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

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8
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1answer
282 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
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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?
1
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4answers
373 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 ...
15
votes
7answers
4k 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
415 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
3k 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
147 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
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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
votes
1answer
152 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
818 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
162 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
64 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
334 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
505 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
226 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
114 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
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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
146 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
506 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 ...
54
votes
5answers
11k 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
181 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
65 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?
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2answers
151 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
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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
253 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 ...
7
votes
1answer
246 views

Can someone clarify Example I.I.2 from Hardy's Course of Pure Mathematics?

"If $\lambda, m,$ and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^2 − n^2), 2\lambda mn$, and $\lambda(m^2 + n^2)$ are positive rational numbers. Hence show how to determine any ...
2
votes
2answers
473 views

Proof f(x) is continuous given $x$ rational and irrational.

How can I resolve the task below: Given $f(x)= \begin{cases} x, &x\in \mathbb{Q}\text{ }\\ 1-x, &x\notin \mathbb{Q}\text{ (irrational)} \end{cases}$, $0 \leq x \leq 1$. Show $f(x)$ is ...
0
votes
1answer
66 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 ...
1
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2answers
215 views

Can all possible angles on a rational triangle be represented as a rational multiplied by Pi?

The reason I ask: I was wondering if it was possible to find the angle of a rational triangle by only using the lengths of its sides and knowledge of $\pi$ (that is, no inverse trig functions). So, ...
1
vote
1answer
87 views

Can you produce a number like 1.01010101… by just addition and subtraction?

I'm working on a program in C# where a Decimal variable can hold negative and positive values including 0 and those values can only change by addition and subtraction. I have a conditional where if ...
3
votes
2answers
139 views
1
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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 $x$ ...
3
votes
1answer
197 views

length of period

Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
6
votes
1answer
119 views

Defining $\Bbb{Q}$ without the axiom of infinity

(TL;DR version: I want a meaningful definition of $\Bbb{Q}$ without $\sf{Inf}$.) In the "conventional" construction of the rationals, we define $\Bbb{Q}$ as follows: $\omega$ is the first limit ...
2
votes
2answers
127 views

Every 10 years the population of a city is five-fourths of what it was 10 years before.

I am working through Serge Lang's "Basic mathematics", currently on chapter 1, Section 5 question 21. The part that troubles me is, when I asked someone I know how to solve this, they suggested ...
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
3answers
188 views

Show that there is no rational number $r=m/n$ such that $r^3=3$ [duplicate]

How do I solve this by prime factorization? I came across a similar problem on MSE just recently, but I can't find it and I thoroughly searched for it. If anyone can find it, please post it in the ...
1
vote
2answers
94 views

Approximating Euclidean geometry, restricted to $\mathbb{Q}$

I'm having trouble putting this into a fully coherent question, so I'll give the broad question, then a few bullet points to give you a better idea of what I'm asking. I'm looking for a line of ...
1
vote
2answers
53 views

Understanding the boundary of a set

I try to understand the boundary of a set. I know the definition (Let $A \subseteq \mathbb{R} $: P is a point of the boundary, if for every small $\epsilon \in \mathbb{R}, \epsilon>0$ are points ...