One of these must exist, so the Turing machine always halts. (The set of $x_1,...x_n$ such that $\phi(x_1, ... , x_n)$ is valid if $\phi$ belongs to $\Sigma_1$ is, by contrast, is only recursively enumerable.) By contrast, since the action of any Turing machine is simulable by existential formulas in first-order logic (i.e. there exists a number $k$ such that $M$ halts in $k$ steps), any language which is recursively enumerable can be expressed by existential formulas. Any language whose complement is recursively enumerable can similarly be expressed by universal formulas (by the analog of deMorgan
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|logic|computability|proof-theory|
| 1
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Is there a real number lookup algorithm or service?
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Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write a program to calculate an approximation to $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$, look up the answer ("looks close to $\displaystyle\frac{\pi^2}{6}$") and then use the likely answer to help find a proof that the sum really is $\displaystyle \frac{\pi^2}{6}$. Does such a thing exist?
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I've long used Simon Plouffe's inverse symbolic calculator for this purpose. It is essentially a searchable list of "interesting" numbers. Edit: link updated (Mar 2022).
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|math-software|experimental-mathematics|
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How are we able to calculate specific numbers in the Fibonacci Sequence?
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I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position. Is there a way to do this? If so, how are we able to apply these formulas to arrays?
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A lot of people have mentioned Binet's formula. But I suspect this is not the most practical way to compute the nth Fibonacci number for large n, because it requires either having a very accurate value of $\sqrt{5}$ and carrying around lots of decimal places (if you want to do floating-point arithmetic) or expanding large powers of $1+\sqrt{5}$ using the binomial formula. The latter comes out to writing the Fibonacci number as a sum of binomial coefficients. The following formulas hold, though: $$F_{2n-1}=F_n^2+F_{n-1}^2$$ $$F_{2n}=(2F_{n-1}+F_n)\cdot F_n$$ which you can find derivations of in the Wikipedia article on Fibonacci numbers . This lets you find $F_k$ , for any $k$ even or odd, in terms of two Fibonacci numbers with approximately half the index. The result is faster than Binet's formula.
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|combinatorics|generating-functions|fibonacci-numbers|
| 0
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Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
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How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ is prime? Thanks.
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[NOTE: it seems that there is some difference between preview and actual output, so instead if using (mod p) I stick with (p)] to show that $(p-1)! \equiv -1 (p)$ without explicitly use group theory, maybe the simplest path is: (the following assumes $p$ is odd, but if $p=2$ then the result is immediate) given $n \ne 0$, all values $n, 2n, ... (p-1)$ $n$ are different mod $p$. Otherwise, if $hn \equiv kn (p)$ then $(h-k)n \equiv 0 (p)$ against the hypothesis that $p$ is prime. this means that each $n$ has an inverse mod $p$, that is for each $n$ there is a $m$ such that $mn \equiv 1 (p)$. the equation $x^2\equiv 1 (p)$ may be written as $(x+1)(x-1) \equiv 0 (p)$; therefore its only solutions are $x \equiv 1 (p)$ and $x \equiv -1 (p)$. For each other number $n$, an inverse $m$ must exist (because of the pigeonhole principle) but $m \neq n$. we are nearly done. Let's couple every number from $2$ to $p-2$ with its own inverse. Their product is $1 (p)$, so they don't count in the overall t
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|group-theory|elementary-number-theory|modular-arithmetic|
| 0
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Calculating the probability of two dice getting at least a $1$ or a $5$
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So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question. This question comes from the game farkle.
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Go backwards: Calculate the probability that neither of them shows a 1 or 5. That means both show a 2, 3, 4, or 6. Thats $(4/6)^2$. Hence the probability that at least one shows a 1 or 5 is $1-(2/3)^2=5/9$.
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|probability-theory|
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Unital homomorphism
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What is a unital homomorphism? Why are they important?
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A unital homomorphism between rings R and S is a ring homomorphism that sends the identity element of R to the identity element of S. Homomorphisms (between objects in any algebraic category like groups, rings, vector spaces, etc.) preserve the algebraic structure, and if you want a map between rings with an identity element, it is natural to require this to preserve this element (since it satisfies unique properties).
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|abstract-algebra|terminology|definition|
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Books on Number Theory for Layman
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Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math)
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I would still stick with Hardy and Wright, even if it is quite old.
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|number-theory|reference-request|soft-question|book-recommendation|
| 0
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Unital homomorphism
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What is a unital homomorphism? Why are they important?
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A lot of results about rings just won't work otherwise: for instance, a unital homomorphism of rings sends units to units. A nonunital homomorphism doesn't have to do that. Nonunital homomorphisms can be very degenerate, e.g. the zero homomorphism. Another reason you want homomorphisms to preserve the unit is that this is how you get a map $\operatorname{Spec S} \to \operatorname{Spec} R$ from a ring-homomorphism $R \to S$.
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|abstract-algebra|terminology|definition|
| 0
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Calculating the probability of two dice getting at least a $1$ or a $5$
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So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question. This question comes from the game farkle.
