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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Why does there exist no real number such that it is equal to an integer multiple of any other number?

Was reading about waves in my Physics textbook and a mathematical fact was invoked which I was curious about: If we combine an infinitely large number of sinusoidal component waves, each with ...
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Set of numbers as a summation of two subsets

Denote by $\mathbb{F}$ one of the set of (all) integer, rational, real or complex numbers. We are looking for necessary and/or sufficient conditions for $\emptyset\neq C\subseteq \mathbb{F}$ such that ...
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Intervals $( +\infty , - \infty ) = \mathbb R$

In intervals we say $( +\infty , - \infty ) = \mathbb R$. In this case what about zero because it is neither positive not negative
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How do you solve $a^2b^2c-8abc+c^2+ac+bc-ab+8c+8=0$?

What is the general solution inf $\mathbb{R}_+$ of $a^2b^2c-8abc+c^2+ac+bc-ab+8c+8=0$, where $0<ab<8$? I can see the trivial solution 2,2,2, but what is the easiest way to write down the general ...
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2 votes
1 answer
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Relation between connectedness and Dedekind Completeness

In my calc class we constructed the real numbers using the following 5 axioms: The set is nonempty. The set has an ordering. The set has no first/last point. The set is connected. The set contains a ...
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Show $\mathbb Q^c\;\cap\;(0,1)=\left\{x+y:x\in\mathbb Q, 0<x<\frac 12 \text{ and } y \in\mathbb Q^c,0<y<\frac 12\right\}$ [closed]

Denseness of $\mathbb Q$ : Let $\mathbb Q^c$ be the set of irrational numbers in $\mathbb R$. Show $$\mathbb Q^c\;\cap\;(0,1)=\left\{x+y:x\in\mathbb Q, 0<x<\frac 12 \text{ and } y \in\mathbb Q^c,...
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sequence of functions not converging to $f$ are contained in sigma algebra

Given $f_{i}, f,(X, A) \rightarrow (\mathbb{R}, B(\mathbb{R}), i \in \mathbb{N}$ show that $$\{x \in X: f_{i}(x) \not\rightarrow f(x)\} \in A$$ Attempt: We know that $f_{i}(x) \rightarrow f(x) \...
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-2 votes
1 answer
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Solving $(p-n)\bmod\:\left(\sqrt{n}\right)\:=\:0$ for $n$ [closed]

I have the equation $$(p-n)\bmod\:\left(\sqrt{n}\right)\:=\:0$$ where $p$ is known. n is a perfect square. Is there a fast algorithm or method to find solutions to this equation?
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Prove that $\forall a,b {\in} \mathbb{Q} \:\:\exists x {\in}\mathbb{R}{\setminus} \mathbb{Q}\:\:a \lt x \lt b$

I'm trying to use cardinalities to prove that $$\forall a,b {\in} \mathbb{Q} \:\:\exists x {\in}\mathbb{R}{\setminus} \mathbb{Q}\:\:a \lt x \lt b.$$ Is the following proof of correct? Let $ I = \...
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I’m using what I want to demonstrate? Inverse of the product equals product of inverses

I want to prove that the inverse of a product equals the product of inverses, but I’m not quite sure if to I am using what I want to demonstrate. If so, can you explain me a way to avoid this kind of ...
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What does the $I$ mean in Tensor notation?

I'm quite new to tensors. I notice many math sources show a tensor, T, in this way for example: $$ T \in \mathbb{R}^{I_{1} \times I_{2} \times \ldots \times I_{N} \times K}$$ I believe that $I$ is a ...
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4 votes
1 answer
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Prove point wise inequality

How to prove: $$|x+y| \cdot \mathbb{I}_{\{|x+y| \ge 2a \}} \le 2\big( |x| \cdot \mathbb{I}_{\{ |x| \ge a\}} + |y| \cdot \mathbb{I}_{\{|y| \ge a\}} \big) $$ My attempt, I know that $ \{ |x + y| \ge ...
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Hausdorff measure as a limit

We define $H_{\delta}^{d}(S):= \inf\Big\{\sum_{i=1}^{\infty}diam(U_{i})^{d}: S \subset \bigcup_{i=1}^{\infty}U_{i}, diam(U_{i}) < \delta\Big\}$ where $diam(U):= \sup\{d(x,y): x,y \in U\}$. $$H^{d}(...
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Computation and understanding some wording around the Hardy Littlewood Maximal function

I have just started to read Stein's Singular Integrals and Differentiability properties of functions. The Hardy-Littlewood maximal function has just been introduced i.e. $$M(f)(x):= \sup_{r > 0} \...
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How to turn a finite union of half-open intervals into a disjoint union of half-open intervals?

