All Questions
26
questions
2
votes
1
answer
234
views
were Irrational numbers discovered at Archimedes's age?
Archimedes axiom states a property of real numbers, while the real numbers include all the rational numbers and all the irrational numbers.
I wonder were Irrational numbers discovered at Archimedes's ...
- 2,267
11
votes
2
answers
978
views
Is there a mathematical statement that is linking integer limits to real limits?
I saw a question asking for the limit
$$\lim_{n \to \infty}\frac{\tan(n)}{n}.$$
At first I thought that the limit assumed $n$ to be a real number. So I gave the advice to use $\pi/2+2\pi k$ and $2\...
- 15.9k
1
vote
4
answers
1k
views
If $b-a>1$ then there is a $k\in \mathbb{Z}$ such that $a<k<b$
Given $a, b \in \mathbb{R}$, such that $b-a>1$, there is at least one $k\in \mathbb{Z}$ such that $a<k<b$.
My attempt:
Consider $E:=(a,b)\cap \mathbb{N}$. We need to show that $E$ is not ...
- 9,708
7
votes
2
answers
4k
views
Let $x$ be a real number. Prove the existence of a unique integer $a$ such that $a \leq x < a+1$
Let $x\in \mathbb{R}$ , Using the Well-Ordering Property of $\mathbb{N}$ and the Archimedean Property of $\mathbb{R}$, show that there exist a unique $a \in \mathbb{Z}$ such that $a \leq x < a+1$
...
- 2,157
1
vote
0
answers
38
views
how to find continued fractions with terms less than or equal to -2?
How can one find a continued fraction with all the terms less than or equal to -2? meaning that $x=[a_0,a_1,...,a_n]$ with $a_i\leq -2$.
- 963
9
votes
0
answers
138
views
If $n^x\in\Bbb Z,$ for every $n\in\Bbb Z^+,$ then $x\in\Bbb Z$ [duplicate]
Let $x$ is a real number such that $n^x\in\Bbb Z,$ for every positive integer $n.$ Prove that $x$ is an integer.
I got that problem here and it looks difficult, I tried writing $x$ as $\lfloor x\...
- 3,457
4
votes
2
answers
587
views
Cantor Sets in perfect sets in the Real numbers
My thesis is related with the Cantor sets. I was reading a lot of papers, blogs, etc, in order to look for the mean properties of these sets.
In one blog a read a proposition.
''Every perfect set ...
- 171
1
vote
1
answer
49
views
Can we find a relation between the three integrers $m$, $j$ and $k$?
Let $r>4$ and $n>1$ positive integers and let $α$ be a positive real number.
Let us define the following three positive integers:
$$
\begin{align*}
m &= \lfloor r^{(n+1)^2} \alpha \rfloor \...
- 3,021
0
votes
2
answers
94
views
Find the number of digits of the number $k$ in function of $r$ and $n$
Let $α∈(0,1)$ be an irrational number with infinitely digits after the decimal point. Let $r>4$ and $n>1$ be positive integers. Let $$k=⌊r^{n²}α⌋$$
where $⌊.⌋$ is the floor function.
My ...
- 3,021
4
votes
6
answers
389
views
Why is $0^0$ undefined when $x^x=1$ as $x$ approaches $0$?
This question was born in another post available here. I believe $0^0=1$, because $x^x$ is continuous as $x$ approaches $0$.
Consider $\lim_{x \to 0}x^x$. Let $$f(x_n)=\bigg(\frac{1}{x}\bigg)^{\...
- 618
1
vote
2
answers
113
views
Proving any $x \in [0,1]$ belongs to infinitely many $S^{k}_{n}:=[\frac{k-1}{2^n},\frac{k}{2^n}]$
I am trying to prove the following:
Define $S^{k}_{n}:=[\frac{k-1}{2^n},\frac{k}{2^n}]$
i) Given any $x \in [0,1]$, then $x$ belongs to infinitely many $S^{k}_{n}$
ii) Any $x \in [0,1]$ also belongs ...
- 293