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To visually see the answer given by balpha above, you could write out the entire set of dice rolls [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6] [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [2, 6] [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6] [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [4, 6] [5, 1], [5, 2], [5, 3], [5, 4], [5, 5], [5, 6] [6, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6] Total number of possible dice rolls: 36 Dice rolls that contain 1 or a 5: 20 20/36 = 5/9
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|probability-theory|
| 0
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Best Maths Books for Non-Mathematicians
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I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is " Journey Through Genius ", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking abo
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I think any book by John Allen Paulos would be something any Math enthusiast could enjoy and learn from.
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|reference-request|soft-question|big-list|book-recommendation|
| 0
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Books on Number Theory for Layman
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Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math)
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A concise introduction to the theory of numbers by Alan Baker (1970 Fields medalist) covers a lot of ground in less than 100 pages, and does so in a fluid way that never feels rushed. I love this little book.
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|number-theory|reference-request|soft-question|book-recommendation|
| 0
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Best Maths Books for Non-Mathematicians
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I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is " Journey Through Genius ", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking abo
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John Derbyshire's Prime Obsession is about Riemann's hypothesis. One of the stated goals of the author is to explain what "all non-trivial zeros of the zeta function have real part one-half" means to readers who have no background in calculus. Odd-numbered chapters tell the story of how Riemann came to his hypothesis, and even-numbered chapters are more mathematical in nature.
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|reference-request|soft-question|big-list|book-recommendation|
| 0
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Why are $3D$ transformation matrices $4 \times 4$ instead of $3 \times 3$?
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Background: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually has a value of $1$. Given the extra computing effort required to multiply $4\times 4$ matrices instead of $3\times 3$ matrices, there must be a substantial benefit to including that extra fourth term, even though $3\times 3$ matrices should (?) be sufficient to describe points and transformations in 3D space. Question: Why is the inclusion of a fourth term beneficial? I can guess that it makes the computations easier in some manner, but I would really like to know why that is the case.
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To follow up user80's answer, you want to get transformations of the form v --> Av + b, where A is a 3 by 3 matrix (the linear part of transformation) and b is a 3-vector. We can encode this transformation in a 4 x 4 matrix by putting A in the top left with three 0's below it and making the last column be (b,1). Multiplying the 4-vector (v,1) with this matrix will give you (Av + b, 1).
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|linear-algebra|matrices|geometry|
| 0
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Books on Number Theory for Layman
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Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math)
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Serre's "A course in Arithmetic" is pretty phenomenal.
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|number-theory|reference-request|soft-question|book-recommendation|
| 0
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
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The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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Similar to the Monty Hall problem, but trickier: at the latest Gathering 4 Gardner, Gary Foshee asked I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? We are assuming that births are equally distributed during the week, that every child is a boy or girl with probability 1/2, and that there is no dependence relation between sex and day of birth. His Answer: 13/27. This was in the news a lot recently, see for instance BBC News . ( Later analysis showed the answer depends on why the parent said that.)
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|soft-question|big-list|examples-counterexamples|
| 0
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A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
|
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently the Banach-Tarski paradox was not a good example.) Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes." "Impossible! "Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't
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You have two identical pieces of paper with the same picture printed on them. You put one flat on a table and the other one you crumple up (without tearing it) and place it on top of the first one. Brouwer's fixed point theorem states that there is some point in the picture on the crumpled-up page that is directly above the same point on the bottom page. It doesn't matter how you place the pages, or how you deform the top one.
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|soft-question|big-list|examples-counterexamples|
| 0
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List of Interesting Math Blogs
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I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog
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Math-blog is one I have in my google reader and I just found this one through google reader but it's a little over my head: The Unapologetic Mathematician
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|soft-question|big-list|online-resources|
| 0
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Why is $x^0 = 1$ except when $x = 0$?
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Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
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If $a$ and $b$ are natural numbers, then $a^b$ is the number of ways you can make a sequence of length $b$ where each element in the sequence is chosen from a set of size $a$ . You're allowed replacements. For example $2^3$ is the number of 3 digit sequences where each digit is zero or $1$ : $000, 001, 010, \ldots, 111.$ There is precisely one way to make a zero length sequence: the empty sequence. So you'd expect $0^0=1$ .
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|definition|exponentiation|
| 0
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Simple numerical methods for calculating the digits of $\pi$
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Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits?
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Some of the easiest algorithms to use are the spigot algorithms. They allow you to jump rapidly to the nth digit without computing the n-1 digits before. The catch is that they only really work in binary. Here are some implementations in Python. You can see how short they are.
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|approximation|numerical-methods|pi|
| 0
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Where can I find a review of discrete math
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I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
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This is the discrete math course one in my school. It contain lecture notes, homework and previous exams. http://www.cs.sunysb.edu/~cse547/
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|soft-question|reference-request|education|discrete-mathematics|
| 1
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Best Maths Books for Non-Mathematicians
|
I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is " Journey Through Genius ", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking abo
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One's that were suggested to me by my Calculus teacher in High School. Even my wife liked them and she hates math now: The Education of T.C. Mits: What modern mathematics means to you Infinity: Beyond the Beyond the Beyond Written and illustrated(Pictures are great ;p) by a couple: Lillian R. Lieber, and Hugh Gray Lieber. These books were hard to find before because they went out of print but I have this new version and like it a lot. The books explains profound topics in a way that is graspable by anyone without being dumbed down. Godel's proof is one I enjoyed. It's was a little hard to understand but there is nothing in this book that makes it inaccessible to someone without a strong math background. Keeping with Godel in the title, Godel, Escher, Bach: An Eternal Golden Braid while not just about math was a good read (a bit long ;p). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics It describes the Riemann Hypothesis and people who were involved with
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|reference-request|soft-question|big-list|book-recommendation|
| 0
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Best Maths Books for Non-Mathematicians
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I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is " Journey Through Genius ", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking abo
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Possibly this may not really qualify as presenting much interesting maths, but I think Hardy's A Mathematician's Apology should be on the must-read list.
