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".

2009 questions
43
votes
5 answers

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
syeh_106
  • 2,673
  • 19
  • 29
39
votes
2 answers

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 $x$ is some product of a fraction of $\pi$. Is…
Queequeg
  • 842
  • 1
  • 9
  • 14
37
votes
9 answers

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 as $\frac{p}{q}$, where $p$ and $q$ are integers…
jamaicanworm
  • 3,908
  • 10
  • 32
  • 50
37
votes
6 answers

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 rational number. So is the difference. As well as…
35
votes
5 answers

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 bijection $a$ from $\mathbb N$ onto $\mathbb Q$…
Taladris
  • 8,900
  • 3
  • 27
  • 50
32
votes
7 answers

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). But I never saw any axiomatic characterization of…
AlexE
  • 1,795
  • 13
  • 28
31
votes
4 answers

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 appreciated.
30
votes
4 answers

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 $1/3=0.3333...=0.(3)$ has a period of length 1. For example: how to…
Milingona Ana
  • 1,244
  • 2
  • 11
  • 21
29
votes
2 answers

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 inital solution was like this: let's take a…
29
votes
2 answers

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? (Note that of course $f \equiv 0$ is a solution…
26
votes
5 answers

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 in a regular discussion I have over this t-shirt of…
DoubleMalt
  • 363
  • 3
  • 5
25
votes
3 answers

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

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is irrational}\end{cases} $$ How is it called and why is it so? I'm really…
25
votes
2 answers

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 the sum will…
24
votes
4 answers

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 number can be written in terms, i.e., in a closed…
chtenb
  • 1,243
  • 1
  • 11
  • 20
24
votes
6 answers

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 make a distinction between these two types of…
mage
  • 359
  • 2
  • 7