Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
2,623 questions
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1answer
19 views
Proof of Density of Rational Numbers as given in Howie
I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following:
Let $x$ and $y$ be real numbers, with $ x < y $. Show that, if $x$ and $y$ are ...
0
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0answers
14 views
States of the world/Game theory and Beliefs
This post consists on 3 parts: the question itself, hint and a table.
The question will make sense to you only after you have read the tables and the hint attached. The problem is about beliefs of a ...
0
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2answers
20 views
Dividing an integer by an infinite decimal that extends irregularly
The existance of the multiplicative inverse of a nonzero real number, proves the existance of fractions, such as, $\frac{1}{\sqrt{2}}$ and $\frac{1}{\pi}$.
So, how to compute such fractions, where an ...
0
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3answers
31 views
Dividing an integer by a repeating decimal
What I know is that the existance of multiplicative inverses of a nonzero real number $a$ is $\frac{1}{a}$ and that holds when $a$ is a repeating decimal.
So, how to divide an integer by a repeating ...
0
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1answer
20 views
Rounding off to nearest tenths
Round off $0.2545$ to nearest tenths.
Following my instinct, the answer is $0.3$ because that's what it is more close to. Following the rules, as told by my teacher, it comes out to be $0.2$. Steps I ...
0
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3answers
33 views
Prove that $0< \frac{1}{2^{m}} <y$
If $y$ be a positive real number, show that there exists a natural number $m$ such that $0< \frac{1}{2^{m}} <y$
I think I have to use Archimedean property to prove it. The Archimedean property ...
0
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0answers
17 views
Operations While Solving Equation Systems in $(\mathbb{R},+,.)$
It is easy to understand why we don't do simplification in $(\mathbb{R},+, .)$
without having a good look at while solving equation systems since $(\mathbb{R},+, .)$ is a domain. So I also know ...
1
vote
2answers
53 views
Please explain the proof of the validity of the Cauchy convergence criterion for real number sequences.
My question pertains to BBFSK, Vol I, Pages 143 and 144.
The following appears in the context of developing the real numbers as limits of sequences of rational numbers.
It is also easy to prove ...
0
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1answer
56 views
Irrationality of square root of 2
I was trying to study Understanding Analysis by Stephen Abbott and I am stuck at the very second page. I feel stupid please help me. It says this:
Given two line segments $ab$ and $cd$ it would ...
0
votes
1answer
27 views
mapping of the real line to interval [0,1) [duplicate]
What is a mapping (bijection) of the real line (−∞,∞) to the interval [0,1)? I'm trying out logs and exponentials but they don't seem to work?
1
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1answer
49 views
Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$
Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$.
Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$.
Is that ...
0
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1answer
25 views
The cartesian product of a well-ordered set with $[0,1)$ is a linear continuum in dict. order
Definition: A linear continuum is a simply ordered set $L$ such that:
(1): $L$ has the least upper bound property;
(2): For every $x<y$ in $L$ there is a $z$ sucht that $x<z<y$.
I ...
0
votes
1answer
22 views
Help with a function problem- Define a function $f$: $\mathbb{R^2}$ $\rightarrow$ $\mathbb{R}$ by $f(x,y)= x-y$. [closed]
I am struggling understanding what these two parts of this question are asking for:
Define a function $f$: $\mathbb{R^2}$ $\rightarrow$ $\mathbb{R}$ by $f(x,y)= x-y$.
(1) Find $f([0,1]\times[0,1])$.
...
0
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0answers
30 views
$3^m+7^n$ is a perfect square!! [duplicate]
Determine all pairs $(m,n)$ of positive integers such that $3^m+7^n$ is a perfect square
when n and m are both odd $3^m+7^n$ isn't a perfect square. and when n and m are both even $3^m+7^n$ isn't a ...
0
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1answer
53 views
Which stage in the Neumann hierarchy do powers of the reals fit in?
To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages.
For the sake of ...
