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

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0answers
41 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 ...
4
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2answers
346 views

Proving the rationals are dense in R

I know this is a common proof. I'm following Rudin's proof and I'm following everything except for one step. Suppose $x, y \in \Bbb R$ and $x < y$. Then there exists an $n \in \Bbb N$ such that ...
3
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1answer
269 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 ...
-2
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1answer
156 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
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1answer
79 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
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2answers
35 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 $n$?
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3answers
127 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
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1answer
46 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
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3answers
90 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
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3answers
295 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
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1answer
41 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 ...
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1answer
181 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 ...
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2answers
178 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
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1answer
90 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
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1answer
135 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
82 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 ...
0
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3answers
137 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. all this means that we can find out which digit exist in any position of rational number. but what about ...
4
votes
1answer
149 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
272 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...
4
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4answers
185 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
76 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 ...
2
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1answer
146 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 ...
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2answers
123 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
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2answers
218 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
287 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
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0answers
87 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
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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
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1answer
143 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
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1answer
251 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
108 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 ...
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3answers
1k 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?
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4answers
273 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
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2answers
1k 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 ...
15
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7answers
3k 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 ...
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1answer
69 views
18
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1answer
382 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
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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? ...
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4answers
1k views

Is a Whole Number A Rational Number

Is a Whole Number part of A Rational Number or a whole number??
5
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2answers
139 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) ...
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2answers
76 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
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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
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1answer
131 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 ...
11
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1answer
179 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 ...
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1answer
672 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
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1answer
135 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 ...
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1answer
58 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 ...
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2answers
110 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
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1answer
61 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" ...
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0answers
324 views

Rationals and irrationals on the real number line [closed]

Could you prove that there is a rational between every irrational on R?
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2answers
61 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 $f(n) = n +k; ~~ k \in \mathbb{Z}$ ...