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|reference-request|soft-question|big-list|book-recommendation|
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What is the single most influential book every mathematician should read?
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If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
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A Mathematician's Apology by G H Hardy. I did in fact read this in high school, and it raised my view of mathematics from a thing of utility to a thing of beauty and wonder. It inspired me to go on to study mathematics at Cambridge myself. It's a pity that the "introduction" by C P Snow is longer than the original and contains a rather depressing view of Hardy's later life. I would recommend readers to skip the introduction altogether and concentrate on Hardy's own words.
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|soft-question|big-list|reference-request|
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History of the Concept of a Ring
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I am vaguely familiar with the broad strokes of the development of group theory, first when ideas of geometric symmetries were studied in concrete settings without the abstract notion of a group available, and later as it was formalized by Cayley, Lagrange, etc (and later, infinite groups being well-developed). In any case, it's intuitively easy for me to imagine that there was substantial lay, scientific, and artistic interest in several of the concepts well-encoded by a theory of groups. I know a few of the corresponding names for who developed the abstract formulation of rings initially (Wedderburn etc.), but I'm less aware of the ideas and problems that might have given rise to interest in ring structures. Of course, now they're terribly useful in lots of math, and $\mathbb{Z}$ is a natural model for elementary properties of commutative rings, and I'll wager number theorists had an interest in developing the concept. And if I wanted noncommutative models, matrices are a good place
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Edit: Bill Dubuque has pointed out that much of this answer (specifically, the part about FLT) is essentially a mathematical urban legend, albeit a pervasive one. I cannot delete an accepted answer, so here is a link to an answer of his on MO explaining it. Here is also a link to a related question . There's some of the history here in Bourbaki's Commutative Algebra, in the appendix. Basically, a fair bit of ring theory was developed for algebraic number theory. This in turn was because people were trying to prove Fermat's last theorem. Why's this? Let $p$ be a prime. Then the equation $x^p + y^p = z^p$ can be written as $\prod (x+\zeta_p^iy) = z^p$ for $\zeta_p$ a primitive $p$th root of unity. All these quantities are elements of the ring $Z[\zeta_p]$. So if $p>3$ and there is unique factorization in the ring $Z[\zeta_p]$, it isn't terribly hard to show that this is impossible at least in the basic case where $p $ does not divide $xyz$ (and can be found, for instance, in Borevich-Sha
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|math-history|
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Best Maths Books for Non-Mathematicians
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I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story. My favorite example of this is " Journey Through Genius ", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level. Edit: A few more details on what I'm looking for. The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking abo
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Paul Nahin has a number of accessible mathematics books written for non-mathematicians, the most famous being An Imaginary Tale: The Story of $\sqrt{-1}$ Dr. Euler's Fabulous Formula (Cures Many Mathematical Ills!) Professor Ian Stewart also has many books which each give laymen overviews of various fields or surprising mathematical results Professor Stewart's Cabinet of Mathematical Curiosities Professor Stewart's Hoard of Mathematical Treasures Cows in the Maze: And Other Mathematical Explorations Does God Play Dice? The New Mathematics of Chaos
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|reference-request|soft-question|big-list|book-recommendation|
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Where can I find a review of discrete math
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I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
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When wanting to know about a particular mathematics subject, I often find that starting with the " further reading " section of the relevant wikipedia page is a good way in.
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|soft-question|reference-request|education|discrete-mathematics|
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Tiling a $3 \times 2n$ rectangle with dominoes
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I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences f(0) = g(0) = 1 f(n) = f(n-1) + 2g(n-1) g(n) = f(n) + g(n-1) where $f(n)$ is the actual answer and $g(n)$ is a helper function that represents the number of ways to tile a $3 \times 2n$ rectangle with two extra squares on the end (the same as a $3 \times 2n+1$ rectangle missing one square). By combining these and doing some algebra, I was able to reduce this to f(n) = 4f(n-1) - f(n-2) which shows up as sequence A001835 , confirming that this is the correct recurrence. The number of ways to tile a $2 \times n$ rectangle is the Fibonacci numbers because every rectangle ends with either a verticle domino or two horizontal ones, which gives the exact recurrence that Fibonacci numbers do. My question is, is there a similar simple explanation for this recurrence for tiling a $3 \times 2n$ rectangle ?