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1answer
77 views
+50
Exercise about sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R})$
Let $C=\{(-a, a): a \in \mathbb{R}\}$ and $F=\sigma(C)$.
Prove that $F=\mathcal{B}(\mathbb{R})\cap\{A\subseteq\mathbb{R}: A=-A\}$.
I don't have problems in proving $F\subseteq \mathcal{B}(\...
0
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1answer
30 views
Showing a set is an ideal in a ring of real-valued functions
If $F$ is a ring of all real-valued functions defined on $\mathbb{R}$, is $S = \{f ∈ F | f(0) = 1\}$ an ideal?
What I'm thinking is $(f+g)(0) = f(0)+g(0) = 1+1 = 2$
and hence $f + g$ is in $S$? Is ...
5
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0answers
98 views
Baire space $\mathbb{N}^\mathbb{N}$ written as $\mathbb{R}$ [duplicate]
I'm writing my bachelor thesis on various topics from set theory and descriptive set theory (mainly topological games), and I just read a paper in which the Baire Space $\mathbb{N}^\mathbb{N}$ is ...
4
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1answer
60 views
Show $\alpha^{-1}$ is a Dedekind Cut
Problem statement:
Given an arbitrary positive Dedekind Cut $\alpha = A|B$, prove that the following
$\begin{align*}\alpha^{-1} &= C|D \\
&= \{r\in\mathbb{Q}: r\leq 0 \text{ or } \exists b\...
1
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0answers
21 views
Archimedean ordering and a greatest $r_n<x\in{M}$ shows every number less than a lower bound is also a lower bound?
This question pertains to BBFSK Vol I, page 143. The topic is the definition of the greatest lower bound of a non-empty set $M\subset{\mathbb{R}}$ which is bounded from below.
For $1<g\in\mathbb{...
0
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2answers
43 views
Maximum and minimum values of $\left\lfloor \frac{x}{nm}\right\rfloor - \left\lfloor \frac{1}{m}\left\lfloor \frac{x}{n}\right\rfloor \right\rfloor$
I feel like I need some additional pointers on the following questions as I am unable to come up with a solution for it:
If $m$ and $n$ are any integers, and $x$ is any positive real number, what are ...
1
vote
3answers
65 views
Maximum and minimum value of ⌊2x⌋ − 2 ⌊x⌋ for any real number x?
Working through some sample problems on flooring from a guide book and I'm stuck on the following questions:
What is the maximum and minimum value of $\lfloor 2x\rfloor − 2 \lfloor x\rfloor$ for any ...
0
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0answers
9 views
If $\left[ a^d\left( \frac{1}{e^{pd}} - 1 \right) + b^d \right]^c = \left[ a^c\left( 1 - c^{pc} \right) + b^c \right]^d,$ then $e=1$ or $c=d?$
Question: Assume that $a,b,c,d,e\in \mathbb{R}$ and $p\in \mathbb{N}$ such that $0<a,b<1,$ $c,d\geq 1,$
$$a^d+b^d<1, \quad a^c+b^c<1, \quad c\leq d \quad\text{and}\quad e\leq 1.$$
If ...
0
votes
2answers
43 views
Is Wolfram Alpha wrong? (trig identity)
I have the following identity
$$ \sqrt{\cos(2x)\sec^4(x)}$$
I use the property $ \sqrt{a} \sqrt{b} = \sqrt{ab} $
which then yields
$$ \sqrt{\cos(2x)} \, \sec^2(x) $$
However, Wolfram tells me ...
0
votes
1answer
31 views
What are the greater uses of classifying numbers in advanced mathematics [closed]
In high school throughout college, we have been thought what is rational, integers etc. But how does is fit to the greater scheme of things. Trigonometry, Calculus, Mensuration have very obvious ...
0
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1answer
30 views
$\Sigma_{\alpha \in A}x_{\alpha}\lt \infty$, show that $x_{\alpha}=0$ for all but at most countably many $\alpha \in A$ [duplicate]
If ($x_\alpha)_{\alpha \in A}$ is a collection of numbers $x_{\alpha}\in [0, \infty)$ such that $\Sigma_{\alpha \in A}x_{\alpha}\lt \infty$, show that $x_{\alpha}= 0$ for all but at most countably ...
