Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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70
votes
24answers
13k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never ...
61
votes
5answers
15k 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 ...
41
votes
1answer
734 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that ...
25
votes
5answers
1k views

Is the square root of -1 rational?

This is not a deep question, but if there is a definite answer then here is the place where I will find it. Is it justified to say that $i =\sqrt{-1}$ is rational? The origin of this question lies ...
23
votes
6answers
4k 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 ...
22
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 ...
22
votes
6answers
1k views

Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
22
votes
2answers
516 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
21
votes
4answers
2k views

Can every irrational number be written in terms of finitely many rational numbers?

Consider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...
21
votes
3answers
2k views

GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ ...
18
votes
2answers
706 views

Order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$

Do you have a simple example of order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$? On $\mathbb Q$, I use the usual order and on $\mathbb Q\times\mathbb Q$, I define the ...
18
votes
1answer
471 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 ...
18
votes
0answers
384 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 ...
17
votes
8answers
6k views

Proof that every repeating decimal is rational

Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ...
17
votes
7answers
7k 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 ...
17
votes
4answers
518 views

What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
16
votes
4answers
7k views

Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
15
votes
2answers
451 views

Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
14
votes
1answer
295 views

Why is $x^3-5x$ injective on the rationals?

I've found the statement on the internet that the polynomial $x^3-5x$ is injective on the rational numbers, but without any comments on how to prove it. I think it means it must be easy, but I don't ...
14
votes
2answers
345 views

Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$

Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ? $$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
13
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? ...
13
votes
6answers
830 views

Is there a function that gives the same result for a number and its reciprocal?

Is there a (non-piecewise, non-trivial) function where $f(x) = f(\frac{1}{x})$? Why? It would be nice to compare ratios without worrying about the ordering of numerator and denominator. For example, ...
13
votes
2answers
2k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
13
votes
1answer
161 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 ...
12
votes
3answers
357 views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is ...
12
votes
1answer
191 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 ...
11
votes
4answers
1k views

Are there any bases which represent all rationals in a finite number of digits?

In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in ...
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 ...
11
votes
3answers
219 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
11
votes
1answer
137 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
11
votes
3answers
257 views

Prove that x is rational

Let $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the ...
11
votes
1answer
116 views

Combinatorial question about sets of rational numbers

The following question came up in my research. Since lots of clever people post here, I thought I'd ask it. Recall that the group ring of a group $G$ is the abelian group $\mathbb{Z}[G]$ consisting ...
10
votes
7answers
508 views

When is an Integer a Rational Number, and are All Ratios Rational, Even $\frac{\sqrt{7}}{2}$?

$$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$ In other words, a rational number is a number that can be written as one integer over another. ...
10
votes
4answers
873 views

How do I rewrite -100+1/2 as the mixed number -99 1/2?

This has been bugging me for some time now, so I ask you to try to help me realize what is going on here. I just can't get my brain around this. I have a proper fraction and a negative integer. The ...
10
votes
4answers
475 views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
9
votes
4answers
1k views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
9
votes
3answers
249 views

Half the rationals?

Let $\mathbb{Q}[n]$ be the set of rational numbers with denominator $\le n$ and for any $X\subseteq \mathbb{Q}$, let $X[n]=X\cap \mathbb{Q}[n]$. Is there a set of rational numbers, X, such that for ...
9
votes
3answers
1k views

If $ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x $, then $f(x)=x$

Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property: $$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$ Prove that $ f(x) = ...
8
votes
4answers
520 views

How to obtain all the rational numbers without repetitions?

Some days ago I've seen Cantor's diagonal argument, and it presented a table similar to the following one: $$\begin{matrix} ...
8
votes
3answers
463 views

“Length” of rationals in an interval

For $x \in \mathbb{R}$, define $r(x)$ as follows: $$ r(x)= \begin{cases} 1 &\text{if $x$ is rational},\\ 0 &\text{if $x$ is irrational}. \end{cases} $$ Q. What is $\int_0^1 r(x) dx$ ? I ...
8
votes
1answer
317 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 $1/221$, ...
8
votes
6answers
3k views

Infinite number of rationals between any two reals.

Let $a$ and $b$ be reals with $a<b$. Show that there are infinitely many rationals $x$ such that $a<x<b$. My plan of action was to assume that $x$ is the smallest such rational and find ...
8
votes
2answers
858 views

Is there a sequence that contains every rational number once, but with the “simplest” fractions first?

The Calkin-Wilf sequence contains every positive rational number exactly once: 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, …. I'd consider 5/1 to be a "simpler" ratio than ...
8
votes
2answers
1k views

How can we find and categorize the subgroups of R?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
8
votes
4answers
391 views

Equation with an infinite number of solutions

I have the following equation: $x^3+y^3=6xy$. I have two questions: 1. Does it have an infinite number of rational solutions? 2. Which are the solutions over the integers?($ x=3 $ and $ y=3 $ is one) ...
8
votes
2answers
216 views

Is this graph based on rationals familiar?

Has anyone come across a graph like this? The black circles represent rationals in $(0,1)$ and their heights are roughly proportional to the reciprocal of the square of their lowest terms ...
8
votes
2answers
120 views

Given dividend and divisor, can we know the length of nonrepeating part and repeating part?

$13/92=0.14\overline{1304347826086956521739}$ In this example, the length of nonrepeating part is $3$. The length of repeating part (repeating period) is $21$. I collected some properties related to ...
8
votes
2answers
227 views

Conditions that $\sqrt{a+\sqrt{b}} + \sqrt{a-\sqrt{b}}$ is rational

Motivation I am working on one of the questions from Hardy's Course of Pure Mathematics and was wondering if I could get some assistance on where to go next in my proof. I have attempted rearranging ...
8
votes
1answer
41 views

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+\text{ such that } a^2-b=k^2 $?

This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: $$\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb ...
7
votes
7answers
582 views

Is the number 0.2343434343434.. rational? [duplicate]

Consider the following number: $$x=0.23434343434\dots$$ My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...