Questions tagged [real-numbers]

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

455 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
24
votes
0answers
293 views

Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
23
votes
2answers
5k views

Showing that $\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$?

$$\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$$ In the second nested radical, the repeating pattern is $(-,-,+)$. I approached this ...
14
votes
0answers
420 views

Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
12
votes
0answers
151 views

What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
7
votes
0answers
124 views

Both variables algebraic/ both transcendental?

Suppose $X\subset \Bbb Z_{<0}$, and $a=\sum_{n\in X} 2^n$ is algebraic, then is $b=\sum_{n\in X}3^n$ also algebraic? (To clarify, what I mean is: does it hold for all such $X$?) We know that $a$ ...
5
votes
0answers
132 views

Direct construction of the real numbers using only the integers (c.f. Eudoxus reals)

Let the natural numbers $\Bbb N = \{0,1,2,3,\dots\}$ and the integers $\Bbb Z$ be given. We define a function $\gamma: \Bbb Z \to \Bbb Z$ by $$ \gamma(n) = \left\{\begin{array}{lr} \...
5
votes
0answers
409 views

Geometric basis for the real numbers

I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients. But I never actually think of real numbers in this way. I ...
5
votes
0answers
154 views

Rational numbers as angles - where do irrationals fit in?

If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. For ...
4
votes
0answers
46 views

An extension of $\mathbb{F}_p$ to an interval $[0,p]$ with $p \equiv 0$

I had a curious question bouncing around my head the other day. I asked myself if there were numbers with similarly nice properties to something like $e$ in a finite field, along the lines of $\mathbb{...
4
votes
0answers
180 views

Comparing elements of sets

Let $a_1, a_2, a_3, a_4$ be real numbers. Consider the following sets $$ \mathcal{U}_1\equiv \{-a_1, a_1, -a_2, a_2, 0, \infty, -\infty\} $$ $$ \mathcal{U}_2\equiv \{-a_3, a_3, -a_4, a_4, 0, \infty,...
4
votes
1answer
149 views

Pick out true statements about the limit of $f_n(x)=\frac{1}{1+n^2x^2}$

For the sequence of functions $f_n(x)=\frac{1}{1+n^2x^2}$ for $n \in \mathbb{N}, x \in \mathbb{R}$ which of the following are true? (A) $f_n$ converges point-wise to a continuous function on $[0,...
4
votes
0answers
77 views

Equality of Floors of some Partial Sums

Let $S_n=\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}$ denote the $n$th partial sum in the series expansion for $e=\sum\frac{1}{k!}$. I want to prove that $\lfloor n\cdot(S_n+1/n!)\rfloor=\lfloor n\...
4
votes
1answer
113 views

How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the (relatively)...
4
votes
0answers
306 views

Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
4
votes
1answer
1k views

Existence and uniqueness up to isomorphism of the real numbers from axioms

Pretty much what the title says: how does one prove the existence and uniqueness of the real number system from the ordered field axioms together with the least-upper-bound property (or maybe some ...
3
votes
0answers
50 views

Prove that $\forall x \in \Bbb R^+:x+0=x\cdot 1=x$

Let $\Bbb R^+=\{x\in\Bbb R \mid x>0\}$ and $x,y\in\Bbb R^+$. We define multiplication operation $(\cdot)$ on $\Bbb R^+$ and addition operation $(+)$ on $\Bbb R$ by $$x\cdot y:=\inf\{r\cdot s\mid r,...
3
votes
0answers
36 views

Outer measure induced by a Jump function

This is from exercise 4.4 of Elstrodt's measure theory textbook. By a jump function $F:\mathbb{R}\rightarrow\mathbb{R}$ we mean a function which can be written in the form $$F(x)= \begin{cases}\...
3
votes
1answer
207 views

The axiomatic method to real number system VS the constructive method(genetic method)

According to book Georg Cantor: His Mathematics and Philosophy of the Infinite - Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the ...
3
votes
0answers
47 views

Expectation of a matrix over random vectors

Suppose I have a matrix A which is for argument's sake 3 by 3 and has different eigenvalues, let's say 1, 2 and 3. If I was to generate vectors on a unitary sphere $\eta$ and then calculate the dot ...
3
votes
0answers
100 views

Existence of two subsets of $\mathbb R$ with a certain property, undecidable in $ZF$?

