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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1answer
298 views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental ...
-1
votes
1answer
202 views

Is this a rational or irrational number?

It is given that $$z=\sqrt\frac{\sqrt{3x+1}}{\sqrt{3x-1}}$$ How does one find whether $z$ is a rational or irrational number?
3
votes
1answer
91 views

Is the infinite table argument for the countability of Q unsound?

The first "proof" I learned for why the rationals are countably infinite relied on arranging the rational numbers in a two-dimensional array and using the well-known traversal shown below to construct ...
1
vote
2answers
38 views

A necessary and sufficient on the co-efficients of a quadratic to give an integer

Le $f(n) := an^2 + bn + c$ for all integers $n$, where $a$, $b$, $c$ are rational. What are the necessary and sufficient conditions on $a$, $b$, and $c$ such that $f(n)$ be an integer for all $n$?
0
votes
3answers
140 views

Finding Rational numbers

Please help with the following question: Find rational numbers a and b such that: $$\left(7 + 5\sqrt2\right)^{\frac13} = a + b \sqrt2$$ Thank you
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) ...
0
votes
3answers
112 views

If a triangle has rational coordinates, does it have rational area?

Basically, the topic says it all. If a triangle has rational coordinates (say, in $\mathbb Q^2$), must it have rational area? I realize the side-lengths are usually irrational; that's fine. Heron's ...
2
votes
3answers
318 views

What condition that if imposed on $\alpha$ imply that $\cos^{-1} \alpha$ is a rational multiple of $\pi$?

It is well known that if $x$ is a rational multiple of $\pi$ then $\cos x$, $\sin x$, etc, are algebraic numbers. What is known about the inverse problem? That is, is there a set of conditions that ...
0
votes
1answer
49 views

Linear order, Löwenheim-Skolem

LO is the theory of linear ordering. Suppose T a theory, which contains at least the symbol {<} in her language and T $\vDash$ LO. Suppose T has an infinite model. Prove that there's a model M for ...
1
vote
1answer
260 views

Hausdorff dimension of the set of rational numbers within a certain interval?

Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ...
-2
votes
2answers
199 views

Generalization: $x=\text{sup}\{q\in \mathbb{Q}:q<x\}$

How do I prove that $x=\text{sup}\{q\in \mathbb{Q}:q<x\}$? Provided that $x\in\mathbb{R}$...
1
vote
1answer
91 views

Least Upper Bound: $\text{sup}\{r\in\mathbb{Q}:r<5\}$

What is $\text{sup}\{r\in\mathbb{Q}:r<5\}$ and why? I would suspect that no least upper bound is possible here, and hence no maximum is attainable either...
5
votes
1answer
138 views

What are the positive rational solutions of $x^{(x+y)} = (x+y)^y$?

I saw this problem in the Problem-Solving through Problems book by Larson (# 3.3.25b). I got to here: $$x \log(x) = y\log\left(1+ \frac yx\right)$$ But I can't seem to find a way to reduce this ...
3
votes
1answer
91 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
4
votes
1answer
159 views

Algebraic structure of a set of Egyptian fractions of a positive rational?

It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions). I am struggling to understand in a formal way, ...
3
votes
5answers
380 views

Question on compact metric space.

Is the set of rationals in $[0,1]$ compact? I seems like every open covering should have a finite sub-covering, yet I have read that compact metric spaces are complete...
5
votes
4answers
220 views

What is a suitable name for numbers like $a + b\sqrt{c}$

The motivation for this is to find a succinct name for a data type in a Python module. Suppose I choose an integer $c$ and I want to talk about the set of numbers of the form $a + b\sqrt{c}$, where ...
-2
votes
1answer
92 views

set of terminating decimals

Let $T\subset\mathbb Q$ be the set of all positive rational numbers that can be represented by a terminating decimal (in base 10), that is, a decimal whose tail consists of an infinite sequence of ...
1
vote
2answers
152 views

How to solve a ratio question

Studying for the GRE. In the GRE guide, it says that If the ratio is $2x:5y$, and this equals the ratio $3:4$, what is the ratio of $x:y$? I tried cross multiplying but I don't get the answer. ...
2
votes
2answers
248 views

Ratio GRE question

Cashews cost 4.75 per pound and hazelnuts cost 4.50 per pound. What is larger, the number of pounds of cashews in a mixture of cashews and hazelnuts that costs $5.50 per pound, or 1.25? ...
4
votes
5answers
352 views

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: ...
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 ...
0
votes
2answers
59 views

What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$?

