Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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2
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3answers
95 views

Proving f(x)=0 for all x in [a,b] when we only know that f is continuous and f(x)=0 when x is rational. [duplicate]

The question is as follows a.) Let $f(x)$ be continuous function on an interval [a,b] and suppose that $f(x)=0$ for each rational value $x$ in [a,b]. Prove that $f(x) = 0$ for all $x \in [a,b]$. b.) ...
4
votes
3answers
74 views

Rational solutions $(a,b)$ to the equation $a\sqrt{2}+b\sqrt{3} = 2\sqrt{a} + 3\sqrt{b}$

Find all rational solutions $(a,b)$ to the equation $$a\sqrt{2}+b\sqrt{3} = 2\sqrt{a} + 3\sqrt{b}.$$ I can see that we have the solutions $(0,0), (2,0), (0,3), (3,2), (2,3)$, and I suspect that ...
2
votes
0answers
51 views

Is it possible to reduce Theory of Rationals to Theory of Natural Numbers?

Is the following possible ? $$ Th( \mathbb{Q}, +, \leq ) \leq^{\log}_m Th( \mathbb{N}, +, \leq )$$ I believe it is not possible since Natural Numbers are not dense. It is also not possible $$ Th( ...
1
vote
1answer
58 views

Finding multiplicative factors of $p\in[\sqrt{2},2)\cap\mathbb{Q}$

Given $p\in[\sqrt{2},2)\cap\mathbb{Q}$, how to find $q,r\in(0,\sqrt{2})\cap\mathbb{Q}$ such that $p=qr$ ? Context: When constructing $\mathbb{R}$ with Dedekind cuts, this question arises when trying ...
6
votes
2answers
64 views

Uses of vector spaces over $\mathbb Q$

I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$: Hilbert's third problem; and The Buckingham pi ...
0
votes
0answers
51 views

When is a finite sum of rational numbers an integer?

Asking in "Which radical equations transformable into a polynomial equation by exponentiating the equation?" and in "When is the number of radicals in a power of a sum of radicals less than or equal ...
1
vote
1answer
39 views

Continuity of a function defined on rationals

So I have a function $$f: \mathbb{Q} \rightarrow \mathbb{R}, f(x)=x$$ and need to state with justification whether or not it is continuous. I seem to be having trouble actually interpreting the domain ...
1
vote
0answers
43 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
-3
votes
1answer
383 views

(22/7) is a rational number and (π) is irrational number [closed]

Why (22/7) is a rational number and (π) is irrational number. please explain. Edit: How can you say that $22/7=\pi$, when one number if rational and the other is irrational?
1
vote
0answers
27 views

Proof that $\zeta_P$ never does the following…

Let's assume that $\zeta_P(s)$ is the prime-zeta function or: $$\zeta_P(s)=\sum_{n\in P} \frac{1}{n^s}$$ I noted that if $\forall s\in \Bbb{Q},s\not =0, \zeta_P (s)\not\in \Bbb{Q}$ I cant really ...
2
votes
4answers
107 views

Can all irrational numbers be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational? [closed]

I am curious to know whether all irrational numbers can be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational. (Almost similar to how all complex numbers can be written as $x + iy$, ...
1
vote
0answers
127 views

Interesting facts/ proofs about rational and irrational numbers

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.
1
vote
2answers
68 views

In $Q[x]/(x^2-2)$ find inverses of $[3x-2]$

In $\mathbb{Q}[x]/(x^2-2)$ Find inverses of $[3x-2]$
2
votes
1answer
55 views

Examining the nature of mapping of $f(x) = \frac{x}{x^2 - 2}$.

Let f: $\mathbb{Q} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2 - 2}$, $x \in \mathbb{Q}$. Examine the nature of mapping Attempt: I think f(x) is not ...
3
votes
2answers
67 views

Intersection of union of crazy intervals in $\mathbb{R}$

I am looking at two sets $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For each $n,k \in \mathbb{N}_{\ge1}$ I define an interval $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Now I ...
7
votes
1answer
108 views

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Question. Let $\mathbb A=\mathbb R\!\smallsetminus\!\mathbb Q$ be the irrational numbers. Is there a continuous function $\,f:\mathbb A\to\mathbb Q$, which is nowhere locally constant? – i.e., for ...
3
votes
1answer
69 views

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of ...
2
votes
2answers
55 views

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

Assume $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $\mathbb{R}$ and such that $f(r)=0$ for every rational number $r$. Show that $f(x)=0, \forall x\in\mathbb{R}$ using the $\varepsilon-\delta$ ...
1
vote
2answers
26 views

$\mathbb{Q}$: Unique operation s.t. $1\star q=q$ and right distributivity hold?