Inspired by Question # 2420627 "Prove that there is a bijection $f:\mathbb R\times \mathbb R\to \mathbb R$ in the form of $f(x,y)=a(x)+b(y)$" and the answer and comments by Thomas Andrews. ...
3
votes
0answers
117 views

How to prove that the intersection of countably many open sets of reals is either at most countable or of cardinality continuum?

I've learned that any Borel set in $\mathbb R^n$ is either countable or of cardinality continuum from Springer's Problems and Theorems in Classical Set Theory. I wonder if there exists some more ...
3
votes
0answers
63 views

Sequence in 10-base decimal representation of $\pi$

For some time ago Wolfram launched MyPiDay, which lets you find your 6-digit birthday string in the 10-base decimal representation of $\pi$. For example, my birthday string "970524" starts at the ...
3
votes
0answers
171 views

infinitely small vs arbitrarily small

In a question I recently asked, Definitions of supremum it was pointed out that my usage of infinitly small was misguided, and the proper terminology was arbitrarily small. Could someone explain what ...
3
votes
0answers
125 views

Coloring the real plane with a countable number of colors.

We color the real plane with a countable number of colors (each vertex gets a color). Can we always find a rectangle such that all of its vertices have the same color? (the edges of the rectangle need ...
3
votes
0answers
66 views

Notation for the interval between two arbitrary numbers

Does anyone know of a non-awkward way to notate the interval between two numbers when you don't know which number is larger? For example when describing the remainder term in Taylor's theorem, one ...
3
votes
0answers
60 views

$f(f(…f(x)…))$ $a$ times, where $a\in\mathbb{R}$

Take $f(x)$ and do a "double-call": $f^2(x)=f(f(x))$ I use this notation here to explain my problem. This can be easy calculated for any function. Also $f^{100}(x)$ is not really a problem. This ...
3
votes
0answers
50 views

Proof of If $A \subseteq \Bbb{R}$ with $A$ inductive set then $\Bbb{N}\subseteq A$

In calculus class, I saw a proof of this, but I am not convinced. Note: A$\subseteq \Bbb{R}$ is inductive if and only if 1$\in A$ and $\forall x \in A \Rightarrow x+1 \in A$. I tried to do a proof ...
3
votes
0answers
198 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
3
votes
2answers
108 views

Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc

I wonder what is the name of a mathematical system extending the real numbers that includes signed zero along with unsigned zero as well as other "limit targets", such as $1^+=1+0^+$, $5^-$ etc, so ...
3
votes
0answers
96 views

Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
3
votes
0answers
99 views

There are infinitely many $\frac{m}{n}\in\mathbb Q$ such that $|x- \frac{m}{n} |<\frac{1}{n^2}$

Let $x$ be an irrational number. Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that $$|x- \frac{m}{n} |<\frac{1}{n^2}$$ It's clear that $-1<1/n^2 <1$ with $n \...
3
votes
0answers
45 views

How I can calculate this product

How I can calculate this product: $$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$ for a natural number $p>3$.
3
votes
1answer
835 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
3
votes
0answers
169 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
3
votes
2answers
108 views

Find the smallest $n$ such that an integer lies between $nx$ and $ny$ for real numbers $x$ & $y$.

Say I'm given two real numbers as inputs: $x$ and $y$, with $x < y$. I want to find the smallest natural number n such that there's at least one integer between $nx$ and $ny$ (inclusive of $nx$ and ...
3
votes
1answer
374 views

What's this number called and what are its properties?

What is the following number called and what are its known properties? $$.12233344445555566666677777778888888899999999910101010101010101010...$$ (I think you get the pattern.) P.S.Also, see what I ...
2
votes
0answers
25 views

A question about a proof that there is a unique $x \in \mathbb{R}^{+}$ such that $x^{n}=a$ .