Let $s$ be complex and $\zeta(s)$ the Riemann Zeta function. What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$ ? I want a series expansion such that ...
0
votes
1answer
168 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 ...
8
votes
1answer
288 views

Predicting the number of decimal digits needed to express a rational number

The number $1/6$ can be expressed with only two digits (and a repeat sign denoted as $^\overline{}$), $$ \frac{1}{6} = \,.1\overline{6}$$ Meanwhile, it takes 49 digits to express the number $1/221$, ...
4
votes
1answer
110 views

A sequence of rationals whose continued square roots are also rational

For a given sequence $A_n=\{a_1,a_2,\cdots\}$, let $$b_1=\sqrt{a_1},b_2=\sqrt{a_1+\sqrt{a_2}},\cdots,b_k=\sqrt{a_1+\sqrt{a_2+\sqrt{\cdots+\sqrt{a_k}}}},\cdots,b_0=\lim_{n\to \infty}b_n.$$ Do there ...
1
vote
3answers
2k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
1
vote
4answers
461 views

How can I expand mathematical induction to rational numbers?

I know mathematical induction can be used to prove that a statements is true for all natural numbers (or those belonging to a certain subset of N). However, it is pretty obvious, unless I'm terribly ...
21
votes
2answers
2k views

Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
16
votes
7answers
5k views

What rational numbers have rational square roots?

All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$). My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer ...
1
vote
1answer
75 views
18
votes
1answer
419 views

How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?

Open problem in Geometry/Number Theory. The real question here is: Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational? The ...
12
votes
5answers
1k views

Are there infinitely many rational outputs for sin(x) and cos(x)?

I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? ...
1
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4answers
5k views

Is a Whole Number A Rational Number

Is a Whole Number part of A Rational Number or a whole number??
5
votes
2answers
151 views

Problem from Hardy's _Pure Mathematics_

If $a$, $b$, $x$, $y$ are rational numbers such that $$(ay-bx)^2+4(a-x)(b-y) = 0 $$ then either (i) $x = a, y = b$ or (ii) $1-ab$ and $1-xy$ are squares of rational numbers. (Math. Trip. 1903) ...
1
vote
2answers
78 views

How would I compute a fractional exponent, such as $\bigl(\frac{9}{16}\bigr)^{3/2}$?

How would I determine $\left(\frac{9}{16}\right)^{\frac{3}{2}}$? I've never encountered an exponent that was a fraction, or a least never without a calculator. What steps would I take to simplify ...
11
votes
6answers
2k views

Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that ...
4
votes
1answer
159 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 ...
12
votes
1answer
186 views

Is there a math field that studies something like this?

I was having a blurry thinking of differences about rational and irrational numbers, then I had the idea of ploting them in a specific way: $$\frac{1}{2}=0.5$$ Getting that value, I've thought about ...
1
vote
1answer
847 views

Converting decimal ratio to a percentage

Note before; I have searched the site and can't find an answer to this question, so that I wouldn't make a duplicate. I couldn't find the answer already so I either didn't search properly, or this ...
2
votes
1answer
164 views

Show that a rational number has no good rational approximations

This is homework question. The teacher proved that if $a$ is irrational, there are infinitely many rational numbers $\frac{x}{y}$ such that $|\frac{x}{y} - a|<\frac{1}{y^2}$. What we need to ...
1
vote
1answer
65 views

How well do we need to do in the remaining months to meet an annual percentage goal?

I am required to have a employment team meet a yearly percentage rate of 50% for a process. The team is currently short of the goal with a rate of 43%. My boss wants to know what percentage is ...
6
votes
2answers
118 views

Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?

Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$? Club challenge problem. I don't think it's possible to do with only high ...
2
votes
1answer
65 views

Order type of the real algebraic numbers

As a countable, everywhere-dense, totally ordered set without minimal or maximal elements, $\Bbb{A}$, the set of real algebraic numbers, must be order isomorphic to $\Bbb{Q}$. I'm wondering how "nice" ...
-3
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0answers
339 views

Rationals and irrationals on the real number line [closed]

Could you prove that there is a rational between every irrational on R?
1
vote
2answers
63 views

Order automorphisms of rationals

Is it right that all order automorphisms of rational numbers have form $~~f(r) = ar+b; ~~ a,b\in \mathbb{Q}; ~~ a>0$? As i know all order automorphisms have form $f(n) = n +k; ~~ k \in \mathbb{Z}$ ...
4
votes
4answers
527 views

On comparing fractions , fraction with smaller difference between numerator and denominator is greater than the other

A text book proposed that "when comparing fractions ,if the compared fractions's are such that numerator is smaller than denominator ,then fraction with more difference(absolute) between numerator ...
7
votes
2answers
236 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
1
vote
2answers
116 views

Which sets of positive rationals are closed under addition?

This question evolved because I was interested in generalizing power series so the exponents were rational numbers instead of integers, i.e., $\sum_{i=1}^{\infty} a_n x^{r_n}$, with the $a_i$ real and ...
2
votes
2answers
50 views

which parameters always make this rational equation evenly divisible?

Hi guys I have the following equation: $$x = \dfrac{a + b \times c - b}{c}$$ This is what I know about each variable: $$a \ge 64$$ $$b \ge 0$$ $$8 \le c \le a$$ My questions is there a concise way ...