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

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2answers
188 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
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
156 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
368 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
219 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
89 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
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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
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2answers
244 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
345 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
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 ...
8
votes
1answer
284 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
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?
1
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4answers
439 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
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1answer
75 views
18
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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? ...
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4answers
4k views

Is a Whole Number A Rational Number

Is a Whole Number part of A Rational Number or a whole number??
5
votes
2answers
149 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
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
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 ...
12
votes
1answer
185 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
843 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
163 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
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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" ...
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0answers
338 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
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
520 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
232 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
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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 ...
0
votes
1answer
148 views

Drawing a triangle in a unit circle

This is a question that I derived for a long time ago. It asks if we draw a triangle in a unit circle does all arc lengths $(\alpha ,\beta ,\theta)$ and sides of triangle $(a,b,c)$ can be rational ...
2
votes
1answer
73 views

True or false statements

Two of the following statements are true and one is false a) For all rational numbers $q$, there exists an integer $n$ so that $q+n=271$. b) For all integers $n$, there exists a rational ...
23
votes
6answers
3k views

Why must we distinguish between rational and irrational numbers?

The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians ...
4
votes
3answers
545 views

Are all integer fractions rational?

Any repeating decimal can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers. But is the reverse true. Will any fraction $\frac{a}{b}$ where $a$ and $b$ are integers produce a ...
55
votes
5answers
12k views

Why is 987654321/123456789 = 8.0000000729?

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since ...
0
votes
3answers
182 views

I'm just curious, what exactly is $\mathbb{R}\setminus\mathbb{Q}$? [duplicate]

What exactly is $\mathbb{R}\setminus\mathbb{Q}$? How many different kinds of things live in this place? For $n>1$ how does $$ q_1x_1+\cdots+q_nx_n=p $$ have a solution for $q_i,p\in \mathbb{Q}$ ...
2
votes
4answers
72 views

What would be the value of $a$ and $b$ in following rational expression?

If $(5 + 2\sqrt{3})/(7 + \sqrt{3}) = (a - \sqrt{3b})$, How do I find the value of $a$ and $b$ where $a$ and $b$ are rational numbers?
1
vote
2answers
165 views

How do I evaluate the following expression?

How to evaluate the following expression: $\displaystyle \frac{1}{\sqrt{2}+1}+ \frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}} +\cdots +\frac{1}{\sqrt{9}+\sqrt{8}}$
1
vote
3answers
1k views

real numbers a vector space over rational numbers? [duplicate]

Let $V$ be set of real numbers and $K$ the field of rational numbers. Is $V$ a vector space over $K$, with ordinary addition of real numbers and multiplication by rational numbers?
4
votes
4answers
257 views

Definition(s) of rational numbers

The definitions of rational numbers are somewhat confusing for me. The definition of rational numbers on wikipedia and most other sites is: In mathematics, a rational number is any number that ...