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Here's my best shot at the sort of explanation you're asking for, although it's not nearly as clear as the $2 \times n$ case. The negative sign makes combinatorial proofs difficult, so let's rearrange this as: $$f(n) + f(n-2) = 4f(n-1)$$ Then you want to show that the number of $n$-tilings, plus the number of $(n-2)$-tilings, is four times the number of $(n-1)$-tilings. (An "n-tiling" is a tiling of a $3 \times 2n$ rectangle by dominoes.) In bijective terms, then, we want a bijection between the set of $n$-tilings and $(n-2)$-tilings and the set of $(n-1)$-tilings, where the $(n-1)$-tilings are each tagged with the number $1, 2, 3,$ or $4$. Given an $(n-1)$-tiling, there are three "obvious" ways to obtain an $n$-tiling from it, namely by adding one of the three $1$-tilings on the right end. These generate tilings which have a vertical line passing all the way through, two units from the right end; call these "faulted" tilings, and those which don't have a vertical line in that position
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|combinatorics|discrete-mathematics|recurrence-relations|tiling|
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Is there a closed-form equation for $n!$? If not, why not?
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I know that the Fibonacci sequence can be described via the Binet's formula. However, I was wondering if there was a similar formula for $n!$. Is this possible? If not, why not?
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The relative error of Stirling's approximation gets arbitrarily small as n gets larger. $$n!\sim\sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$ However, it is only an approximation, not a closed-form of $n!$
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|combinatorics|algorithms|factorial|closed-form|
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Simple numerical methods for calculating the digits of $\pi$
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Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits?
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There is also the option of approximating $\pi$ using Monte Carlo integration . The idea is this: If we agree that the area of a circle is $\pi r^2$, for simplicity we build a circle of area $\pi$ by setting $r=1$. Placing this circle wholly inside of another region of known area, preferably by inscribing it in a square of side length 2, then we have a ratio of the circle's area to the total area of the square (in this example that ratio is $\frac{\pi}{4}$). The Monte Carlo method works by approximating areas based on the ratio of the number of sample points lying within our region of interest and the total number of sample points we choose to try. If we spread a uniformly distributed sequence of $N$ points over our square of area 4, and call the number of points that land inside of the inscribed circle $p$, then we can say $\frac{\pi}{4} = \frac{p}{N}$. My implementation of this in Matlab requires tens of thousands of test points to achieve 3.14159xxxx, but I have not tried it for low
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|approximation|numerical-methods|pi|
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Good books and lecture notes about category theory.
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What are the best books and lecture notes on category theory?
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Categories for the Working mathematician by Mac Lane Categories and Sheaves by Kashiwara and Schapira
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|reference-request|soft-question|category-theory|big-list|book-recommendation|
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Is there a closed-form equation for $n!$? If not, why not?
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I know that the Fibonacci sequence can be described via the Binet's formula. However, I was wondering if there was a similar formula for $n!$. Is this possible? If not, why not?
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If you're willing to accept an integral as an answer, then $n! = \int_0^\infty t^n e^{-t} \: dt$.
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|combinatorics|algorithms|factorial|closed-form|
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Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?
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Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
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I will assume this is in the context of rings (e.g., real numbers, integers, etc). In this case, the axiom defining $0$ is that $x + 0 = x$ for all $x$. $x*0 = 0$ is a result of this since we have $x*0 = x*(0+0) = x*0 + x*0$ which implies $x*0 = 0$ (canceling one of the $x*0$'s). I am guessing that for the second one you mean $x*1 = x$. This is a definition (axiom). The third one is a consequence of the definition of $-x$ being the element such that $x + (-x) = 0$. For then we have $(-x) + x$ is also zero so that $x$ is the negative of $-x$.
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|abstract-algebra|definition|axioms|
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Good books and lecture notes about category theory.
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What are the best books and lecture notes on category theory?
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Lang's Algebra contains a lot of introductory material on categories, which is really nice since it's done with constant motivation from algebra (e.g. coproducts are introduced right before the free product of groups is discussed).
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|reference-request|soft-question|category-theory|big-list|book-recommendation|
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Where can I find a review of discrete math
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I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
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When it comes to textbooks, the Kenneth Rosen text Discrete Mathematics and its Applications is highly recommended. I was first introduced to it at my university, but I've seen it cited in several places.
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|soft-question|reference-request|education|discrete-mathematics|
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Good books and lecture notes about category theory.
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What are the best books and lecture notes on category theory?
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Arbib, Arrows, Structures, and Functors: The Categorical Imperative More elementary than MacLane. I don't know very much about this, but some stripes of computer scientist have taken an interest in category theory recently, and there are lecture notes floating around with that orientation.
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|reference-request|soft-question|category-theory|big-list|book-recommendation|
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Applications of the Fibonacci sequence
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The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?
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They are sometimes used or occur in financial applications, e.g. Elliot Wave Theory . This seems more like magic than math to me, but I am not a trader.
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|combinatorics|big-list|applications|fibonacci-numbers|
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Applications of the Fibonacci sequence
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The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?
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It isn't exactly an application as such, but the upper bound of the size of a subtree in a Fibonacci heap whose root is a node with degree $k$ is $F_{k+2}$ where $F_n$ is the $n^{th}$ Fibonacci number.
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|combinatorics|big-list|applications|fibonacci-numbers|
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Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?
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Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
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Answering what you asked, the question is poorly formed because you didn't specify the theory you are talking about. To do that, well, you must define symbols, deduction rules... and axioms. Answering what you probably meant to ask, no. Those are properties that $-$ and $\times$ have in the $\mathbb Z$ set, not axioms. They can be deduced from the operations themselves.
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|abstract-algebra|definition|axioms|
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Applications of the Fibonacci sequence
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The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?
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They're useful in determining the amortized run time of the appropriately named Fibonacci Heap in computer science.
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|combinatorics|big-list|applications|fibonacci-numbers|
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Sum of reciprocals of numbers with certain terms omitted
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I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$. But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?
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EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series. An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't say what it converges to in the introductory paragraph). The article goes on to generalize the result. A Curious Convergent Series Frank Irwin The American Mathematical Monthly, Vol. 23, No. 5 (May, 1916), pp. 149-152 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2974352
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|sequences-and-series|convergence-divergence|
| 1
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Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
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One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$ is at a maximum, $\sin x$ is zero and moving upwards. But these "matching points" only work for multiples of $\pi/4$. Let us move back towards the original definition(s) of sine and cosine: At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle. To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis. The definition of $\cos x$
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This isn't exactly what you asked, but look at the Taylor series for the polynomials: $$ \sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x\!$$ $$\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\text{ for all } x\! $$ The relationships between the derivatives are clear from this.
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|calculus|trigonometry|
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Sum of reciprocals of numbers with certain terms omitted
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I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$. But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?
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It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases. Here's a proof. Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + \ldots$, where $S_i$ is the sum of all terms whose denominator contains exactly $i$ digits, all different from $k$. Now, the number of $i$-digit numbers that do not contain the digit $k$ is $8\cdot9^{i-1}$ (there are $8$ choices for the first digit, excluding $0$ and $k$, and $9$ choices for the other digits). [Well, if $k=0$ there are $9$ choices for the first digit, but the proof still works.] So there are $8\cdot9^{i-1}$ numbers in the sum $S_i$. Now each number in $S_i$ is of the form $\frac1a$, where $a$ is an $i$-digit number. So $a \geq 10^{i-1}$, which implies $\frac1a \leq \frac1{10^{i-1}}$. Therefore $S_i \leq 8\cdot\dfrac{9^{i-1} }{10^{i-1}} = 8\cdot\left(\frac9{10}\right)^{i-1}$.
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|sequences-and-series|convergence-divergence|
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Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
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One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$ is at a maximum, $\sin x$ is zero and moving upwards. But these "matching points" only work for multiples of $\pi/4$. Let us move back towards the original definition(s) of sine and cosine: At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle. To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis. The definition of $\cos x$
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I don't think you can get an intuitive feel for the derivatives without looking at the plots personally. When you consider that a derivative is a rate of change, you need to be looking at a function that is varying, which implies you are looking at the plot/graph of the function. When you further consider that a derivative (by definition of it being a rate of change) is a gradient function, the intuitive answer is that cos is the gradient function of sin, and -sin is the gradient function of cos (and so on). So if you calculate the gradient of the sin curve at any point, the value you get will be the cosine value for that point.
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|calculus|trigonometry|
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What is larger -- the set of all positive even numbers, or the set of all positive integers?
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We will call the set of all positive even numbers E and the set of all positive integers N . At first glance, it seems obvious that E is smaller than N , because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E , two items in N could be matched (the item x and x-1). This implies that N is twice as large as E On second glance though, it seems less obvious. Each item in N could be mapped with one item in E (the item x*2). Which is larger, then? Or are they both equal in size? Why? (My background in Set theory is quite extremely scant)
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They are both the same size, the size being 'countable infinity' or 'aleph-null'. The reasoning behind it is exactly that which you have already identified - you can assign each item in E to a single value in N. This is true for the Natural numbers, the Integers, the Rationals but not the Reals (see the Diagonal Slash argument for details on this result). -- Added explanation from comment -- The first reasoning is invalid because the cardinality of infinite sets doesn't follow 'normal' multiplication rules. If you multiply a set with cardinality of aleph-0 by 2, you still have aleph-0. The same is true if you divide it, add to it, subtract from it by any finite amount.
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|elementary-set-theory|infinity|
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What is larger -- the set of all positive even numbers, or the set of all positive integers?
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We will call the set of all positive even numbers E and the set of all positive integers N . At first glance, it seems obvious that E is smaller than N , because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E , two items in N could be matched (the item x and x-1). This implies that N is twice as large as E On second glance though, it seems less obvious. Each item in N could be mapped with one item in E (the item x*2). Which is larger, then? Or are they both equal in size? Why? (My background in Set theory is quite extremely scant)
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Each item in N could be mapped with one item in E (the item x*2). Yes. Both sets have cardinality aleph-0.
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|elementary-set-theory|infinity|
| 0
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Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
|
One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$ is at a maximum, $\sin x$ is zero and moving upwards. But these "matching points" only work for multiples of $\pi/4$. Let us move back towards the original definition(s) of sine and cosine: At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle. To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis. The definition of $\cos x$
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One of the main ways that sine and cosine come up is as the fundamental solutions to the differential equation $y'' = -y$, known as the wave equation. Why is this an important differential equation? Well, interpreting it using Newton's second law it says "the force is proportional and opposite to the position." For example, this is what happens with a spring! Now that's a 2nd degree equation, so it has a 2-dimensional space of solutions. How to pick a nice basis for that space? Well, one way would be to pick $f$ and $g$ such that $f' = i f$ and $g' = -i g$. However, that involves too many imaginary numbers, so another option is $f' = -g$, and $g' = f$. Thus if you're trying to find two functions which explain oscillatory motion you're naturally lead to picking functions that have $f' = g$, $g' = -f$, etc. (On the other hand it's totally unclear from this point of view why Sine and Cosine should have anything to do with triangles...)