Let $a<b$, $c< d\in\mathbb{R}$ and $(a,b]\cap(c,d]\ne\emptyset$. Non-disjointness implies that $c<b$ and $a<d$. There are two possibilities either one interval is contained in the other or ...
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The sequence $(\frac {\sin nx}{ \sqrt{n}})$ is uniformly convergent on $\mathbb R$

Show that the sequence $(\frac {\sin nx}{ \sqrt{n}})$ is uniformly convergent on $\mathbb R$. $|\sin nx / \sqrt{n}| \leq |1/ \sqrt{n}|$ $\forall x \in \mathbb R$ since $|\sin nx| \leq 1$. Let us ...
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-2 votes
1 answer
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Is it true that you can add empty sets as $∅ + ∅ = ∅$? [duplicate]

Is it true that you can add empty sets as $∅ + ∅ = ∅$? More specifically, Letting, $∅\neq A\subseteq \Bbb{R}$ and $∅\neq B\subseteq \Bbb{R} $ and assuming that $A$ and $B$ are each bounded by their ...
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2 votes
2 answers
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Finding the set of parameters for which the inequality holds for all $x,y$

I encountered the following question about inequalities which I am curious how to solve. The simplest case is to consider the inequality $$|x|+|y|+|x+y|+ax+by\geq 0$$ where $x,y,a,b\in\mathbb{R}$. The ...
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0 answers
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Converse of Bolzano Weierstrass Theorem

Bolzano Weierstrass Theorem (for sequences) states that Every bounded sequence has a limit point. However the converse is not true i.e. there do exist unbounded sequence(s) having only one real limit ...
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Exercise 6, Section 2.2 of Hoffman’s Linear Algebra

(a) Prove that the only subspaces of $\Bbb{R}$ are $\Bbb{R}$ and the zero subspace. (b) Prove that a subspace of $\Bbb{R}^2$ is $\Bbb{R}^2$, or the zero subspace, or consist of all scalar multiples of ...
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Given $0\leq a\leq b\leq c$ and $b > 0$, prove that: $2\sqrt{\frac{a}{c}}\leq \frac{b}{c} + \frac{a}{b}\leq 1 + \frac{a}{c}$

This is an exercise that got me stuck. It's from the book "Functions of Several Real Variables 1.1 - exercise 10" by Martin Moskowitz and Fotios Paliogiannis. Exercise: Given $0\leq a\leq b\...
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4 votes
1 answer
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Find real number $x$ such that both $x+\sqrt{2022}$ and $\frac{3}{x}-\sqrt{2022}$ is an integer

I have some difficulty to do this problem: Find real number $x$ such that both $x+\sqrt{2022}$ and $\frac{3}{x}-\sqrt{2022}$ is an integer My attempts: I tried to addition and multiple $x+\sqrt{2022}$ ...
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All open convex sets are homeomorphic [duplicate]

Prove that all the open convex (non-empty) subset of $\mathbb{R}^n$ are homeomorphic to each other. First of all I'm not sure if the statements is correct in the first place or not. I sketched a proof,...
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Borel's Hierarchy of Real numbers

I'm trying to find the place of the Real numbers with the usual topology in the Borel Hierarchy, $\Delta_{3}^{0}$ is what my intuition says since $\mathbb{Q}$ is $F_{\sigma}$ and $\mathbb{I}$ is $G_{\...
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2 votes
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Why Real Numbers are called Real Numbers?

I have been searching on internet for the meaning that why real numbers are called real numbers, I found many information on different websites like in https://www.cuemath.com/numbers/real-numbers/ , ...
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Proof $ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le \lfloor{x + y}\rfloor $

I want to prove that $ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le \lfloor{x + y}\rfloor $, I started proving but I got stuck. Let $ x, y \in \mathbb{R} $. Therefore: $ \lfloor{x}\rfloor \le x $ $ \...
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1 answer
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Definitions of subtraction and division

I'm reading some lecture notes, but I don't understand their definitions of subtraction and division. It states that subtraction is the inverse operation of addition, which I believe, but states that ...
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1 vote
1 answer
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Sets with Unique Subset Summing to Every Real

Do there exists sets of reals such that every real has a unique subset that sums to it. Formally, do there exists sets $S\subset\mathbb{R}$ such that every $r\in\mathbb{R}$ has a unique (up to ...
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Which Rearrangements Always Leave a Sum Fixed?

The Riemann Rearrangement Theorem is very well-known. It states that a conditionally convergent series may have its terms permuted to give any real sum as well as $\infty$ and $-\infty$ and we can ...
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2 answers
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Terence Tao, Analysis I, Theorem 5.5.9

I am working through Tao's analysis and I am trying to prove theorem 5.5.9 However, I have some trouble with the following line: "Let $n \geq 1$ be a positive integer. We know that $E$ (where $E \...
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3 votes
1 answer
470 views

Is the following statement true: "Any algebraic number can be raised to some integer power and become rational"?

I was recently reading a math book that was listing facts about $\pi$ and it said: "$\pi$ is irrational, meaning it cannot be expressed as a fraction. $\pi$ is also transcendental, meaning it is ...
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1 answer
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This formula creates all real numbers, no?

Claim: $r = \dfrac z { 10^n}$ given: $r ∈ ℝ, z ∈ ℤ, n ∈ ℕ.$ $r$ is an element of the set of real numbers and can therefore be expressed as a decimal numeral. $z$ is an element of the set of integers ...
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2 votes
0 answers
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Who wins the Even/Odd game, and by how much?

