The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [rational-numbers]

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.

Filter by
Sorted by
Tagged with
103
votes
24answers
17k 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 ...
101
votes
9answers
8k views

Division by $0$ and its restrictions

Consider the following expression: $$\frac{1}{2} \div \frac{4}{x}$$ Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$. But if we rearrange the ...
94
votes
12answers
14k views

Why can't calculus be done on the rational numbers?

I was once told that one must have a notion of the reals to take limits of functions. I don't see how this is true since it can be written for all functions from the rationals to the rationals, which ...
81
votes
7answers
28k views

Why is $\frac{987654321}{123456789} = 8.0000000729?!$

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 ...
69
votes
2answers
2k 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 $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{...
68
votes
6answers
36k views

Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
64
votes
2answers
3k views

What is the average rational number?

Let $Q=\mathbb Q \cap(0,1)= \{r_1,r_2,\ldots\}$ be the rational numbers in $(0,1)$ listed out so we can count them. Define $x_n=\frac{1}{n}\sum_{k=1}^nr_n$ to be the average of the first $n$ rational ...
55
votes
3answers
4k views

Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?

The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? ...
46
votes
4answers
4k views

Are rational points dense on every circle in the coordinate plane?

Are rational points dense on every circle in the coordinate plane? First thing first I know that rational points are dense on the unit circle. However, I am not so sure how to show that rational ...
43
votes
1answer
1k 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, y\...
42
votes
12answers
5k views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
41
votes
5answers
5k views

Is it possible to construct a strictly monotonic sequence of all rational numbers?

I know that the set of all rational numbers is countable, and can be enumerated by a sequence, say $\{a_n\}$. But can we construct a monotonic $\{a_n\}_{n=1}^{\infty}$, e.g. with $a_k<a_{k+1}$? ...
40
votes
7answers
5k views

Why does Rudin say “the rational number system is inadequate as a field”?

In the INTRODUCTION of chapter 1 of Baby Rudin, he says The rational number system is inadequate for many purposes, both as a field and as an ordered set. Addition and multiplication of rational ...
39
votes
10answers
6k views

Can sum of a rational number and its reciprocal be an integer?

Can sum of a rational number and its reciprocal be an integer? My brother asked me this question and I was unable to answer it. The only trivial solutions which I could think of are $1$ and $-1$. ...
39
votes
3answers
7k 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: $$ \gcd\left(\frac{13}{...
35
votes
5answers
40k 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 ...
35
votes
6answers
3k views

Test of being a rational number for $(1-\frac13+\frac15-\frac17+\cdots)/(1+\frac14+\frac19+\frac1{16}+\cdots)$

Is the following expression a rational number? $$\frac{1-\dfrac13+\dfrac15-\dfrac17+\cdots}{1+\dfrac14+\dfrac19+\dfrac1{16}+\cdots}$$ My thoughts: The sum and product of two rational numbers is a ...
34
votes
5answers
2k views

Total distance traveled when visiting all rational numbers

My students found an old problem given in my school in 2007 (probably from a Honor Calculus class) and had been trying to solve for some time. Here is the problem: Prove or disprove: there exists a ...
32
votes
9answers
25k 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 ...
28
votes
2answers
8k 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 ...
28
votes
2answers
9k 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 $x$ ...
28
votes
6answers
3k 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). ...
28
votes
2answers
2k views

Solutions to $f'=f$ over the rationals

The problem is as follows: Let $f: \mathbb{Q} \to \mathbb{Q}$ and consider the differential equation $f' = f$, with the standard definition of differentiation. Do there exist any nontrivial solutions?...
25
votes
5answers
6k 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 ...
25
votes
2answers
665 views

Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?

