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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Discretized Distributions on Rationals?
Consider the measure space $(\mathbb{Q}, 2^{\mathbb{Q}}, \nu)$, with $\nu$ being the counting measure. The space is then $\sigma$-finite. Is there any attempts made to define analogues of continuous ...
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Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $
Define a function $f:\Bbb Q \to \Bbb Q$ by the following $$ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $$
where $\pi(\cdot)$ is the prime counting function and $N\in \Bbb N.$ I would like to find ...
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Constructing the interval [0, 1) via inverse powers of 2
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
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Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$
Prove $$\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$$
My effort:
$$\begin{aligned}
& \frac{1}{11}>\frac{1}{42} \\
& \frac{1}{12}>\frac{1}{42} \\
& \frac{1}{13}>\frac{1}{42} \\...
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Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? ("Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki)
I am reading "Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki.
Problem 1.17
Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$?
My attempt:
...
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Proof that Dedekind Cuts are isomorphic to decimal expansions?
The real numbers $\mathbb{R}$ are defined as the superset of the $\mathbb{Q}$ with additional elements to yield Dedekind Completeness. Specifically, every subset with an upper bound in $\mathbb{R}$ ...
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Rational point on a pythagorean rectangle
Consider the rectangle shown in the above diagram with vertices $(0,0)$, $(a,0)$, $(0,b)$ and $(a,b)$. The sides and diagonals of this rectangle are integers i.e., $a, b, c$ are integers such that $a^...
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Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$
Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational
numbers.
$f$ is strictly increasing in both
arguments.
Can $f$ be one-to-one?
This question is related to many ...
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Rationals are incomplete and naturals are complete
Why are rationals incomplete and natural complete? I have been going through Analysis textbooks but I don't totally get the reason why.
So, naturals are complete because you can divide them into two ...
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Are there more than two rational solutions to a certain system $abcd=a+b+c+d=K$ ($K$ a given constant)?
This question is a follow-on of a previous question asked some days ago which has been deleted due to its lack of precision. In fact, I found it well explained here
But, implicitly, the domain of ...
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Rational solutions to Rational Equation
I am looking for rational solutions to
$$x\frac{x^2-1}{(x^2+1)^2}+y\frac{y^2-1}{(y^2+1)^2}=2z\frac{z^2-1}{(z^2+1)^2}$$
besides $(x, x, x)$, $(-x, 1/x, 0)$, and other similar non interesting solutions.
...
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Is there a $f: \mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^{\infty} \frac{1}{n^2f(n)} \in \mathbb{Q}$?
Take by convention $0 \not \in \mathbb{N}$, and let $f: \mathbb{N} \to \mathbb{N}$. Define the real number $N(f)$ by
$$N(f) = \sum_{n=1}^{\infty} \frac{1}{n^2f(n)}.$$
$N(f)$ is well-defined because, ...
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Prove that the number $0.a_1a_2a_3\ldots$ is a rational number.
Let $a_1$ be any number from the set {$0, 1, 2, \ldots, 9$}. For each $n \in \mathbb{N}$, denote by $a_{n+1}$ the last digit of the number $19a_n + 98$ in decimal notation. Prove that the number $0....
user1315539
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are there 2 or more irrational numbers between any 2 rationals?
… in general, but also related to a calculus problem I have before me which is about continuity.
The question regards continuity wrt the function
$$
f(x) =
\begin{cases}
x, x \in \mathbb{Q} \\
0, ...
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Simplifying Gauss’s Lemma
Hello Math StackExchange Community,
I am revisiting Gauss's Lemma in my lecture notes and considering a simplification in its proof. I am proposing to remove the necessity of proving that $\lambda$ is ...
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