For questions about mathematical constants, that are "significantly interesting in some way".

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5
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3answers
120 views

a simple formula linking the value of $e$ to the Golden Ratio $\phi$

These last days, I was looking for an approximation formula to $\pi$. But, surprisingly, the formulas led to this other one: $$ e = \left (\frac {\phi} {\phi - 1} \right)^{\frac {1} ...
1
vote
2answers
28 views

Countability - Constant Functions

I am learning about countability. I know about diagonalization and I am confused about constant functions and whether or not they are countable. A constant function in my case would be: $f(0) = 1,$ ...
3
votes
2answers
38 views

Can I say that a fixed constant is less or equal infinity?

Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$? E.g., is $10 \leq \infty$ a correct mathematical statement? I know this comparison is true in computer arithmetic, ...
1
vote
2answers
29 views

Given that equation is a positive constant, equal roots, find value of k

I am having trouble solving this equation. It reads... Given that the equation $kx^2+12x+k = 0$, where $k$ is a positive constant, has equal roots, find the value of $k$. I am not sure where to ...
0
votes
4answers
115 views

what is $e$ really? what is its meaning? [duplicate]

I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ ...
2
votes
2answers
69 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
0
votes
0answers
18 views

Paperfolding Constant

In the Wikipedia article about the regular paperfolding sequence, it says (more or less quoted): Taking $G(t_n;x) = G(t_n;x^2) + \sum_{n=0}^{\infty}x^{4n+1} = > G(t_n;x^2) + \frac{x}{1-x^4}$ ...
0
votes
1answer
12 views

Does a constant of integration changes the shape of a distribution?

Let $f(x)$ be the frequency distribution of the variable $x$. Let assume that $\int^{\infty}_{-\infty} f(x) ≠ 1$. Let $g(x) = C f(x)$ such as $C$ is the constant of integration so that ...
0
votes
2answers
54 views

Twin Prime Constant

How would one prove that the twin prime constant $$C_2 = \prod_{p > 2}1-\frac{1}{(p-1)^2} > 0$$ Simply computing the product for a large number of terms isn't rigorous, and simply establishes ...
0
votes
2answers
86 views

Is Planck's constant a mathematical or a physical constant?

Planck's constant $h$ came out of his considerations of black body radiation and features prominently in quantum physics. Recently I came across the statement that $-i\hbar = pq - qp$ for elements ...
0
votes
0answers
18 views

how i could show that $f$ is a constant if it has intermediate value and local extremum properties? [duplicate]

let $ f\colon R\to R $ be a function with intermediate value property . if $f$ has a local extremum at every point $x\in R $ . my question is how i could show that $f$ is constant ? I would be ...
5
votes
1answer
73 views

Is $e$ involved in some geometric figure in any way?

Let's take some popular numbers in math: $\pi$, $e$, $\sqrt{2}$ and $\phi$. The number $\pi$ is the ratio between the circumference and the diameter of a circle; $\sqrt{2}$ is the length of a diagonal ...
1
vote
2answers
150 views

Is there any reason why $4-\pi$ is quite close to $\frac{\sqrt{3}}{2}$?

In this question obviously the error of our "approximation" is $4-\pi=0.858...$ . I tried to reconstruct the false argument with $\tau=2\pi$, and the error in that case would be $8-\tau=1.716...$, ...
1
vote
0answers
41 views

Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
0
votes
1answer
16 views

Landau Constants precision

Looking at an article in Wikipedia on Landau constants it indicates that the actual values are not known except that they are within a certain interval. This seems surprising to me since for most ...
1
vote
1answer
84 views

How is PI used to predict weather patterns?

I've heard that using PI to predict weather patterns is possible. I would like verification on this, and how this is possible. I can't seem to find any other sources explaining this concept. My ...
0
votes
0answers
30 views

calculating the position of a given digit in a constant (e.g. $\pi$)

I'm aware that there are a lot BBP type formulas out there which extract the n-th digit of the observed constant. I'm asking for the reverse action, namely, is it possible to find the first ...
1
vote
1answer
60 views

Why we can't define more mathematical constant?

I would like to know how many mathematical constant are there? I saw this link and I know the names. Who can define a mathematical constant? Someone can say that ...
3
votes
1answer
36 views

A misconception about arbitary constant

Given a function $f(x)$ from $\mathbb R$ to $\mathbb R$, If $f'(x)=0$ $\text{ for all } x\in \mathbb R$. Then $f(x)=C$.(This is my understanding) Question: I think that $C$ has to remain constant ...
8
votes
0answers
275 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 ...
2
votes
4answers
197 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
0
votes
2answers
65 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
1
vote
1answer
53 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
0
votes
3answers
42 views

How to prove that $\frac{d}{dx}\epsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant

How to prove that $\frac{d}{dx}\varepsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant? Thanks in advance for your immense help.
0
votes
1answer
37 views

What is the constant $c$ in $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy}$?