I'm reading Proposition 10.9 in Section The Real Numbers from textbook Analysis I by Amann/Escher. and its proof: In the case $s^n < a$, the authors define $b$ by $b :=\sum_{k=0}^{n-...
2
votes
1answer
51 views

Archimedean field with non total order

I am currently looking for the simplest way to characterize the real numbers. Usually they are described as the complete archimedean field, showing that all such fields are isomorphic. Archimedean ...
2
votes
0answers
56 views

Proof verification about a property of the topological space $[0,1]$ Part 2

Suppose $A_1,\dots,A_k$ are connected open subsets of $[0,1]$ such that $[0,1]=\bigcup_{i=1}^k A_i$ and $A_i \not\subseteq A_j$ for each $i\ne j$. By characterization of connected subsets of $\mathbb{...
2
votes
0answers
40 views

Let $x\in\Bbb R$, $a=\inf \{r \mid r\in\Bbb Q,x<r\}$, and $b=\inf\{-s\mid s\in\Bbb Q,s<x\}$. Prove that $a+b=\inf\{r-s \mid r,s\in\Bbb Q,s<x<r\}$

Let $x,y\in\Bbb R$. We define addition operation $(+)$ by $$x+ y:=\inf\{r+ s\mid r,s\in\Bbb Q \text{ and } x<r \text{ and } y<s\}$$ Theorem: Let $x\in\Bbb R$, $a=\inf \{r \mid r\in\Bbb Q,...
2
votes
0answers
37 views

An introduction book for analysis with the axioms of Tarski

Briefly: Is there an introduction book for analysis with The axioms of Tarski as basis? (I found them very elegant. And the most important thing for me is that the axioms are "obvious", not like the ...
2
votes
1answer
44 views

System of six equations in real numbers

Let $a,b,c,d,e,f$ be real numbers. Solve the following system of equations: $\begin{cases}a+b=-e\\ ab=f\\ c+d=-a\\ cd=b\\ e+f=-c\\ ef=d\end{cases}$ I got stuck trying to solve this problem by ...
2
votes
0answers
95 views

Vector field in $\Bbb Q^2$ and $\Bbb R^2$

I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a ...
2
votes
2answers
353 views

Prove there is a square of a rational number between any two positive real numbers

I'm doing this problem and the solution is below. I don't understand why this solution proved $m^2 \le x$ firstly? Any help, thanks.
2
votes
0answers
234 views

Is there a Dedekind-theory of real numbers?

Someone said that Dedekind had a "theory of real numbers", and I wonder about the truth level of that statement. He published the well-known construction of real numbers in 1872, partly inspired by ...
2
votes
0answers
42 views

Existence of a Precise axiomatization of Eudoxus theory of magnitude

Is there a precise axiomatization of the Eudoxus theory of proportions? For example, a) (D, +, <) is a structure such that < is a strict linear order, b) + is an order-preserving ...
2
votes
0answers
90 views

Is it true that every open set in Euclid space is a countable union of open (closed) cubes?

There is a theorem in the book but it only applies to $\mathbb{R}$ and open intervals. The proof seems to be unnecessarily complicated. Can I just say I can cut the space countably many times (...
2
votes
2answers
84 views

Dedekind Cuts and Rationals

I have a question regarding how you might construct the reals from the rationals by taking Dedekind cuts. My basic understanding is that a Dedekind cut is a bipartition of the rationals such that ...
2
votes
0answers
75 views

Is an equivalence relation (= sign) needed for the real number system or is a consequence of the other axioms?

My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further. Generally, the Real Number System is said ...
2
votes
2answers
372 views

Prove that ℚ is disconnected as a subset of ℝ

This is the proof I've come up with... Using the following definition: Let S be a metric space. E ⊂ S is disconnected iff there are disjoint, non empty subsets A and B that are open in E and such ...