# Tagged Questions

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### Adding a natural number to a normalized fraction

I am currently writing yet another rational number class where the fraction should always be normalized. When adding a natural number to a normalized fraction, it possible to get a non-normalized ...
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### 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 ...
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### Confusing sum of fractions

Question is to find the sum of: $$(\frac{1}{2^2-1})+(\frac{1}{4^2-1})+(\frac{1}{6^2-1})+(\frac{1}{20^2-1})$$ I know that $a^2-b^2=(a+b)(a-b)$, and that with this I can find the LCM to be 1995, ...
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Some days ago I've seen Cantor's diagonal argument, and it presented a table similar to the following one: $$\begin{matrix} ... 0answers 122 views ### Find all \theta such that sin\theta and cos\theta are both rational number. Find all \theta such that sin\theta and cos\theta are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ... 7answers 282 views ### Why is a repeating decimal a rational number?$$\frac{1}{3}=.33\bar{3}$$is a rational number, but the 3 keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ... 1answer 242 views ### Hausdorff dimension of the set of rational numbers within a certain interval? Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ... 0answers 91 views ### Lower bound for the length of continued fraction Define \mathscr L: \mathbb Q \mapsto \mathbb N as the minimal number of terms in the continued fraction of a rational number. Example: the continued fraction of \frac{5}{8} is ... 1answer 287 views ### Prove that {\sqrt2}^{\sqrt2} is an irrational number without using a theorem. Prove that {\sqrt2}^{\sqrt2} is an irrational number without using the Gel'fond-Schneider's theorem. I'm interested in this problem because I knew that {\sqrt2}^{\sqrt2} is a transcendental ... 1answer 73 views ### True or false statements Two of the following statements are true and one is false a) For all rational numbers q, there exists an integer n so that q+n=271. b) For all integers n, there exists a rational ... 1answer 73 views ### Looking for name of theorem: “rational \Leftrightarrow fractional part terminates or repeats” I am looking for the name of the theorem that says that a number x is rational if and only if its fractional part terminates or repeats (where "fractional part" refers to the representation of x ... 0answers 143 views ### Must be rational number Let a, b positive rational number. Suppose that there exist two odd positive integers p, q such that \sqrt[p]{a}+\sqrt[q]{b} is rational. Prove that both \sqrt[p]{a} and \sqrt[q]{b} are ... 1answer 109 views ### Analytical function taking rationals to rationals. Suppose f:I \rightarrow \Bbb R is an analytic function defined on the interval I\subset \Bbb R with the property that for every q \in \Bbb Q:f(q)\in \Bbb Q. Does this already imply that f\in ... 5answers 344 views ### Odd divided by even is a fraction How can we prove that an odd number divided by an even number is a fraction? I started with odd =2m+1 and even =2n and get left with with (m+2)/n. 1answer 402 views ### Prove that given any rational number there exists another greater than or equal to it that differs by less than \frac 1n I am currently attempting to prove a claim in Hardy's Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this. ... 2answers 123 views ### A question about rational. Is that true : Every positive rational number q can be written as q = \sum_{i=0}^{k}1/n_i , where n_i,k are positive intergers and n_i\not=n_j if i\not=j. 3answers 240 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 ... 3answers 545 views ### Why (directly!) does every number divide 9, 99, 999, … or 10, 100, 1000, …, or their product? A curiosity that's been bugging me. More precisely: Given any integers b\geq 1 and n\geq 2, there exist integers 0\leq k, l\leq b-1 such that b divides n^l(n^k - 1) exactly. The ... 3answers 1k 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:$$ ...
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 ...