In the manner in which $\frac{d}{dx} e^{x} = e^{x}$. What is the value of the constant $c$ for which $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy} $ ?
1
vote
1answer
71 views

Asymptotic Big Omega Proof

Let $f(n) = 2n^4 − 4n^3 + 16n^2 − 64^n + 3$. (a) Using the constant $c = 1.9$, prove that $f(n) \in \Omega(n^4)$. Just by looking at it it is clear to me that this is true as for $f(n) \in ...
1
vote
4answers
368 views

Calculus Derivative - Finding unknown constants

Determine the constants $a$, $b$, $c$, and $d$ so that the curve defined by $y = ax^3 + bx^2 + cx + d$ has a local maximum at the point $(2, 4)$ and a point of inflection at the origin. Sketch the ...
0
votes
2answers
51 views

Statistics - scaling sets of data - mean & S.d

A set of numbers has a mean of 22 and a standard deviation of 6. If 3 is added to each number of the set, and each resulting number is then doubled, find the mean(50) and standard deviation(12) of the ...
5
votes
2answers
108 views

If $\pi $ is a normal number, is $\tau $ one?

If $\pi$ is a normal number, would that imply that $\tau =2\pi $ is also a normal number? If so, why? Something tells me that it should be, but I have no idea how to prove it. If all digits of $\pi$ ...
1
vote
5answers
105 views

Limits to infinity Finding Constant Number

Hi I have a question regarding of limits to infinity please help which I need to find the constant number for a and b. Please help! Thank You! The question states the user to find the following ...
0
votes
3answers
662 views

What are the uses of Euler's number $e$?

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ...
0
votes
1answer
41 views

Integration involving multiple constants…

So I've been tackling the rather nasty integral of... $\int^R_s\frac{2r}{\sqrt{r^2-s^2}}.\frac{1}{2}(R-r)^2dr$ ...where R and s are constants. However, every method I try I seem to get stumped by ...
6
votes
2answers
114 views

How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can prove that these two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and $\frac{d}{dx}\ln x$; some use the former ...
1
vote
0answers
29 views

Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...
0
votes
0answers
45 views

$f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
5
votes
1answer
137 views

On a constant defined by Ramanujan.

In the second letter to Hardy Ramanujan writes about the number of prime numbers less than $n$ there he writes. Here this constant $\mu$ facinated me . What is its closed form? and How to compute ...
3
votes
0answers
92 views

Finding relatives of the series $\varphi =\frac{3}{2}+\sum_{k=0}^{\infty}(-1)^{k}\frac{(2k)!}{(k+1)!k!2^{4k+3}}$.

Consider $\varphi=\frac{1+\sqrt{5}}{2}$, the golden ratio. Bellow are series $(3)$ and $(6)$ that represent $\varphi$ $$ \begin{align*} \varphi &=\frac{1}{1}+\sum_{k=0}^{\infty}\cdots&(1)\\ ...
2
votes
1answer
129 views

Express the mathematical constant $e$ in terms of a limit that goes to zero.

The mathematical expression of the mathematical constant $e$ in terms of a limit that goes to infinity is $$e = \lim\limits_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$ But can we express the ...
8
votes
3answers
745 views

On the “Look-and-Say” sequence and Conway's constant

The look-and-say sequence starting with $S_1=1$ is, $$S_n = 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,\dots$$ If $L_n$ is the number of digits of the $n$th term then, ...
-1
votes
1answer
143 views

Why is the number Pi more popular than any other constant? [closed]

What is so special about the number $\pi$? There are many more interesting constants, such as e, $\gamma, \sqrt{2}$ or the catalanian number. $\pi$ has been calculated to more digits than any other ...
1
vote
3answers
77 views

General form for the series expansion of $e$

I've found a lot of series expansions of the Napier's constant. I was wondering if a general form for this could be devised. They all have a trend and similarities. I've been trying but I've been ...
3
votes
1answer
124 views

Why $\operatorname e$ was named e ? What is the history and reason behind it?

Why the constant that Euler discovered has symbol $\operatorname e$ and not other symbols? What is the history and reason behind it? Thanks.
12
votes
4answers
465 views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
0
votes
1answer
57 views

Prove that the function is constant $f(x)=\arcsin2x\sqrt{1-x^{2}} - 2\arcsin x$ [closed]

Prove that this function is constant: $$f(x)=\arcsin\left(2x\sqrt{1-x^{2}}\right) - 2\arcsin x$$
2
votes
1answer
45 views

Antiderivative where resulting constant depends on x?

Everything was going really well until one week before the exam when the teacher gave us this problem: $\int \frac{2x^2+13x+19}{x^2+5x+6} dx$ For which I and Wolfram Alpha finds the solution: ...
23
votes
2answers
303 views

On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as ...
2
votes
0answers
105 views

In the Hunt for Kaprekar's Constants for more than 4 digits.

Kaprekar's constant is $6174$ . Take any four digit number with at least two different digits; create two four digit numbers by writing the digits in descending order and in ascending order; subtract ...
1
vote
1answer
53 views

Mathematical constants

Am I mistaken, or is there a mistake on the mathematical constants Wikipedia page that describes the Conic constant / Schwarzschild constant in terms of Napier's constant?
0
votes
1answer
70 views

Real Analysis constant function

Suppose that $f:(a,b) \to \Bbb R$ satisfies $|f(x)-f(y)|=M|x-y|^\alpha$ for some $\alpha > 1$, some $M \geq 0$, an all $(x,y)$ in $(a,b)$. Prove $f$ is constant on $(a,b)$ I have absolutely no ...
3
votes
0answers
330 views

MRB constant proofs wanted

This article has been edited for a bounty. $C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html . There is an excellent 56 page paper who's author has passed away. ...