-1
votes
1answer
40 views
why is the following U-Substitution wrong?
It is known that
$$ \int \frac{1}{\sqrt{1-x^2}} dx = arcsin(x)+c$$
this can be done utilizing u-substitution $ x = sin(u) $
However,
i can let $ u = 1-x^2 $
$dx = -2u \, du $
which gives ...
2
votes
1answer
88 views
If $(2^b-1+c^b)^a = (2^a-1+c^a)^b,$ then is it true that $a=b?$
Question: Assume that $a,b,c$ are real numbers such that
$$1\leq a\leq b, \quad\text{and}\quad 0<c<1.$$
If we have
$$(2^b-1+c^b)^a = (2^a-1+c^a)^b,$$
then is it true that
$$a=b?$$
...
2
votes
2answers
43 views
Relation defined on the set of real numbers by xRy when $x^2 + y^2 = 1$. Show whether or not R is reflexive, symmetric, antisymmetric or transitive.
Let R be the relation defined on the set of real numbers by $xRy$
whenever $x^2 + y^2 = 1.$ Show whether or not $R$ is reflexive,
symmetric, antisymmetric or transitive.
All right so I think I've ...
1
vote
2answers
70 views
Solve $\left(\sqrt{\sqrt{2}-4-x}\right)+x^{\frac{1}{4}}=2^{\frac{-1}{4}}$
Solve the Equation in real/Complex numbers:
Solve $$\left(\sqrt{\sqrt{2}-4-x}\right)+x^{\frac{1}{4}}=2^{\frac{-1}{4}}$$
My try:
Letting $x=t^4$ we get
We get
$$\left(\sqrt{\sqrt{2}-4-t^4}\right)+...
0
votes
1answer
21 views
Family of sets proof with archimedean principle
How to proceed with this proof
Let $A=\left\{A_{i}\right\}_{i\in I}$ a family of sets in $\mathbb{R}$ such that verifies the following properties:
$\forall a\in \mathbb{R}, \; (a,+\infty ) \...
2
votes
1answer
10 views
Problem explanation regarding subspaces of $\mathbb{R}$
Does the word "subspace" here imply "linear subspace"?
A subspace $Y$ of $\mathbb{R}$ is said to be a retract of $\mathbb{R}$ if there exists a continuous map $r: \mathbb{R} \to Y$ such that $r(y)= ...
1
vote
2answers
94 views
There are no other clopen sets in $\mathbb{R}$ except for $\mathbb{R}$ and $\emptyset$
Proof attempt:
Let there be another clopen set $S$ in which is a proper subset of $\mathbb{R}$. Hence, $ S^c \neq \emptyset $.
We can assert the following statements:
No point of $S$ lies in $S^c$...
2
votes
1answer
70 views
Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$
If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that :
$$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$
My attempt
...
4
votes
3answers
120 views
A series such that $\sum {a_n}$ converges, but $\sum {a_{3n}}$ diverges.
Give an example of a convergent series $\sum {a_n}$ such that the series $\sum {a_{3n}}$ is divergent.
Give an example of a divergent series $\sum {b_n}$ such that the series $\sum {b_{3n}}$ is ...
0
votes
1answer
45 views
$x_i=\prod_{j \neq i}x_j$ for $i=1,2,..,6$
Find all real $x_i^s$ satisfying the system of equations
$x_i=\prod_{j \neq i}x_j$ for all $i=1,2,..,6$
It is obvious that $(x_1,x_2,..x_6)=(0,0,..0),(1,1,...,1),(-1,-1,-1,...-1)$ are obvious ...
1
vote
2answers
54 views
Definition of the sum of 2 Dedekind cuts
I was recently studying the construction of real numbers. I read that the sum of 2 reals using the left Dedekind sets was the set of sum of all the rational numbers contained within those two sets.
