Define the function $g(x)$ to take the value 0 at irrational values of $x$ and to take the value $1/q$ when $x=p/q$ is a rational number in lowest terms, $q>0$. At which points is $g$ continuous? At which points is the function discontinuous? It seems like it would be discontinuous everywhere because between each irrational […]

This question already has an answer here: Can there be two distinct, continuous functions that are equal at all rationals? 4 answers

How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?

What are all the values of $a$ and $b$ such that $a, b\notin\mathbb{Q}$ and $a^b\in\mathbb{Q}$? For instance, the classic example is $\left(\sqrt2^\sqrt2\right)^\sqrt2=2$. We also have $a,b\notin\mathbb{Q}$ and $b=\log_ak$, where $k\in\mathbb{Q}$, (making sure that $b$ is irrational), then we can see $a^b=k$. What other ways are there to find these $a$’s and $b$’s? I’m not asking […]

This question already has an answer here: Can there be two distinct, continuous functions that are equal at all rationals? 4 answers

Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part: Since $n!f(x)$ has integral coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and its derivatives (…) have integral values for $x=0$; also for $x=\pi=\frac{a}{b}$, since $f(x)=f(\frac{a}{b}-x)$ Assuming this, the rest I understand. But why is this […]

Just hints but solution thx. Any hints for me? I simply suppose that $a = \dfrac mn$ then $\sqrt a = \sqrt{\dfrac mn}$ But this does not make sense ..

Problem: Prove that $\sqrt{2}+\sqrt{3}$ is irrational. The book where I encountered this problem had the following hint: We make a polynomial with integer coefficients called $f(x)$ that $f(\sqrt{2}+\sqrt{3})=0$. (Why?) Accepting this I solved the problem like this: If $x=\sqrt{2}+\sqrt{3}$ then $x^2=5+2\sqrt{6}$ and so $(x^2-5)^2=24$ thus: $$x^4-10x^2-1=0$$ But I want to know the reason that we […]

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?

For integers $n \neq 0$ is $\sin n$ irrational or transcendental? This arose from another question. I would hypothesize yes and yes, possibly with proof for irrationality existing and but not for the more difficult property of transcendentality. Any ideas?

Intereting Posts

Does universal set exist?
Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?
Closed-form of $\int_0^1 B_n(x)\psi(x+1)\,dx$
How to estimate the following integral: $\int_0^1 \frac{1-\cos x}{x}\,dx$
$\forall m,n \in \Bbb N$ : $\ 56786730\mid mn(m^{60}-n^{60})$
Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:\to
Guide to mathematical physics?
How can I prove that $e^x \cdot e^{-x}=1$ using Taylor series?
Does convergence in distribution implies convergence of expectation?
Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables
What is $L^p$-convergence useful for?
Algorithms for mutually orthogonal latin squares – a correct one?
Optimization of $2x+3y+z$ under the constraint $x^2+ y^2+ z^2= 1$
Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$
Continuity of an inverse function.