For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.

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3
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3answers
64 views

The supremum of rationals that are less than a given number is equal to that number

I have the following theorem to prove. Given a real number $a$, define the set $S$ such that $S = \{x \in \mathbb Q: x < a\}$. Show that $a = \sup S$. My attempt at a proof is as follows ...
0
votes
3answers
47 views

Correctness of proof that every positive rational with square $>2$ is an upper bound for those with square $<2$

I would like to know whether my proof makes sense or not, and if not where should it be corrected. Let $E=\{x \text{ is rational }: x>0 \text{ and } x^2<2\}.$ Claim: Every member of $F=\{x ...
12
votes
1answer
152 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
4
votes
1answer
146 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
4
votes
1answer
158 views

Linear independence over rationals

I am trying to figure out for what values of $n$, the numbers $\sin\left(\frac{2\pi k}{n}\right)$, for $k = 1,\dots,n-1$, are linearly independent over the rationals. Any thoughts on how I may want ...
2
votes
1answer
35 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
2
votes
1answer
61 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
2
votes
1answer
58 views

Analysis, Density of Rational Numbers

Suppose p/q and k/l are rational numbers with abs(p/q - k/l) < 1/ql. Prove p/q = k/l. Similarly, let p/q be a fixed rational number and suppose k/l is a rational number with 0 < abs(p/q - k/l) ...
2
votes
1answer
195 views

Counting Real Numbers

Forgive me if this is a novice question. I'm not a mathematics student, but I'm interested in mathematical philosophy. Georg Cantor made an argument that the set of rational numbers is countable by ...
1
vote
1answer
62 views

Proof by contradiction problem on rational numbers

Using proofs by contradiction, show that there is no smallest negative rational number and no largest positive rational number. Assume that there is a smallest negative rational number. Therefore, ...
1
vote
1answer
69 views

Difference between density and measure

In terms of definition, I know the difference between the two. However, the set of rationals $\mathbb{Q}$ has measure zero but is dense in $\mathbb{R}$. Whenever I envision this, I see a set of ...
1
vote
1answer
53 views

Prove $k!(e-s_k)$ is irrational.

Given that $\frac{p}{q} = e = 1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{k!} + \frac{e^{z}}{(k+1)!}$ for some $z$ in $[0,1]$ (using Taylor's theorem), and that $s_k = 1 + \frac{1}{1!} + ...
1
vote
1answer
73 views

Looking for name of theorem: “rational $\Leftrightarrow$ fractional part terminates or repeats”

I am looking for the name of the theorem that says that a number $x$ is rational if and only if its fractional part terminates or repeats (where "fractional part" refers to the representation of $x$ ...
0
votes
1answer
26 views

Given that $a>1$, show that the exponential function $a^x$ is increasing for $x\in\mathbb{Q}$

The assumption one can make here is that it is increasing for $x\in\mathbb{Z}$. I have tried to make a proof but I'm not sure if it is valid. Here it goes. Say $x,y\in\mathbb{Q}$. Then they can be ...
0
votes
1answer
69 views

For any $a \in \mathbb R$ and any $n \in \mathbb N^+$ there exists $q \in \mathbb Q$ such that $|a-q|< \frac{1}{n}$.

For any $a \in \mathbb R$ and any $n \in \mathbb N^+$ there exists $q \in \mathbb Q$ such that $|a-q|< \frac{1}{n}$. I think i can prove this is false, let $a=2,n=2,q=1/2$ so $|2-\frac{1}{2}|< ...
0
votes
1answer
167 views

Ratio and Proportion.

A girl went to market to buy brinjal,onion and coconut.........she gives Rs 2 and buys 40 brianjals Rs 1 and buys 01 onion Rs 5 ...
0
votes
1answer
68 views

Continuous variable defined over Rational numbers only?