Given $\mathbb{Q}$ and the usual addition $+$ on it, do we have unicity of a binary operation $\star$ such that \begin{align*} \tag{1}1\star q&=q\\ \tag{2}(q+r)\star s&=q\star s+r\star s ...
1
vote
5answers
45 views

Log laws proof using only rational exponents [closed]

For all real $a>0$ and rational $b>0$, Show that $\ln(a^b)=b\ln(a)$
1
vote
0answers
25 views

Smallest Rational Number Proof [duplicate]

Can anybody help me out with solving this mathematical proof? Prove the statement “There is no smallest rational number greater than 2” by contradiction. Contradiction: There is a smallest ...
0
votes
1answer
25 views

What is the value of $x$ in $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$?

The value of $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$ is rational for only one positive integer $x$ that is not a perfect square. What is the value of $x$? I tried $x=1$ and i get ...
2
votes
2answers
51 views

A group automorphism of real numbers that is not the identity

Is it possible to have a group automorphism of the additive group of the real numbers that fixes a subset of real numbers but is not the identity? The subset might be infinite. I'm thinking of using ...
-1
votes
1answer
220 views

There is no smallest rational number greater than 2

I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why. ...
1
vote
1answer
33 views

How to find the closest bounded rational approximation to a rational number?

Say I have a rational number $a/b$ and I want to find its closest rational approximation $x/y$ where $$x_- \leq x \leq x_+$$ $$y_- \leq y \leq y_+$$ for some constants $x_\pm$, $y_\pm$. How can I ...
2
votes
1answer
33 views

Number of ordered positive rationals (x,y,z) satisfying following conditions.

How many ordered triples $(x,y,z)$ of positive rational numbers satisfy the conditions: $x+y+z$, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, and $xyz$ are all integers.
2
votes
3answers
114 views

proving $ \sqrt 2 + \sqrt 3 $ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
1
vote
1answer
45 views

Exchangeability of union and intersection of open balls around all rational numbers in $[0,1]$

Let $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For $n,k \ge1$ set $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Is it true that $$ \bigcup_{n\ge1} \bigcap_{k\ge1} I_{n,k} = ...
3
votes
2answers
68 views

Is there a mathematical definition for the “divisibility” of rational numbers?

The term divisibility usually refers to integer numbers only. I want to define the divisibility of a rational number $q$ by an integer number $z$ as follows: $q$ is divisible by $z$ if and only if ...
3
votes
4answers
421 views

Integral of rationals

Define $f(x)$ as $$f(x)=\begin{cases}0,&\text{if }x\in \mathbb{Q}\\ 1,&\text{if }x\notin \mathbb{Q}\;. \end{cases}$$ Considering the fact that there is a countable infinity of rationals yet an ...
4
votes
3answers
195 views

Prove that the product of an irrational number and a rational number is irrational.

If $x$ is an irrational number and $r$ is a rational number then $xr$ is an irrational number. Proof. Suppose that $xr$ is a rational number. By defintion of a rational number $xr= m/n$ where ...
3
votes
2answers
67 views

Numerical polynomials by means of prime powers

Let $\mathbb{E}$ be the set of prime powers (except $1$). Let $f \in \mathbb{Q}[x]$ be a rational polynomial with $f(\mathbb{E}) \subseteq \mathbb{Z}$. Does it follow that $f$ is numerical, i.e. ...
0
votes
2answers
21 views

Proof about rational neighbors

Two rational numbers $\frac{a}{b}$ < $\frac{c}{d}$ will be called neighbors if $\frac{c}{d}$ - $\frac{a}{b}$ = $\frac{bc-ad}{bd}$ = $\frac{1}{bd}$. Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are ...
1
vote
2answers
85 views

How many distinct equivalence classes does this equivalence on rationals have?

