# 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.

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### Definition of rational numbers from real numbers

Usually the set of numbers are introduced starting from integers, from wich the rational numbers are defined using equivalence classes of couples of integer numbers. Than, from these rational numbers, ...
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### Showing the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ in $\mathbb{Q}[x]$

I would like to show the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ and $x^8 - 120 x^6 + 4360 x^4 - 45600 x^2 + 15376$ in $\mathbb{Q}[x]$. In both cases Eisenstein criterion fails. ...
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### $\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]

BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$ Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
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### Finding irrational entries such that the determinant will never be zero

Context. The main goal is to find whether or not a subspace of $\mathbb R^5$ of dimension $3$ intersects a rational subspace of dimension $2$. By rational subspace, we mean a subspace of $\mathbb R^5$...
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### For which x the following sequences converges

If $q_n$ be an enumeration of rational numbers, for which $x$ the following sequence converges? $$\sum_{n=1}^{\infty}e^{-n^2|x-q_n|}.$$ I guess that for no $x$ the sequence converges. I tried to ...
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### How is it that there are 'gaps' in rational numbers and yet between any two rational numbers, there exists another rational number?

If there are gaps in rational numbers then lets assume we have a gap between a and b, both being rational. Then we have $\frac{a+b}{2}$ which is inside the gap which essentially makes it a non-gap. ...
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### Apostol proof for $\mathbb{Q}$ being countable.

I am trying to understand a proof from Mathematical Analysis by Apostol for the following theorem: The set of rationals $\mathbb{Q}$ is countable. Here is the proof (I rewrote a few things): ...
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### Proving the Set of Rational Numbers is the collection of equivalence classes of ratios of integers with nonzero denominators.

I read in a textbook that the set of Rational Numbers is the collection of equivalence classes of ratios of integers with nonzero denominators. I found it interesting and tried to prove it but ended ...
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### Recurring Decimal Expansion

For any natural number $n>1$, we write the infinite decimal expansion of $\frac 1n$ (for example, $\frac 14$ is written as $0.24999$... instead of $0.25$). We need to determine the length of the ...
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### Silly Question about $π$ [closed]

In our junior classes, we learnt that $π$ is an irrational number. Now, also we know about rational numbers. So, if I say that I have a thread of 44cm long and we may convert it into a circle. Then ...
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### Arithmetic laws for rational numbers vs. real numbers

Are there any arithmetic laws that are always true for the set of rational numbers but not always true for the set of real numbers? This came up because I was doing various exercises in different ...
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### Irreducibility of a polynomial over Rationals with condition given on its coefficients.

Let $f = a_nX^n+\cdots+a_1X\pm p \in \mathbb{Z}[X]$ with $\sum_{i=1}^n |a_i| < p$. Show that $f$ is irreducible in $\mathbb{Q}[X]$. Hint: Show that every root of $f \in \mathbb{C}$ has modulus ...
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### Given $x^2=2$ prove for any rational number $\frac{p}{q} < x$,there exists $\frac{m}{n}$ such that $\frac{p}{q}<\frac{m}{n}<x$

Without using limits or the definition of irrational numbers, how do you solve this? I was thinking proof by contradiction, but I keep running into problems.
It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication ...