Articles of irrational numbers

At which points is the function discontinuous?

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 […]

Continuity question: Show that $f(x)=0, \forall x\in\mathbb{R}$.

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?

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

On $a^b \mid a, b\notin\mathbb{Q}$ and $a^b\in \mathbb Q$

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 […]

Proving f(x)=0 for all x in when we only know that f is continuous and f(x)=0 when x is rational.

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

Understanding proof that $\pi$ is irrational

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 […]

Prove that if $a$ is irrational then $\sqrt a$ is irrational

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 ..

Prove that $\sqrt{2}+\sqrt{3}$ is irrational.

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 […]

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

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?

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?