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|calculus|trigonometry|
| 0
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What is larger -- the set of all positive even numbers, or the set of all positive integers?
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We will call the set of all positive even numbers E and the set of all positive integers N . At first glance, it seems obvious that E is smaller than N , because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E , two items in N could be matched (the item x and x-1). This implies that N is twice as large as E On second glance though, it seems less obvious. Each item in N could be mapped with one item in E (the item x*2). Which is larger, then? Or are they both equal in size? Why? (My background in Set theory is quite extremely scant)
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Mathematics is the art of clever forgetting. The first mathematical breakthrough, numbers, came about when people realized that if you just forgot about whether it was 5 cows + 3 cows, or 5 rocks + 3 rocks or whatever you always got 8. Numbers are what you get when you look at collections of objects and forget what kind of object they are . When you say "as sets" you mean you're forgetting a lot of information, in particular you don't care about what the names of the elements in that set are or what properties those elements have. As sets the positive numbers and the positive even numbers are "the same" (that is are in bijection) because you can take 1,2,3,... and just rename 1 to 2, and 2 to 4, and n to 2n, and you've just renamed all the elements and got the even numbers! However, if you want to remember more about these sets, for example that they're not arbitrary sets they're both subsets of the natural numbers, then they become distinguishable. Depending on how you want to measure
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|elementary-set-theory|infinity|
| 1
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Good books and lecture notes about category theory.
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What are the best books and lecture notes on category theory?
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Another book that is more elementary, not requiring any algebraic topology for motivation, and formulating the basics through a question and answer approach is: Conceptual Mathematics An added benefit is that it is written by an expert!
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|reference-request|soft-question|category-theory|big-list|book-recommendation|
| 0
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Why do complex functions have a finite radius of convergence?
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Say we have a function $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ with radius of convergence $R>0$. Why is the radius of convergence only $R$? Can we conclude that there must be a pole, branch cut or discontinuity for some $z_0$ with $|z_0|=R$? What does that mean for functions like $$f(z)=\begin{cases} 0 & \text{for $z=0$} \\\ e^{-\frac{1}{z^2}} & \text{for $z \neq 0$} \end{cases}$$ that have a radius of convergence $0$?
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If the radius of convergence is $R$, that means there is a singular point on the circle $|z| = R$. In other words, there is a point $\xi$ on the circle of radius $R$ such that the function cannot be extended via "analytic continuation" in a neighborhood of $\xi$. This is a straightforward application of compactness of the circle and can be found in books on complex analysis, e.g. Rudin's. However, it does not mean that there is a pole, branch cut, or discontinuity, though those would cause singular values. Indeed, a "pole" on the boundary would only make sense if you can analytically continue the power series to some proper domain containing the disk $D_R(0)$, and this is generally impossible. For instance, the power series $\sum z^{2^j}$ cannot be continued in any way outside the unit disk, because it is unbounded along any ray whose angle is a dyadic fraction. The unit circle is its natural boundary, though it does not make sense to say that the function has a branch point or pole th
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|sequences-and-series|complex-analysis|
| 1
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What property of certain regular polygons allows them to be faces of the Platonic Solids?
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It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number of sides? Also, why are there more Triangle-based Platonic Solids (three) than Square- and Pentagon- based ones? (one each) Similarly, is this the same property that allows certain Platonic Solids to be used as "faces" of regular polychoron (4D polytopes)?
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The regular polygons that form the Platonic solids are those for which the measure of the interior angles, say α for convenience, is such that $3\alpha Regular (equilateral) triangles have interior angles of measure $\frac{\pi}{3}$ (60°), so they can be assembled 3, 4, or 5 at a vertex ($3\cdot\frac{\pi}{3} Regular quadrilaterals (squares) have interior angles of measure $\frac{\pi}{2}$ (90°), so they can be assembled 3 at a vertex ($3\cdot\frac{\pi}{2} Regular pentagons have interior angles of measure $\frac{3\pi}{5}$ (108°), so they can be assembled 3 at a vertex ($3\cdot\frac{3\pi}{5} 2\pi$). Regular hexagons have interior angles of measure $\frac{2\pi}{3}$ (120°), so they cannot be assembled 3 at a vertex ($3\cdot\frac{2\pi}{3}=2\pi$--they tesselate the plane). Any other regular polygon will have larger interior angles, so cannot be assembled into a regular solid.
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|geometry|polyhedra|platonic-solids|
| 1
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Construct a bijection from $\mathbb{R}$ to $\mathbb{R}\setminus S$, where $S$ is countable
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Two questions: Find a bijective function from $(0,1)$ to $[0,1]$. I haven't found the solution to this since I saw it a few days ago. It strikes me as odd--mapping a open set into a closed set. $S$ is countable. It's trivial to find a bijective function $f:\mathbb{N}\to\mathbb{N}\setminus S$ when $|\mathbb{N}| = |\mathbb{N}\setminus S|$; let $f(n)$ equal the $n^{\text{th}}$ smallest number in $\mathbb{N}\setminus S$. Are there any analogous trivial solutions to $f:\mathbb{R}\to\mathbb{R}\setminus S$?
