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
0
votes
0 answers

Why $\mathbb{Q}(\sqrt{\alpha})\neq \mathbb{Q}(\sqrt{\beta})$ if $\sqrt{\alpha \beta} \notin \mathbb{Q}$?

I was wondering why if we take $\alpha, \beta \in \mathbb{Q}$ such that $\sqrt{\alpha \beta} \notin \mathbb{Q}$, then we have that the field extensions $\mathbb{Q}(\sqrt{\alpha})$ and $\mathbb{Q}(\sqrt{\beta})$ are not the same. And also, it is an…
0
votes
3 answers

Does there exist a positive integer $n$ such that $P^n = I$, where $P$ is a rotation matrix?

Does there exist a positive integer $n$ such that $P^n = I$, where $P$ is a $2 \times 2$ rotation matrix for a rotation of the plane by an angle $2\pi q$ radians? if $q$ is a rational number if $q$ is an irrational number How should I go about…
0
votes
4 answers

Prove that $\sqrt[3]{45+29\sqrt{2}} +\sqrt[3]{45-29\sqrt{2}} $ is rational.

Prove that $\sqrt[3]{45+29\sqrt{2}} +\sqrt[3]{45-29\sqrt{2}} $ is rational. So I assumed, $\sqrt[3]{45+29\sqrt{2}} +\sqrt[3]{45-29\sqrt{2}} = x$ So we have to prove $x$ is rational, I did so and cubed $x$, so I got: $45+29\sqrt{2}…
user907745
0
votes
0 answers

Proving a Dedekind Cut for Rational Numbers

For Dedekind cuts α, β > $0^*$, I need show that α ⊘ β := {p ∈ Q | p · s < r for some r ∈ α, s ∈ $β^c$ such that r,s > 0 and s not the lowest element. I know the properties of Dedekind Cuts are: α is non-empty (and a proper subset); if p ∈ α and q…
Zeta_Y
  • 11
  • 2
0
votes
1 answer

Proof: Cubic Polynomial with Rational Coefficients and Roots

Can some explain this Proposition and proof, please? Maybe breakdown it down for me? Information for the proposition and proof: Let us call polynomials with integer (respectively, rational) coefficients and roots Z-polynomials (respectively,…
0
votes
0 answers

Theorem: Cubic Polynomials with Rational Coefficients and Roots of h' and h''

I need help explaining this theorem and example provided below. $f (x) = (x^2 + bx + c)(x − d)$ $= x^3 + (2u − 3d)x^2 + (u^2 − 4ud + 3d^2 − v^2 )x− d(u − d − v)(u − d + v)$. (2) For brevity, let us call polynomials with integer (respectively,…
0
votes
1 answer

What is a more rigorous definition for powers of rational numbers?

My current understanding of $x^\frac{m}{n}$ is that it is equal to $\sqrt[n]{x^m}$. Now technically, (-1)$^\frac{2}{4}$ is equal to $\sqrt[4]{(-1)^2}$=1. As $\frac{2}{4}$=$\frac{1}{2}$, (-1)$^\frac{1}{2}$ is technically equal to 1. However, the…
Aqeel
  • 57
  • 2
0
votes
1 answer

Linear independence of square root of primes without abstract algebra

I recently found a linear algebra question from the very beginning of my first semester. Without any knowledge of abstract algebra, we were supposed to prove that if $p$ and $q$ are distinct primes, the numbers $1$, $\sqrt{p}$ and $\sqrt{q}$ are…
Meowdog
  • 4,634
  • 4
  • 18
0
votes
1 answer

If $\alpha\in(0,\pi/4)$ and $\tan(\alpha)$ is rational, then $\sqrt{2\tan(2\alpha)}$ is irrational

Let $ \alpha\in(0,\frac{\pi}{4})$. prove that $$\tan(\alpha) \in \Bbb Q \;\implies \;\sqrt{2\tan(2\alpha)}\notin \Bbb Q$$ I tried to prove the contrapositive, but it is still not obvious. Any help will be appreciated.
hamam_Abdallah
  • 1
  • 4
  • 24
  • 43
0
votes
6 answers

Are there any known infinite series of rational terms that are just irrational (not transcendental)?

I have probably encountered hundreds of infinite series where each term is rational. In each case (as far as I can remember), the value of the infinite series was either rational or transcendental. For example, some simple cases…
0
votes
0 answers

Unique ratios of integers

If $m$ is an integer between $0$ and $M$ inclusive, and $n$ an integer between $0$ and $N$ inclusive, there are $(M+1)(N+1)$ non-unique values of $m/n$, including the undetermined form $0/0$ and infinity. For instance, if $M=N=4$, then $2=2/1=4/2$…
Mister Mak
  • 171
  • 4
0
votes
0 answers

Can the existence of multiplicative inverse of real numbers be derived from its other properties?

Is there a construction of some subset of real numbers where the existence of multiplicative inverse of real numbers be derived by its other properties? In particular, I am curious of the following more limited question: consider the set of limits…
Argyll
  • 756
  • 1
  • 5
  • 16
0
votes
0 answers

Do probabilities (frequentist interpretation) always have to be rational numbers?

in frequentist interpretation probability is defined as the ratio of positive outcomes over all outcomes. Since positive outcomes and outcomes over all are count-data and therefor integers, they ratio between those two has to be a rational…
0
votes
1 answer

Doubts regarding Dedekind construction of R starting from Q (Rudin PMA)

I followed the 9 steps given by Rudin but in the end I didn't fully understand. My main questions are the following (in my understanding 2. is the follow up of 1.): since the members of R have been defined as cuts of Q (thus containing rational…
0
votes
1 answer

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 numbers? Intuitively I believe that the rationality…
newzad
  • 4,683
  • 23
  • 49
1 2 3
99
100