Obviously: $$\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=0.1111\dots=\frac{1}{9}$$ is a rational number. Now, if we make terms with demoninators in the form: $$q_n=\sum_{k=0}^{n} 10^k$$ Then ...
24
votes
4answers
3k 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 ...
23
votes
6answers
7k 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 ...
23
votes
1answer
568 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 $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$.}\...
21
votes
3answers
929 views

“Least trivial” function preserving rationality

Is there a "non-trivial" function $f(x,y)$ such that $$f(x,y) \in \mathbb{Q} \iff x,y\in \mathbb{Q}?$$ An example of a "trivial" function would be $$f(x,y) = \begin{cases} 0 & x,y\in \mathbb{...
20
votes
1answer
1k views

Will every rational number eventually be in this set?

Let $A_0=\{0\}$. For every $n\ge 0$, let $B_n=\bigcup_{k\ge 0}A_n+k$, and let $A_{n+1}=f(B_n)$, where $f:[0,\infty)\to[0,1):x\mapsto\frac{x}{x+1}$. Let $q\in\Bbb Q\cap [0,1)$. Can we prove that $q\in ...
20
votes
1answer
780 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 ...
19
votes
9answers
1k views

Function that maps the “pureness” of a rational number?

By pureness I mean a number that shows how much the numerator and denominator are small. E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is ...
19
votes
9answers
4k views

Distributive property on fractions

I'm in seventh grade and my teacher wasn't able to explain this to me. why is $\frac{1}{a+b}$ not equal to $\frac 1b +\frac 1a$? I'm sorry if this is obvious. EDIT: thank you to everyone who ...
19
votes
4answers
1k 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 ...
19
votes
2answers
2k 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 ...
19
votes
1answer
439 views

An interesting way of expressing any real number using the harmonic series.

I recently saw the identity $$ \frac{1}{1} - {1 \over 2} +{1 \over 3} - {1 \over 4} + {1 \over 5} - {1 \over 6} \dotsb = \log(2) $$ which I found rather interesting. I was intrigued by the way a ...
18
votes
4answers
21k views

How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals?

I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly ...
18
votes
3answers
2k 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 irrational}\end{...
18
votes
1answer
344 views

Do all rational numbers repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, $\frac{1}{3}$ in base 10 is $0.33333...$, in ...
17
votes
7answers
2k views

How many sequences of rational numbers converging to 1 are there?

I have a problem with this exercise: How many sequences of rational numbers converging to 1 are there? I know that the number of all sequences of rational numbers is $\mathfrak{c}$. But here we ...
17
votes
2answers
321 views

Orbits of vectors under the action of $\mathrm{GL}_n(\mathbb Q)$

Context. While working on a larger proof, I would love to have the following lemma, but I can't even decide if it's true or not. The question. We consider the action of $\mathrm {GL}_n(\mathbb Q)$ ...
17
votes
1answer
519 views

Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational?

I'm having difficulty with the following problem: Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational, for $0<x<1$? I've tried proving by ...
16
votes
4answers
3k 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 ...
16
votes
4answers
7k views

Show that the curve $x^2+y^2-3=0$ has no rational points

Show that the curve $x^2+y^2-3=0$ has no rational points, that is, no points $(x,y)$ with $x,y\in \mathbb{Q}$. Update: Thanks for all the input! I've done my best to incorporate your suggestions and ...
16
votes
2answers
681 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 $ABC$ ...
16
votes
1answer
244 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 ...
15
votes
4answers
397 views

Show that the numerator of $1+\frac12 +\frac13 +\cdots +\frac1{96}$ is divisible by $97$

Let $\frac{x}{y}=1+\frac12 +\frac13 +\cdots +\frac1{96}$ where $\text{gcd}(x,y)=1$. Show that $97\;|\;x$. I try adding these together, but seems very long boring and don't think it is the right way ...
15
votes
4answers
11k views

Length of period of decimal expansion of a fraction

Each rational number (fraction) can be written as a decimal periodic number. Is there a method or hint to derive the length of the period of an arbitrary fraction? For example $1/3=0.3333...=0.(3)$ ...
15
votes
1answer
384 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 ...
15
votes
3answers
3k views

How can we find and categorize the subgroups of $\mathbb{R}$?

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