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The proof of the Schroeder-Bernstein theorem allows you to get a bijection for 1 , since we have an injection $(0,1) \to [0,1]$ and a bijection $f: [0,1] \to [1/4, 3/4] \subset (0,1)$ (say $x \to x/2 +1/4$ ). The function's definition will be somewhat messy (basically, it depends on how many times you can lift a point under these to injections already defined, and specifically the parity of the number of times), but it'll do it. For 2 , iterate this construction to get a bijection $R \to R - N$ . Then given any countable set $S$ , define the map of $R$ that interchanges $N$ and $S$ and leaves every other point fixed. Then the composition of the first bijection with this second map is your bijection. Continuity considerations imply that the map can't be continuous: in 1 , for instance, we'd otherwise have that $(0,1)$ is compact, which it's not.
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|elementary-set-theory|
| 1
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What can we conclude from correlation?
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I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a strong negative correlation. What can we conclude? Explain your answer. I know that we can't conclude that getting older causes people to become right-handed. Something else might be causing it, not the age. If two things are correlated, we can only conclude association, not causation. So I wrote: We can conclude that many people become right-handed as they grow older, but we cannot tell why. That's exactly what association means, but my teacher marked me wrong! What mistake did I make? Is 40 too small of a sample size to make any conclusions?
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This is wrong: "We can conclude that many people become right-handed as they grow older." We cannot conclude this at all from the given data. For one, the study only takes a sample at one point in time, rather than selecting a sample and monitoring their progress through many decades. This is what would be needed for us to even entertain the possibility that aging causes a change in handedness. Other possible causes include that left handed people might have a shorter life expectancy, or perhaps there was a spike in the birth rate of right handed people in the past. There are many other possibilities that have been mentioned in others answers which would also account for the skewed proportions without requiring people to change handedness with age, which is what you falsely concluded in the test. Also, just an observation, but it appears the "study" was conducted under false pretenses. Handedness is a false dichotomy, people can also be ambidextrous.
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|statistics|
| 1
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What can we conclude from correlation?
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I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a strong negative correlation. What can we conclude? Explain your answer. I know that we can't conclude that getting older causes people to become right-handed. Something else might be causing it, not the age. If two things are correlated, we can only conclude association, not causation. So I wrote: We can conclude that many people become right-handed as they grow older, but we cannot tell why. That's exactly what association means, but my teacher marked me wrong! What mistake did I make? Is 40 too small of a sample size to make any conclusions?
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We can only conclude that the result is interesting and it deserves more research. Without further study we can not say if right-handedness causes age (ie. being left handed causes biological alterations that shorten the life span), age causes right-handedness (ie. as people age they become right handed), they are correlated because they are caused by another variable (ie. older people became educated in a different system that discouraged left-handedness), the study had bad luck selecting its sample, the study sample was bad designed, the study was bad designed, etc.
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|statistics|
| 0
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Intuitive reasoning behind the Chain Rule in multiple variables?
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I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to multiple variables. Even the case of 2 dimensions $$z = f(x,y),$$ where $x = g(t)$ and $y = h(t)$, so $$\frac{dz}{dt} = \frac{\partial z}{dx} \frac{dx}{dt} + \frac{\partial z}{dy} \frac{dy}{dt}.$$ Now, this is easy enough to "calculate" (and figure out what goes where). My teacher taught me a neat tree-based graphical method for figuring out partial derivatives using chain rule. All-in-all, it was rather hand-wavey. However, I'm not sure exactly how this works, intuitively. Why, intuitively, is the equation above true? Why addition ? Why not multiplication, like the other chain rule? Why are some multiplied and some added?
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Changing time is the same as changing x and changing y. If the changes each one caused in z didn't interact, then the total change would be the sum of both changes. If the function is well behaved (differentiatable) then the interaction caused by an infinitesimal change in x with an infinitesimal change in y will be doubly infinitesimal. The proof of the double chain rule just shows this formally.
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|calculus|linear-algebra|
| 0
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Intuitive reasoning behind the Chain Rule in multiple variables?
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I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to multiple variables. Even the case of 2 dimensions $$z = f(x,y),$$ where $x = g(t)$ and $y = h(t)$, so $$\frac{dz}{dt} = \frac{\partial z}{dx} \frac{dx}{dt} + \frac{\partial z}{dy} \frac{dy}{dt}.$$ Now, this is easy enough to "calculate" (and figure out what goes where). My teacher taught me a neat tree-based graphical method for figuring out partial derivatives using chain rule. All-in-all, it was rather hand-wavey. However, I'm not sure exactly how this works, intuitively. Why, intuitively, is the equation above true? Why addition ? Why not multiplication, like the other chain rule? Why are some multiplied and some added?
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The correct context of the chain rule is that taking the tangent bundle is functorial. A more down-to-earth answer is provided by working coordinate free using linear algebra. Suppose $f:X\to Y$ and $g:Y\to Z$ are functions between Banach spaces (these are a generalized version of R^n) such that $f$ is differentiable at $\vec{v}$ and $g$ is differentiable at $f(\vec{v})$ (note that in the general case we must require that their derivatives are toplinear (continuous and linear), since not all linear maps are continuous in the context of infinite dimensional spaces). Then taking the total differential, we see that the chain rule is equivalent to saying that: $$T_\vec{v}(g\circ f)=T_{f(\vec{v})}(g) \circ T_\vec{v}(f).$$ The description you get with coordinates comes from this very much simpler presentation (Where T denotes the total differential) as follows: To derive the formula with coordinates (say, for example, in three dimensions), we present the total differentials (which are linear
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|calculus|linear-algebra|
| 0
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Are the "proofs by contradiction" weaker than other proofs?