...
-1
votes
2answers
20 views
Use Wilson Theorem for the divisor
Use Wilson Theorem to find the smallest possible number which completely divides (12! + 6! + 12! × 6! + 1!).
Wilson Theorem →
Wilson Theorem states that if n is a prime number then n divides [(n-1)!+...
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votes
0answers
26 views
Proving the uniqueness of x=sqrt(r)
Given any $r \in \mathbb{R}_{>0}$, the number $\sqrt{r}$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $x = \sqrt{r}$
I would appreciate any nudge in the ...
0
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0answers
33 views
Help with elementary proof about r in the real numbers
If $r<0$, there exists no $x \in \mathbb{R}$ such that $x^2 = r$.
I'm thinking I need to prove by contradiction assuming there does exist an $x$ such that $x^2=r$, but I'm having trouble finding ...
11
votes
0answers
137 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}$, ...
0
votes
0answers
23 views
difficulty evaluating solving this gamma function value
I am aiming to solve the following
$$\prod_{n=1}^{\infty} \left(1-\frac{1}{(2n)^3} \right) $$
Note its similarity to
$$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3} \right)=\frac{\cosh(\frac{\pi}{2}\...
0
votes
0answers
22 views
Limit crossover problem
If $f$ is a real function satisfying $\lim_{x\to\infty}f(x)=\infty$ and $\lim_{x\to-\infty}f(x)=-\infty$ then there must exist $a\in\mathbb{R}$ such that for all $\epsilon>0$ there exists $0<r&...
2
votes
1answer
51 views
$-33 \leq x^3+4x^2-3x \le 53$. [closed]
Let $-3 \leq x \leq 2$, then $ -2 \le -x \le 3$, $0 \leq x^2 \le 9$ and $-27 \le x^3 \le 8$.
From the above can we conclude that $$-27+0-6\le x^3+4x^2-3x \le 36+8+9 $$
i.e. $-33 \leq x^3+4x^2-3x \le ...
-1
votes
2answers
59 views
Does the bar in $\bar{\mathbb{R}}=\{-\infty,\infty \}\bigcup\mathbb{R}$ mean closure of the reals?
Does the bar in $\bar{\mathbb{R}}=\{-\infty,\infty \}\bigcup\mathbb{R}$ mean closure of the reals? Where Closure is defined as the union of the set and all it's limit points?
Certain sequences in $\...
-1
votes
1answer
26 views
Proofs on inequalities of real numbers [closed]
So I have these inequalities (statements) to prove:
$x,y \in \mathbb{R}$
$\vert xy \vert \leq \frac{1}{2}(x^2 + y^2)$
$x,y \geq 0 \implies xy \leq \frac{1}{4}(x + y)^2$
I know that I have to use ...
2
votes
1answer
64 views
An apparently harmless exercise concerning induction
I'm trying to solve Exercise 12 at page 14 from "A Concrete Introduction to Higher Algebra" by L. Childs. The text is the following.
Let $b \in \mathbb{R}, b \ge 2$. Prove that $$(b^n - 1)(b^n - b)(b^...
2
votes
1answer
79 views
A bijection between the set of all sequences whose terms are either 0 or 1, and the set of all sequences whose terms are 0, 1 or - 1.
Let X be the set of all sequences whose terms are either 0 or 1. Let Y be the set of all sequences whose terms are 0, 1 or - 1.
We know that both are uncountable sets, by Cantor's diagonal ...
1
vote
0answers
41 views
Why multiplication of real numbers is commutative??
In real analysis books, it is taken as axiom. But I want to know the logic behind it. As for example, 32*75 means adding 32 for 75 times and 75*32 means adding 75 for 32 times. How to be sure these ...
4
votes
2answers
116 views
Find the positive root of $100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$
I was struggling with this problem:
$$100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$$
It came up when i was developing a solution to a geometry problem. I've already checked in Mathematica ...
