# rationals

1. ### Question concerning covering the rationals with open intervals (paradox?)

The following seems paradoxical. Consider the following countable collection of open intervals: I1 is ( 0, 1/2 ), I2 is ( 1/2, 3/4 ), I3 is ( 3/4, 7/8 ), etc. The sum of the lengths of the intervals is 1. Now cover each rational number with one of the intervals. Either 1) two successive...
2. ### adding rationals with factorials

Hello, I am brushing up on my mathematics skills, and to do so I am self-studying some algebra at the moment. For basic mathematics I use the book 'Basic Mathematics' by Serge Lang. Now one of the exercises in this book (1.5 ex 7d), involving binomial coefficients, is to show the following...
3. ### Function is irreducible in the rationals.

Is a=3x^3+2x+2 reducible in the rationals? So a is reducible iff we can write a=bc, for bc in the rationals (polynomials) and b,c\neq0. So deg(a)=3, therefore 0<deg(b)<deg(a) and 0<deg(c)<deg(a). So either deg(a)=2 or deg(a)=1, since deg(b)+deg(c)=deg(a)=3 The next step I'm not too sure...
4. ### rationals and union of intervals that covers it

I have a question, if \{q_{n}\}_{n=1}^{\infty} is a sequence of all rational numbers, let I_{n}=[q_{n}-\frac{1}{10^{n}},q_{n}+\frac{1}{10^{n}}] why do we have the following inclusion \mahbb{Q}\subset \displaystyle\cup_{n=1}^{\infty}I_{n}?
5. ### GCD and Cauchy sequences of rationals

Let P be the set of all prime numbers. For any a,b\in \mathbb{Q} with a = \prod_{p\in P}p^{a_p}, a_p \in \mathbb{Z} and b=\prod_{p\in P}p^{b_p}, b_p \in \mathbb{Z}, I will use the standard GCD (a,b) = \prod_{p\in P}p^{\min(a_p,b_p)}. Let a: \mathbb{N} \to \mathbb{Q} be an arbitrary Cauchy...
6. ### Come up with a bijection?

Hello, I need some help with this problem: Let S = {x ∈ : x2 ∈ ) I need to come up with a bijection between S and Q. Any help appreciated. Thanks!
7. ### maximal and prime ideals question

Let p be a fixed prime, and let R be the subring of Q consisting of all rational numbers a/b with (b,p) = 1. a) Show that R has a unique maximal ideal M. b) What are the prime ideals of R? c) Show that R/M is isomorphic to Z/pZ .
8. ### Multiplying and Dividing Rationals

Would I eliminate 3n first or simplify 3n+9 on the bottom?
9. ### Rationals and irrationals - properties

Hey guys! So I have two questions which are similar, but not the same. The first asks me to prove that between any two distinct rational numbers there exists an irrational number - I haven't managed to do this. The question after however, which asks me to show that between any two real numbers...
10. ### irreducible polynomials over the field of rationals

Let p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 be a polynomial with rational coefficients such that a_n and a_0 are nonzero. Show that p(x) is irreducible over the field of rational numbers if and only if q(x) = a_0 x^n + a_1 x^{n-1} + ... + a_{n-1} x + a_n is irreducible over the...
11. ### Proof that the rationals are countable.

Is this proof ok? If I assume that the set of two-touples X \{(a,b):a,b\in \mathbb{N}\} is countable (which I think follows directly from the proof the the countable union of countable sets is countable). Can we make the natural bijection f:\mathbb{Q}^+\rightarrow X s.t...
12. ### nonnegative solution in rationals

Hi, I encounter the following problem. Let A be an m*n matrix with all integer entries (may be positive or nonpositive). Suppose that Ax=0 admits a nonnegative solution x (i.e., xi>=0 for all i=1,...n). Is it true that A also admits a nonnegative solution with all entries being rational...
13. ### Continuity on rationals?

If g:R \rightarrow R is continuous and f:R \rightarrow R is such that f(x)=g(x) for all rational numbers x, is f continuos? I'm not 100% sure about this because f is clearly continuos on the rationals, but what happens at the irrationals? EDIT: Being slow today... consider g(x) = 1...
14. ### Determine if this Ring is a field

Can you help me prove that this ring is a field? R = Q[\sqrt(d)] = { a+b\sqrt(d) ; a,b are rational} So I tried: We must show that every element of R is a unit. So for some element (a+b*\sqrt(d)) there must be some (e+c*\sqrt(d)) such that (a+b*\sqrt(d))*(e+c*\sqrt(d)) = 1 so...
15. ### Reversing building irrationals on the rationals

Has anyone tried to start off with irrational numbers as a foundation, then derive the rational numbers from them?
16. ### Evaluating Limits with rationals.

I got a question on my test today and I spent a great deal of time on the question. lim as x -> 27 (27-x)/[(x^1/3)-3] I was able to approximate the limit by evaluating left and right of 27; however, there was another method where a change of variable was used, for example let u=(x^1/3). I was...
17. ### Probably a simple explanation

Could someone explain why 2x-10 is not an irreducible in Z[x] but is an irreducible in Q[x].
18. ### Rationals

This question is part of a question on Pade approximants, but I can't see why this is true: the rational function r_{km}(x)={p_{km}(x) \over {q_{km}(x)} }where p is a poly in x of degree at most k and q is a poly in x of degree at most m, then f(x)-r(x)=O(x^{k+m+1}), does this not say r has...
19. ### Complete proof that sum of rationals is rational

Hi, I did a search, admittedly brief, trying to find a complete proof that the sum of two rational numbers is a rational number. I have a proof in my discrete mathematics book. When it comes to proving useful stuff, the proofs in the book change from showing each step to being much more...
20. ### [SOLVED] Rationals and countable intersection of open sets

Is it possible to use Baire's theorem to prove that the set of rational numbers isn't a countable intersection of open sets?