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I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons. For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice. Is there a similar case for proofs by contradiction?
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Proof by contradiction is just as logically valid as any other type of proof. If you are unsure, I think it might help to consider exactly what a proof by contradiction entails. Say we have a set of statements $\Gamma$, and that $\Gamma\cup\{(\neg\phi)\}$ is not consistent. That is, the statement $\neg\phi$ contradicts something in $\Gamma$. (In other words, we supposed $\phi$ was false, and reached a contradiction.) Say that statement was $\psi$. Then $\Gamma\cup\{(\neg\phi)\}\vdash\psi$ and $\Gamma\cup\{(\neg\phi)\}\vdash \neg\psi$. By the principle of explosion , we conclude that $\Gamma\cup\{(\neg\phi)\}\vdash\phi$. (We can prove any statement, so we prove $\phi$. By deduction , we know that $\Gamma\vdash(\neg\phi\Rightarrow\phi)$. Most first-order logic systems have an axiom that gives us $((\neg\phi\Rightarrow\phi)\Rightarrow\phi$. I hope you can convince yourself that this is true without trouble. This yields $\Gamma\vdash\phi$. We started with the idea that the negation of the
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|logic|proof-theory|
| 0
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Intuitive reasoning behind the Chain Rule in multiple variables?
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I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to multiple variables. Even the case of 2 dimensions $$z = f(x,y),$$ where $x = g(t)$ and $y = h(t)$, so $$\frac{dz}{dt} = \frac{\partial z}{dx} \frac{dx}{dt} + \frac{\partial z}{dy} \frac{dy}{dt}.$$ Now, this is easy enough to "calculate" (and figure out what goes where). My teacher taught me a neat tree-based graphical method for figuring out partial derivatives using chain rule. All-in-all, it was rather hand-wavey. However, I'm not sure exactly how this works, intuitively. Why, intuitively, is the equation above true? Why addition ? Why not multiplication, like the other chain rule? Why are some multiplied and some added?
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Think of it in terms of causality & superposition. $$z = f(x,y)$$ If you keep $y$ fixed then $\frac{dz}{dt} = \frac{df}{dx} * \frac{dx}{dt}$ If you keep $x$ fixed then $\frac{dz}{dt} = \frac{df}{fy} * \frac{dy}{dt}$. Superposition says you can just add the two together.
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|calculus|linear-algebra|
| 0
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Intuitive reasoning behind the Chain Rule in multiple variables?
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I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to multiple variables. Even the case of 2 dimensions $$z = f(x,y),$$ where $x = g(t)$ and $y = h(t)$, so $$\frac{dz}{dt} = \frac{\partial z}{dx} \frac{dx}{dt} + \frac{\partial z}{dy} \frac{dy}{dt}.$$ Now, this is easy enough to "calculate" (and figure out what goes where). My teacher taught me a neat tree-based graphical method for figuring out partial derivatives using chain rule. All-in-all, it was rather hand-wavey. However, I'm not sure exactly how this works, intuitively. Why, intuitively, is the equation above true? Why addition ? Why not multiplication, like the other chain rule? Why are some multiplied and some added?
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The basic reason is that one is simply composing the derivatives just as one composes the functions. Derivatives are linear approximations to functions. When you compose the functions, you compose the linear approximations---not a surprise. I'm going to try to expand on Harry Gindi's answer, because that was the only way I could grok it, but in somewhat simpler terms. The way to think of a derivative in multiple variables is as a linear approximation. In particular, let $f: R^m \to R^n$ and $q=f(p)$. Then near $p$, we can write $f$ as $q$ basically something linear plus some "noise" which "doesn't matter" (i.e. is little oh of the distance to $p$). Call this linear map $L: R^m \to R^n$. Now, suppose $g: R^n \to R^s$ is some map and $r = g(q)$. We can approximate $g$ near $q$ by $r$ plus some linear map $N$ plus some "garbage" which is, again, small. For simplicity, I'm going to assume that $p,q,r$ are all zero. This is ok, because one can just move one's origin around a bit. So, as bef
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|calculus|linear-algebra|
| 1
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Why does the discriminant of a cubic polynomial being less than $0$ indicate complex roots?
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The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes , but also that there are three distinct, real roots if $\Delta > 0$, and that there is one real root and two complex roots (complex conjugates) if $\Delta Why does $\Delta
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The discriminant of any monic polynomial is the product $\prod_{i \neq j} (x_i - x_j)^2$ of the squares of the differences of the roots (in an algebraic closure, e.g. $C$). Cf. the Wikipedia article on this. Consequently, if the roots are all real and distinct, this must be positive. (If the polynomial is not monic, the factor $a_0^{2n-2}$ is thrown in, for $a_0$ the leading coefficient and $n$ the degree; this is positive for a real polynomial.)
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|algebra-precalculus|polynomials|roots|symmetric-polynomials|
| 0
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