Articles of rational numbers

Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$

I am currently attempting to prove a claim in Hardy’s Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this. Claim: Given any rational number r and any positive integer $n$, there exists a rational number on either side […]

Is there a polynomial $f\in \mathbb Q$ such that $f(x)^2=g(x)^2(x^2+1)$

I was asked the following question: $g\in \mathbb Q[x]$ is a polynomial (not the zero polynomial). Find $f \in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$ or show that such an $f$ does not exist. I really have no idea where to begin and would appreciate all help I can get to solve this.

Proving $\mathbb{Q} = \{f(\sqrt{2}): f(x) \in \mathbb{Q}\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$

I need to prove that: $$\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$$ Well, $ \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} $ is the set of $a_nx^n+\cdots +a_0$ when we take $x\in \mathbb{Q}$, so of course $$\{x+y\sqrt{2}:x,y\in\mathbb{Q}\} \subset \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\}$$ but how to prove $$\{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\}\subset \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$$? Can I just assume that I […]

Digamma equation identification

I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ab\right)=\sum_{\substack{\large\rho^b=1\\\large\rho\ne1}}(\rho^a-1)\ln(1-\bar\rho)-\gamma$$ when $0<\dfrac ab\le1$. It’s unusual in that it sums over the $b$-eth roots of unity (which I don’t see very often). (Note that $\bar\rho=\rho^{-1}$.) It also gives explicit values of the digamma function for all rational arguments, […]

Does $\sin n$ have a maximum value for natural number $n$?

In formal, does there exist $k\in\mathbb{N}$ such that $\sin n\leq\sin k$ for all $n\in\mathbb{N}$?

Determine if $(p/q)^{a/b}$ is rational

I know, in general, that it isn’t true. ${\frac{2}{1}}^{1/2}$ is irrational. I’m only interested in this where $\frac{p}{q}$ and $\frac{a}{b}$ are positive, but to make this even simpler, lets just say that $a,b,p,q \in \mathbb{N}$. I’m curious to know if there’s an algorithm for figuring it out within the bounds I’ve mentioned, but I’m even […]

Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?

Suppose I have a rational number $a$ and $a^2 < 2$. Can I find another rational number $B$ such that $a^2<B^2<2$? Based on the answer to this question, I thought of doing the following: $$ a^2 < 2 \implies a < \frac{2}{a}\\ \text{Let}\hspace{1cm} B=\frac{a+\frac{2}{a}}{2}=\frac{a^2+2}{2a} $$ $B$ is greater than $a$ because: $$ \begin{array} {aa} B>a […]

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is not complete. You can change from increasing to decreasing and vice versa in each irrational “hole”). Is it true that every homeomorphism of $\mathbb{Q}$ is monotone?

Why is epsilon not a rational number?

I was wondering why epsilon, the smallest positive number, isn’t a rational number. I was watching a video a few days ago about surreal numbers, and I’ve learned that, in the field of surreal numbers, o.(9) is not equal to 1, in contrast to the field of the real numbers, where they represent the same […]

Is there a rational surjection $\Bbb N\to\Bbb Q$?

The question is in the title. Is there a one-dimensional rational function $f\in\Bbb R(X)$ which restricts to $\Bbb N\to\Bbb Q$, which is a surjection onto $\Bbb Q$? My guess is no. Expanding the scope a little (at the expense of precision), are there any “nice” functions that enumerate $\Bbb Q$? Here “nice” is meant to […]