Let $$A = \{ r\in \mathbb Q \mid \exists p\in \mathbb Z,\text{ and $q\in \mathbb Z$, with $p$ even and $q$ odd, and $r = p/q$} \}$$ For example, $A$ contains such $2/9, 16/(-34)$, and $4$. $A$ does ...
0
votes
0answers
41 views

In which way to prove that the set has the measure zero in R3?

I can understand this task in the way that we should prove that there are less numbers in the rational set compared to the numbers in the real set ? TO PROVE: The set A is described as follows: ...
0
votes
2answers
48 views

prove or disprove if a number is irrational

Prove or disprove : I'm pretty sure this isn't true yet i can't find a counter example. Thanks in advance !
5
votes
2answers
217 views

Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$

Prove : $$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$ I think the way to go here is to falsely assume that ...
1
vote
2answers
20 views

If you apply the Distributive Property to a Rational and an Irrational number, which will your solution be?

Say that "A" and "B" are Rational, and C is irrational, would the solution to "A(B+C)" be Rational or Irrational? An example for clarification would be wonderful.
1
vote
3answers
81 views

Why rational numbers are dense?

So the books says that rational numbers are dense, meaning that for every two rational numbers there is another rational number in between them. Is it actually true? Why? It feels to me that there ...
-2
votes
1answer
59 views

Prove that the equation $x^2=x$ has the same solutions in rational numbers as in integers

I was wondering if you could help me start in my discrete math homework. I'm asked to prove that A = B: $A =\{x \in \mathbb{Z}\mid x^2 = x\}$ and $B = \{x \in \mathbb{Q}\mid x^2 = x\}$ I'm having ...
0
votes
2answers
23 views

equivalence class of function, picking proper x

Defining R to be the relationship on real numbers given by xRy iff x-y is rational, I've been asked to find the equivalence class of $\sqrt2$. My instincts say that the equivalence class of $\sqrt2$ ...
8
votes
3answers
2k views

subtraction of two irrational numbers to get a rational [duplicate]

Say you have a number like $\pi$ or e. Is it possible to subtract another number from it and end up with a rational number? I mean I guess you could write an equation like $\pi-x=3$ But could there ...
4
votes
1answer
145 views

Prove that $x$ is rational iff $a=b=0$

So the question goes let $x = a\sqrt3 + b\sqrt 5$ where $a,b$ rational. Prove that $x$ is rational iff $a=b=0$. I think I can prove this but I'm not sure if my proof is correct or rigorous. Well ...
8
votes
2answers
956 views

Is there any basis transformation under which all irrational numbers are rationals and vice-versa?

For example, if you change the length of your "unit scale" or basis for numbers to $\sqrt{2}$, then you may represent all fractional multiples of $\sqrt{2}$ as "rational numbers" in the new basis ...
0
votes
1answer
61 views

determine the frontier

determine the frontier of the set R\Q (where R is the real numbers and Q is the rational numbers). I figured R\Q is the same as saying the real line minus all the rational numbers which would just ...
3
votes
2answers
112 views

Write $0.2154154\overline{154}$ as a fraction

Let $x = 0.2154154\overline{154}$ , I have to prove that it is a rational number just by writing it as a fraction with the proper steps. I note that the repeating part, $154$, is composed by 3 ...
1
vote
1answer
18 views

A question of rationality of integral powers

Given $a,b\in\mathbb{Z}$, is $x$ from $a^x=b$ ever rational? More specifically, is $x$ in $2^x=3$ irrational?
1
vote
4answers
60 views

Prove rational nums

For all real number x : R(x) -> there exist two integers k, l such that x = k/l. (i.e. x is a rational number) Prove/Disprove: For all real number x : R(x) -> R(x+1) My answer: Let x be a real ...
3
votes
2answers
97 views

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 $n\in\mathbb{N}$?
2
votes
3answers
48 views

Why proof about a rational on open interval (a,b) works…

For the case $0 < a < b$ which is what I'm interested in, there is a proof that there exists a rational number on the open interval which I've seen many times but I don't really understand it. ...