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37

I've often asked myself the same thing, and this is what I tell myself. $\mathbb R$ is (up to isomorphism) the only totally ordered, complete field. This is pretty big news, because these two nice structures lead to so many others we find useful to study in math. $\mathbb R$ (and more generally $\mathbb R^n$) is so great because a plethora of these ...


7

There's no way to do justice to "Why is mathematics about real numbers?" within the length constraints of a Math.SE post, but here are some relatively philosophical observations and opinions (meant to be a bit provocative, in the spirit of answering a soft question). First, as multiple people have commented, the real numbers are not universally regarded as ...


5

A nice property of real numbers is that they are complete: every Cauchy sequence converges. In analysis, mathematicians like to study spaces that are complete. People study Banach spaces rather than ordinary normed spaces; study Hilbert spaces rather than ordinary inner product spaces. A space that is not complete does not have as nice properties as ...


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There's no conflict between your high school teacher's advice To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side. and your professor's To prove a statement is true, you must not use what you are trying to prove. As in Siddarth Venu's answer, if you prove $a = c$ and $b = c$ ...


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You need a topology to define the notion of continuity. Therefore the crucial notion is the notion of open sets. It is not correct to say that there are no continuous maps in algebra. For instance, if you take a finite dimensional real vector space, you can give it the topology of $\mathbb{R}^n$, where $n$ is the dimension. Then any endomorphism (linear ...


2

A Hilbert space is a set with some structure. It's not hard to show in ZFC that Hilbert spaces exist, but if we wanted to we could consider some extremely weak theory $T\subset ZFC$ which can't prove e.g. that the reals form a set; then the statement "There is a Hilbert space" might not be provable in $T$, and would - if we wanted to use Hilbert spaces - ...


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It is enough.. Consider this example: To prove: $a=b$ Proof: $$a=c$$ $$b=c$$ Since $a$ and $b$ are equal to the same thing, $a=b$. That is the exact technique you are using and it sure can be used.


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usually graduate students study PDE (Partial Differential Equations) rather than ODE. You may want to try doing a project on Sturm–Liouville theory. https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory It is a slightly advanced part of ODE that may be suitable for graduate students. If that is too hard, you can try Bernoulli differential equation: ...


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Many possibilities.. Malthusian population model, Prey predator models, pursuit models, dynamic systems, or Black-Scholes model etc.


2

Topology: Why you can't turn your shirt inside out while wearing it.


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The book named Irresistible Integrals by George Boros and Victor Moll seems to fit to your purpose. I found it several years before after having hard time trying to prove some integral formula in the integral table of G&R and it really was helpful.


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Logically you can think of it as 'removing an absence'. For instance, say someone dug $4$ holes, removing 3 cubic meters of dirt from each. The change in amount of dirt is $4 \times -3 = -12$. Now say you want to undo this work... you want to remove the holes. This can be represented as taking away $4$ holes, each of which is missing $3$ cubic meters of ...


2

I think that whenever I saw anyone using anything stronger than $\sf ZFC$, the axioms were defined explicitly. Except, perhaps, things like $I0,I1,I2,I3$ which are essentially the conventional names for these statements. Most of the time you write either something like "We abbreviate by $\sf IC$ the statement "There exists an inaccessible cardinal", and in ...


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You can find all the quotes here and I recommend you to read it. But for the completeness of the answer I quote some of them which I find most important. In The Map of My Life mathematician Goro Shimura said, I discovered that many of the exam problems were artificial and required some clever tricks. I avoided such types, and chose more ...


2

In summary, I am asking about 2 things: first, what exactly is a "chaotic" sequence? The commenters above gave you some definitions you can work with to find the answer to that. Second, and more important to me, is the sequence $z^{z^{...}}$ really chaotic? It can be, for certain values $c=z_0$. But no it's not, (generally) for all ...


1

By definition, a vector space is a Lie algebra, if the Lie bracket is "closed under commutation relation", i.e., $[x,y]\in L$ for all $x,y\in L$, and satisfies skew-symmetry and the Jacobi identity. This holds for finite-dimensional and infinite-dimensional vector spaces.


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Polynomials can describe geometric objects In high school we learn that some low order polynomials can describe geometric shapes: Basic shapes we all recognize ( as intro ) $$\begin{array}{rr}y=kx+m& (line)\\x^2+y^2 = r^2 & (circle)\\y = x^2+ax+b &( parabola)\end{array}$$ Cool properties consider the rotation ...


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Divergent Series can be visual: from the Wikipedia showing that $(1-1+1-1+\dots)^2=1-2+3-4+\dots$


1

decimally-infinite numbers and negative-square numbers


1

A Garden of Integrals is quite nice.


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Bernstein gave a constructive proof of the Weierstrass approximation theorem based on probability: Bernstein, S. N., Démonstration du Théorème de Weierstrass fondée sur le calcul des Probabilités, Comm. Soc. Math. Kharkov 2.Series XIII No.1 (1912), 1-2. See History of Approximation Theory |Historical Papers.


1

I think one reason Complex Analysis is so nice is because being holomorphic/analytic is an extremely strong condition. As opposed to real analysis, differentiability is a rather weak condition, so we have functions that are differentiable once but not twice etc. Real analysis is full of nasty counterexamples like the Weierstrass function which is continuous ...


1

J is very useful. While it has its quirks, once you get used to them, as gar said, you can really understand the language quite well. I consider myself a novice at J, but being exposed to it, I can definitely tell what the example in the accepted answer's "mysterious code". It's easy enough to say you don't know what a program does, that's the same for many ...


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It may fall into the more "undergraduate mathematics" but the Lotka–Volterra equations are a really good topic for something like this.


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I agree with @Siddharth Venu. If we have to prove a=b, At times, on proceeding from LHS you may end up in a stage(intermediate step) which you cannot solve any further and you continue to solve RHS to arrive at the same stage i.e, a=c and b=c so you can conclude that a=b these kind of equality problems often comes in trigonometry Many chapters like ...


1

Start with a Poisson process $\Pi$ of rate $a+b$ on $(0,\infty)$. Paint each point of the Poisson process either red (with probability $a/(a+b)$) or green (with probability $b/(a+b)$), the colors being i.i.d. and independent of the locations of the points themselves. Let $X:=\min\{t\in\Pi: t$ is painted red$\}$ and $Y:=\min\{t\in\Pi: t$ is painted green$\}$. ...



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