Mathematics Stack Exchange
Mathematics Stack Exchange

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.

A rational number is any number that can be expressed as the quotient or fraction $\frac pq$ of two integers, with the denominator $q$ not equal to zero. Since $q$ may be 1, every integer is a rational number. The set of all rational numbers is usually denoted by $\Bbb Q$; it was thus named in 1895 by Peano after quoziente, Italian for "quotient".

2333 questions
131
votes
9 answers

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 can anyone produce an explicit formula for such…
Alex Basson
  • 4,331
125
votes
13 answers

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 I will denote $f$, that $$\forall…
Praise Existence
  • 1,485
114
votes
24 answers

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 deal with irrational numbers, but only rational…
totalnoob
  • 1,263
104
votes
9 answers

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 expression, then: $$\begin{align} \frac12\div\frac4x &=…
Devansh Sharma
  • 1,145
103
votes
8 answers

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 then:) My questions are: Why is this so? What…
marty cohen
  • 110,450
76
votes
3 answers

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? This would be so easy if these points could be on…
Ahmed Amir
  • 1,043
74
votes
2 answers

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 $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$ For $r=p/q$…
mathlove
  • 151,597
71
votes
6 answers

"Gaps" or "holes" in rational number system

In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$. Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $p^2<2$, and $B$ consists of all positive…
Larry
  • 751
67
votes
2 answers

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 numbers from the list. Questions: What is…
jdods
  • 6,632
66
votes
12 answers

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 seems to be shown as equal to its opposite…
Daniel R. Collins
  • 9,177
63
votes
5 answers

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 a rational number? [I posted this only so that the…
MJD
  • 67,568
  • 43
  • 308
  • 617
55
votes
5 answers

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 points are not dense on every circle. How would one…
Hidaw
  • 1,031
52
votes
4 answers

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: $$ \gcd\left(\frac{13}{6}, \frac{3}{4} \right) = \frac{1}{12} $$ I can do…
stevenvh
  • 2,784
49
votes
1 answer

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 $x, y\in \mathbb{Q}$, does there exist $z \in \mathbb{Q}$…
Prahlad Vaidyanathan
  • 32,330
47
votes
5 answers

Is there a $f: \mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^{\infty} \frac{1}{n^2f(n)} \in \mathbb{Q}$?

Take by convention $0 \not \in \mathbb{N}$, and let $f: \mathbb{N} \to \mathbb{N}$. Define the real number $N(f)$ by $$N(f) = \sum_{n=1}^{\infty} \frac{1}{n^2f(n)}.$$ $N(f)$ is well-defined because, for example, each summand is a positive number…
Robin
  • 6,201
1
2 3
…
99 100