# Tagged Questions

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

7k views

### A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
The following integral, $$\int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi$$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves $\... 14answers 10k views ### Can the golden ratio accurately be expressed in terms of e and$\pi$I was playing around with numbers when I noticed that$\sqrt e$was very somewhat close to$\phi$And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ... 9answers 6k views ### Find the average of$\sin^{100} (x)$in 5 minutes? I read this quote attributed to VI Arnold. "Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ... 4answers 9k views ### Motivation for Ramanujan's mysterious$\pi$formula The following formula for$\pi$was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ... 1answer 614 views ### Is there an integral for$\pi^4-\frac{2143}{22}$? In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just$10^{-7}$), $$\pi^4 \approx 2^4+... 9answers 3k views ### How is the derivative truly, literally the “best linear approximation” near a point? I've read many times that the derivative of a function f(x) for a certain x is the best linear approximation of the function for values near x. I always thought it was meant in a hand-waving ... 4answers 2k views ### Geometric explanation of \sqrt 2 + \sqrt 3 \approx \pi Just curious, is there a geometry picture explanation to show that \sqrt 2 + \sqrt 3 is close to \pi ? 7answers 1k views ### Pi Estimation using Integers I ran across this problem in a high school math competition: "You must use the integers 1 to 9 and only addition, subtraction, multiplication, division, and exponentiation to approximate the ... 1answer 808 views ### A series problem by Knuth I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that$$\sum_{n=1}^\infty \left(\frac{n^n}{n!e^n}-\frac{1}{\sqrt{2\... 2answers 785 views ### On Shanks' quartic approximation$\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere$10^{-82}$, $$\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)\color{blue}{+10^{... 6answers 689 views ### Why is it that this gives a good approximation of \pi? At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that:$$\frac{1}{100} \cdot 11^{\ln(11)} \approx 3.14159789211,$$... 9answers 7k views ### What is the purpose of Stirling's approximation to a factorial? Stirling approximation to a factorial is$$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$I wonder what benefit can be got from it? From computational perspective (I admit I don't ... 4answers 4k views ### How much gas does a car use to carry its own gas? I have always been curious about this one. Since the gas has some weight, the car will have to burn some extra gas to carry it's own fuel around. How can I calculate how much that extra gas is? ... 5answers 6k views ### How do you calculate the decimal expansion of an irrational number? Just curious, how do you calculate an irrational number? Take \pi for example. Computers have calculated \pi to the millionth digit and beyond. What formula/method do they use to figure this out? ... 3answers 1k views ### The right “weigh” to do integrals Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of ... 7answers 17k views ### Simple numerical methods for calculating the digits of \pi Are there any simple methods for calculating the digits of \pi? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ... 4answers 1k views ### Prove that 2^{2^{\sqrt3}}>10 With a computer or calculator, it is easy to show that$$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$How can we prove that 2^{2^{\sqrt3}}>10 without a calculator? 1answer 536 views ### Approximate value of a slowly-converging sum of \sum|\sin n|^n/n In this question on Math.SE there appears this sum:$$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$which converges very slowly. What methods would you suggest for evaluating it ... 6answers 1k views ### Why is e^\pi - \pi so close to 20? e^\pi-\pi\approx 19.99909998 Why is this so close to 20? 7answers 4k views ### How to show this formula to get a square root of a number in “just few seconds” is true? I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation. Let x be a non perfect square number. If y is the closer ... 3answers 715 views ### Approximation for \pi I just stumbled upon$$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$which is \delta = 0.0000481330 different from \pi. Although this is a rather crude approximation I ... 8answers 6k views ### Rapid approximation of \tanh(x) This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ... 3answers 702 views ### Maximum of Polynomials in the Unit Circle Let z_{1},z_{2},\ldots,z_{n} be i.i.d random points in the unit circle (|z_i|=1) with uniform distribution of their angles. Consider the random polynomial P(z) given by$$ P(z)=\prod_{i=1}^{n}(... 6answers 7k views ### Approximation of$e^{-x}$Is there a method to mentally evaluate$e^{-x}$for$x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters. 5answers 2k views ### How best to explain the$\sqrt{2\pi n}$term in Stirling's? I recently showed my Algorithms class how to bound$\ln n! = \sum \ln n$by integrals, thereby obtaining the simple factorial approximation $$e \left(\frac{n}{e}\right)^{n} \leq n! \leq en\left(\... 