Let $x(t)$ be a solution of some first order ODE, which is continuous over $t$. In this case, is the continuous $x(t)$ defined only over Rational numbers? what is the reason behind this? Please ...
16
votes
0answers
308 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Question : For every even $k\ge 4$, is the following $(\star)$ true? $$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb ...
6
votes
0answers
155 views

Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that : $0\le x\le 1$ $x \in \mathbb Q$ $f(x) \in \mathbb Q$ Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
5
votes
0answers
105 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
3
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0answers
48 views

Rational analysis

I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real ...
3
votes
0answers
131 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
3
votes
0answers
146 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
2
votes
0answers
40 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
2
votes
0answers
126 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
2
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0answers
42 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
2
votes
0answers
92 views

Lower bound for the length of continued fraction

Define $\mathscr L: \mathbb Q \mapsto \mathbb N$ as the minimal number of terms in the continued fraction of a rational number. Example: the continued fraction of $\frac{5}{8}$ is ...
2
votes
0answers
77 views

Two quartic polynomials to be made a square?

Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$, $$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$ ...
1
vote
0answers
25 views

How many rationals for a given $n \in \Bbb N \;\backslash \{1\}$?

Fix $n \in \Bbb N, n> 1$. Now choose a two digit base-$n$ number $ab $ say. There's $n^2$ choices for this. Consider the number $0.c_1 c_2 c_3 \ldots$ where the $c_i$ are defined recursively: ...
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vote
0answers
95 views

The topology generated by open intervals of rational numbers

Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$ Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of ...
1
vote
0answers
59 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
1
vote
0answers
48 views

Rational approximation bound for real numbers in (0,1)

I am working on a problem that related to rational approximation of real numbers. I am looking for a bound of the form: Given a positive real number, $\alpha \in (0,1)$, there exist positive ...
1
vote
0answers
93 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
1
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0answers
22 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
1
vote
0answers
60 views

“Rational grids” on manifolds.

Here is something which is bothering me a bit. You have rationals on a line. You can define a rational grid on R^n by taking points with all coordinates having rational values. Is there a ...
0
votes
0answers
12 views

Smallest survey size algorithm

I was inspired to think of this problem when I saw a commercial saying "97.1% of audiences loved this movie!". After fiddling around with some numbers, I realised that at least 34 people must have ...
0
votes
0answers
20 views

Verify if the following demonstration is correct

THEOREM: "There are $a, b$ irrationals, such that $a^b$ is rational." PROOF: " If $\sqrt{2} ^ \sqrt{2}$ is rational, $a = b = \sqrt{2}$, otherwise, $a = \sqrt{2} ^ \sqrt{2}$ and $b = \sqrt{2}$, so ...
0
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0answers
14 views

What is a class of equations? Which class would require the rational numbers to guarantee a solution?

What is a class of equations? Which class would require the rational numbers to guarantee a solution? The biggest problem I am having is that I am unsure what exactly a class of equations is. The ...
0
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0answers
30 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
0
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0answers
17 views

Question on rounding off rule

There is a specific rule(Banker's Rule I think) for rounding of numbers that end in 5. The rule is that we add 1 to the preceding digit of it's odd but keep it as it is if it's even. It's always ...
0
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0answers
18 views

Using LLL to get approximate rational representations of numbers

Does anyone understand how is the LLL algorithm implemented to obtained the values of $(x,y)$ for approximating $\pi$ in this portion of text?
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0answers
59 views

Related to $\pi$ and $\tau$ constants, are they transcendental, irrational, or rational numbers?

Below are three OEIS constant sequences and values. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. Decimal ...
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0answers
26 views

Adding a natural number to a normalized fraction

I am currently writing yet another rational number class where the fraction should always be normalized. When adding a natural number to a normalized fraction, it possible to get a non-normalized ...
0
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0answers
27 views

Closure of divisibility in denumerator, under sum of fractions

I have to prove that for a fixed positive integer n, the subset A of Q consisting of rationals with denumerator that divide n under addition, forms a group under addition. I just did that it's ...
0
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0answers
76 views

Limits as a representation of the Dirichlet function

I read that the Dirichlet function (1 if Rational, 0 else) can be written as: What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
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0answers
456 views

Mathematical problems having rational number solutions

Is there a well defined class of mathematical problems which produce only rational numbers as their solutions?
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0answers
19 views

Question on passing from rational to exponet

it's not a question itself, but I'd like to check if am I doing this passage from rational numbers to the exponent form right: From: $\sqrt{a\sqrt{a}}$ Evaluete to: $\sqrt{a*a^{\frac{1}{2}}}$ = ...