The Even/Odd game is defined as follows: Both players pick a number in $(0,1]$ for positive $y$. Even's number is called $E$. Odd's number is called $O$. The scorer number is $\Big \lceil \dfrac{E+y+1}...
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1 answer
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Smallest value of $n \in \mathbb{N}$ to guarantee a property. [closed]

The real numbers $x_1, x_2, ..., x_n$ are such that there are two between them, say, $x$ and $y$ for which $|x − y | < (2 - \sqrt{3})| 1 + xy |.$ What is the smallest value of $n \in \mathbb{N}$ ...
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2 votes
1 answer
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Unclear of Meaning of Notation $\cap _{n=1}^\infty (0,1/n)=\varnothing$

I am a bit unclear about the notation involved in the following question of my textbook. I am concerned with understanding what is asked by the question - please do not give the answer. $\textbf{1.4.3....
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Most curves in $\mathbb{R}^2$ miss $\mathbb{Q}^2$

I would like a formal statement for the following fact. Let $\gamma$ be an analytic curve from the open interval $(-1, 1)$ to $\mathbb{R}^2$. With probability $1$ (in some sense), the image of the ...
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Formal definition of Real Numbers

I was recently getting started with mathematic formal notation by defining sets of numbers. In Wikipedia, there are some of them definited as: $\mathbb{Z} = \{\ldots -3,-2,-1,0,1,2,3\ldots\}$ $\mathbb{...
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3 votes
1 answer
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Is there a topological way to see that $\mathbb R$ and $\mathbb R^2$ have equal cardinality?

I know the "interleaving" proof sending the tuple $(0.x_1x_2\dots,0.y_1y_2\dots)\mapsto0.x_1y_1x_2y_2\dots$, showing $(0,1)\cong(0,1)^2$, which can be extended to $\mathbb R\cong\mathbb R^2$....
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5 votes
1 answer
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Contradiction of axioms of real numbers

I am just starting out in real analysis, so please bare with me. My questions concerns three specific properties of the real numbers, at least as far as i understand them. Those are: The natural ...
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Let in domain $G⊂R^2$, the $ f:G→R, f∈C^1(G)$, and $\frac{ ∂f}{∂y}(x,y)≡0$ in G. Is it possible to assert that $f$ does not depend on $G$?

Question : Let in the domain $G\subset \mathbb{R}^2$, the function $f:G\rightarrow R, f\in C^1(G)$, and $\frac{\partial f}{\partial y}(x,y)\equiv0$ in G. Is it possible to assert that the function $f$ ...
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0 answers
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Applying the axiom of choice on the reals to use transfinite induction

Transfinite induction is the extension of mathematical induction to well-ordered sets, and the axiom of choice, from what I've read (although haven't seen examples or applications of), may be applied ...
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8 votes
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$A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$?

If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$? I think the answer is yes, but I can't construct it. If $A$ is odd number set, ...
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Prove that $D+(-D)=\tilde{0}$ for every Dedekind cut $D$ ("Supplement for Measure, Integration & Real Analysis" by Sheldon Axler).

I am reading "Supplement for Measure, Integration & Real Analysis" by Sheldon Axler. I proved 0.24(f). But I am not sure that my proof is ok or not. Is my proof ok or not? Even if my ...
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Let $f:\Bbb N^2\to\Bbb R:(m,n)\mapsto a^mb^n$, with $0<a<1<b$. Is $\operatorname{im}\! f$ dense in $\Bbb R_{>0}$?

Given real $a$ and $b$ with $0<a<1<b$, can every positive real number be arbitrarily well approximated by a number of the form $a^mb^n$ ($m,n\in\Bbb N$), provided that $a^mb^n=1$ only when $m=...
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4 votes
1 answer
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Dividing real number into two sets

I wonder the following question: Is there a partition of $\mathbb{R}$ into two disjoint subsets $A$ and $B$ such that $B$ satisfies $B=A+A$, namely $B=$ {$ x+y|x,y \in A $}?. Here, "partition ...
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0 votes
1 answer
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In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the ...
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A positive integer gets reduced by nine times when one of its digits is deleted and the resultant number is divisible by 9.

The question says A positive integer gets reduced by nine times when one of its digits is deleted and the resultant number is divisible by 9. Prove that to divide the resultant number by 9, it is ...
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0 votes
1 answer
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If $f$ non-negative and bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}}f^{2} d \lambda < \infty$

I am trying to show if $f$ is a non-negative function that is bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}} f^{2} d \lambda < \infty$ Where d$\lambda$ denotes ...
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10 votes
1 answer
411 views

Can the real numbers be equally split into two sets of same measure?

The rational numbers $\Bbb Q$ are dense in $\Bbb R$, but they are still a set of measure zero, i.e. $$\begin{align} \mu(\Bbb Q \cap [a,b]) &= 0 \\ \mu((\Bbb R\!\setminus\! \Bbb Q) \cap [a,b]) &...
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1 vote
1 answer
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What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound?

What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound? This is an on-a-review sheet for my final. I thought the completeness axiom ...
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