4answers 480 views ### How could I improve this approximation? In a computer application, I need to solve trillions of times an equation which can be reduced to$$f(x)=\sin(x)-a x=0$$Newton methods (quadratic and higher orders) are used for the solution. ... 3answers 2k views ### Sine Approximation of Bhaskara An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.)$$\sin x \approx \frac{{16x\... 3answers 8k views ### Approximating the logarithm of the binomial coefficient We know that by using Stirling approximation:$\log n! \approx n \log n$So how to approximate$\log {m \choose n}$? 3answers 447 views ### Approximating$1/z$by polynomials Let$C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$and$f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials$p_n(z)$such that $$\lim_n p_n(z)=f(z)$$ uniformly on$C$. Does anyone ... 2answers 2k views ### Ramanujan's approximation for$\pi$In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of$\pi$, such as $$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ... 3answers 501 views ### Approximating 100! I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range [a,b] which should contain ... 1answer 381 views ### Nested solutions of a quadratic equation. A quadratic equation of the form x^2+bx+c=0 can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ... 2answers 223 views ### Elegantly Proving that ~\sqrt[5]{12}~-~\sqrt[12]5~>~\frac12 \qquad How could we prove, without the aid of a calculator, that ~\sqrt[5]{12}~-~\sqrt[12]5~>~\dfrac12 ? I have stumbled upon this mildly interesting numerical coincidence by accident, ... 3answers 2k views ### How to convert \pi to base 16? According to this Wikipedia article \pi is approximately 3.243F in base 16 (i.e. hexadecimal). Can someone explain this? (Note: I understand how to convert an integer to base 16) Thanks 2answers 220 views ### A Mathematical Coincidence, or more? According to the paper "Ten Problems in Experimental Mathematics",$$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times 10^{... 2answers 544 views ### Intuitively, why is the Euler-Mascheroni constant near sqrt(1/3)? Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni ... 2answers 217 views ### why$ \rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$Let$n\ge 7$be positive integers,show that $$f(n)=\rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$$ Anyone know this problem background?or maybe have best proof or best result? 1answer 428 views ### A cute approximation for$\cot(2\pi x)$(!?) Numerical calculations and some theory leads to the suggestion that $$\cot(2\pi x) \rightarrow\frac{1}{2\pi}\sum_r \frac{1}{x-r}$$ where$r$ranges over all the roots of$B_{2n+1}$(Bernoulli ... 2answers 2k views ### Approximation to the Lambert W function If: $$x = y + \log(y) -a$$ Then the solution for$y$using the Lambert W function is: $$y(x) = W(e^{a+x})$$ In a paper I'm reading, I saw an approximation to this solution, due to "Borsch and Supan"(?)... 1answer 11k views ### What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline? I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-... 1answer 291 views ### Showing that$\int_0^\infty x^{-x} \mathrm{d}x \leq 2$. This integral is very closely related to the sophmores dream that states $$\int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots$$ For example here http://en.wikipedia.org/wiki/... 1answer 502 views ### Why$e^{\pi}-\pi \approx 20$, and$e^{2\pi}-24 \approx 2^9$? This was inspired by this post. Let$q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ... 1answer 488 views ###$\pi^4 + \pi^5 \approx e^6$is anything special going on here? Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit$e$using$\pi$'s Given enough integers and$\pi$'s we can ... 2answers 932 views ### Application de Stone-Weierstrass Bonjour, J'ai rencontré le problème suivant dans le livre "Real and Functional Analysis" de Lang, au chapitre$3$. J'explique d'abord le contexte, puis j'en viendrai à la question précise. Il faut ... 1answer 647 views ### Approximation of Semicontinuous Functions Assume that$k \in \mathbb{N}$and$f : \mathbb{R}^d \rightarrow [0,\infty)$is lower semicontinuous, i.e.$f(x) \leq \liminf_{y \rightarrow x} f(y)$for all$x \in \mathbb{R}^d$. Does there exist a ... 8answers 1k views ### Approximation of$e$using$\pi$and$\phi$? $$e \approx \frac{4 \phi +3 \pi-5}{4}$$ where$~\phi~$is a Golden ratio . Is it possible to construct better approximation of$e$using$\pi$,$\phi$and integers ? 9answers 14k views ### How to calculate$e^x$with a standard calculator Is there a simple method for calculating the$e^x$($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ... 9answers 412 views ### How do I Approximate$\log{2}\approx 0.693$without using the Maclaurin series? How do I approximate the value$\log{2}\approx 0.693$without using the Maclaurin series? The book gives the hint: consider$f(x)=e^x-e^{-x}-2x\$.
I recently stumbled across the formula: $$\pi=20\arctan\frac{1}{7}+8\arctan\frac{3}{79}$$ developed by Euler, for approximating pi. I evaluated it to several thousand decimal places and up to that ...