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Questions tagged [real-numbers]

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

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2answers
31 views

LCM of irrational numbers

So i read in a book that irrational and rational numbers do not have a common multiple and it said that lcm of irrational numbers is also only possible when both the irrational numbers have the same ...
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1answer
38 views

Absurd/Odd Numbers

We have a set comprising complex numbers. The square root of -1, as I understand it, doesn't have a physical significance. However, I do know that it is quite useful in cryptography. My question Are ...
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4answers
27 views

The set of all values of m for which $mx^2 – 6mx + 5m + 1 > 0$ for all real x is

The set of all values of m for which $mx^2 – 6mx + 5m + 1 > 0$ for all real x is? The answer given is $0<=m<1/4$ My working: $D>=0$ $=> (-6m)^2 -4(m)(5m+1)>=0$ $=> m(4m-1)>=...
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1answer
29 views

Describe the ‘intervals’ $[a, b]$ and $(a, b)$, in the case where $a = b.$

This is a question from my textbook, however there are no solutions, would $[a,b] = a$ and $(a,b) = \text{undefined}$? I'm not sure if $(a,b)$ is right.
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0answers
29 views

Different proofs of irrationality of roots of rational numbers

Most of us know the classical proof of the irrationality of $\sqrt{2}$. In fact there is a list of different proofs of said irrationality that can be found here. Some of them are qeometrically ...
1
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1answer
34 views

number of zeroes of arbitrary function

Sorry if I misused/mixed up some maths terms. I barely know any maths lingo, especially not in English. I was thinking about programmatically solving equations (or rather, approximating their roots), ...
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2answers
16 views

I have the idea of using the transitive property and/or the the integer combination property. I am stuck tho.

\begin{equation} a, b, \text { and } c \text { are integers. Prove that if } a |(b-1) \text { and } 5 a |(c+2), \text { then } a |(2 b+c) \end{equation}
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1answer
23 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 ...
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1answer
54 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 ...
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2answers
21 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 ...
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3answers
32 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 ...
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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 ...
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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 ...
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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 ...
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2answers
56 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 ...
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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 ...
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1answer
29 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?
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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 ...
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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 ...
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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 ...
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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 ...
3
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1answer
98 views

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}(\...
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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 ...
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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
61 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\...
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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{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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1answer
89 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 $$a=b?$$ ...
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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 ...
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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)+...
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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 ) \...
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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)= ...
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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 $S^c$...
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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
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3answers
125 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 ...
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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
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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. ...
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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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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 ...
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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
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0answers
138 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
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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
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1answer
52 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 ...