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My advice would be: $\bullet $ Do many calculations $\bullet \bullet$ Ask yourself concrete questions whose answer is a number. $\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!) $\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in ...


$$\frac{2^{\frac{(2\cdot2)!}{2+2}}+22+2^{2^2}-\sqrt 2^2}{2^{2-\frac{2}{2}}}$$


Almost all real numbers are transcendental. Almost all real numbers are not constructible. Almost all real numbers are not computable. Almost all real numbers are normal. Almost all real numbers are not definable. Almost all real numbers are not periods.


$$50 = 2\cdot(2\varphi - 1)^4$$ where $\varphi$ is the Golden Ratio. $$50 = \sum_{i=0}^{+\infty} (0.98)^i$$ (Geometric series) $$50 = \left(\left(\frac{5^5-5}{5}+5^0\right)\cdot\left(5-5^0\right)\right)^{0.5}$$ $$50 = 0.5 \cdot (5+5)^{\frac{5^0}{0.5}}$$ $$50 = 5\cdot\left(\frac{5}{0.5}+5^0\right)-5$$ (Using only the digits "5" and "0") $$50 = ...


No fancy math here, but if you want to emphasize how old your friend is getting, nothing says it better than implying he's halfway to the century mark: $$100\over2$$


The Weierstrass approximation theorem was published in 1885, when Karl Weierstrass was 70. As T.W. Körner writes in his Fourier analysis book (pg. 294): Fejér discovered his theorem at the age of 19, Weierstrass published this theorem at the age of 70. With time the reader may come to appreciate why many mathematicians regard the second circumstance as ...


We can use only two famous numbers in mathematics, $\large\pi$ and $\large e$, to produce number $50$. $$\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\Large \lfloor e^\pi \rfloor + \lfloor \pi^e \rfloor + \lfloor \pi \rfloor + \lfloor e \rfloor = 50}} $$ Click the box to see Wolfram Alpha's output to confirm the result


I was a teaching assistant in Linear Algebra previous semester and I collected a few applications to present to my students. This is one of them: Google's PageRank algorithm This algorithm is the "heart" of the search engine and sorts documents of the world-wide-web by their "importance" in decreasing order. For the sake of simplicity, let us look at a ...


Let $[a_0; a_1, a_2, \ldots]$ be the continued fraction expansion of $x$. Then there exists a constant $\kappa\approx 2.685452001$ with the property that, for almost all $x$, $$\lim_{n\to\infty} (a_0a_1\ldots a_n)^{1/n} = \kappa.$$ (Many properties about continued fraction expansions are analogous to properties about the normality of base-$n$ expansions. ...


Quoting Wikipedia Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $50 = 1^2 + 7^2 = 5^2 + 5^2$. So you could write something like \begin{align} 50 = \min_{n\in \mathbb{N}}\{n =p_i^2+p_j^2=p_k^2+p_l^2 \quad | \quad p_i,p_j,p_k,p_l\in\mathbb{N} \quad \wedge\quad p_k \not =p_i \not = p_l \} \end{align} ...


Another very useful application of Linear algebra is Image Compression (Using the SVD) Any real matrix $A$ can be written as $$A = U \Sigma V^T = \sum_{i=1}^{\operatorname{rank}(A)} u_i \sigma_i v_i^T,$$ where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix. Every greyscale image can be represented as a matrix of the intensity ...


Possibly Abbott, Understanding Analysis


Louis de Branges proved Bieberbach's Conjecture at age 53. LINK wikipedia (I found the picture when researching his birthdate)


This 'misconception' is not proven or disproven by quoting specific counter examples, but by examining a fair number of mathematical discoveries and determining a trend. 17 equations that changed the world I will be going by this list based on this book by Ian Stewart of the 17 equations that changed the world, I will be taking the earliest possible date ...


We also have \begin{align*} 50 &= 11+12+13+14 \\ &= (8+4)+(8-4)+(8\cdot 4)+(8/4) \\ &= 4^2 + 4^2 + 3^2 + 3^2\\ &= 6^2 + 3^2 + 2^2 + 1^2\\ &= (7+i)(7-i) \\ &= (10-\color{red}{5})(10-\color{red}{0})\\ &= 10(\color{red}{5}+\color{red}{0})\\ &= \sqrt{30^2+40^2}\\ &= \sqrt[3]{170^2+310^2}\\ &= \sqrt[3]{146^2+322^2}\\ ...


In financial mathematics one encounters index numbers, such as consumer price indexes, where a base year (such as 1992) is commonly specified by truly ghastly expressions such as 1992=100. And no, they're not working modulo a divisor of 1892: they're referring to the fact that the index number for the base year is 100. It feels wrong to even bring this ...


The word "trivial." The many uses of this word include: 1.) The colloquial usage as a synonym for "easy." 2.) The trivial group which consists only of the identity element. 3.) The trivial ring which consists only of the multiplicative and additive identities. 4.) A trivial solution to an equation, often when a variable equals 0 (or constant in the cases ...


Grigori Perelman was 36 when he released his proof of Thurston's geometrization conjecture.


I always love to prove that: If $\{a_n\}_{n\in\mathbb{N}}$ is a bounded real sequence, it has a converging subsequence. with the Erdos-Skeres', or Dilworth's, theorem: (Erdos-Szekeres, finite version) Every sequence with $n^2+1$ terms admits a weakly monotonic subsequence with $n+1$ terms. (Dilworth, infinite version) Every infinite POset ...


Whenever one of my friends has a birthday, I find out how old they get and then I visit their number on Wikipedia For your friend, I would write something like this: 50 is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $1^2 + 7^2$ and $5^2 + 5^2$. It is also the sum of three squares, $3^2 + 4^2 + 5^2$. It is ...


I hope he doesn't calculate it by adding all numbers ;) \begin{align} 50 = \sum_{k=0}^{100} (-1)^k k \end{align} And a last one, involving only $4$s and $9$s: \begin{align} 4^9 \mod 49 + \sqrt{49} \end{align}


This is a simpler example, but maybe that'll be good for undergraduate students: Linear algebra is a central tool in 3-d graphics. If you want to use a computer to represent, say, a spaceship spinning around in space, then you take the initial vertices of the space ship in $\mathbb{R}^3$ and hit them by some rotation matrix every $.02$ seconds or so. Then ...


The course Measure Theory by D.H.Fremlin includes TeX source. Topology Course by Aisling McCluskey and Brian McMaster in HTML. Diverse lecture notes by Conor Houghton. Cryptography homework by Boaz Barak. Digital Image Processing. Abstract Algebra handouts and Number Theory lecture notes.




A bit of self publicity, but the reason that A Primer on Hilbert Space Theory was written is precisely to give what you refer to as an 'honest' introduction to the foundations of analysis. Edit: OP's comment below clarifies this book is not at the intended introductory level.


$\forall x\in\mathbb R$: Almost all real numbers are different from $x$.


binary code : 50 is given by 110010


Calculus I,II,III: The '$dx$'s in integrals and derivatives are just notation to help keep track of the important variables in a given problem, and they're otherwise meaningless in isolation. Real Analysis: In fact, when the integration variable is unambiguous we may as well dispense with the differentials altogether and just denote the integral of $f$ over ...

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