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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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What is the minimum value of the expression? [duplicate]

Given reals $x_i \ge 0$ for $i=1,2,3,...,2n$ and $\sum_{i=1}^{2n}x_i=1$ find the minimum value of $\sum_{i=1}^{n}x_i+\sum_{i=n+1}^{2n}{x_i}^2$. I tried the case $n=1$ and got the value $\frac{3}{4}$. ...
Suprativ Mondal's user avatar
1 vote
2 answers
94 views

Necessity of universal quantifier to represent a theorem with logic symbols

I have a preference to reduce the proof steps of a theorem, and the theorem itself, into logic symbols as much as possible. Not just because it is aesthetically appealing, but because it makes makes ...
Davi1399's user avatar
3 votes
1 answer
99 views

Conditions that a sequence should satisfy to be an eventually monotone sequence

Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that: $a_n\in[0,1]$, $\forall n\in\mathbb{N}$ $\lim_{n\to\infty}a_n = 0$ $\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = 1$ $a_{n+1} \leq a_n$...
MathRevenge's user avatar
2 votes
1 answer
54 views
+100

Additive real semigroups with an unbounded below part

Look at $$(*)\;\;\; H=\{ m\sqrt{2}+k|m,k\in \mathbb{Z}, m\geq 2\}\cup (\mathbb{Z}_++\sqrt{2}).$$ It is obvious that $H$ is an additive sub-semigroup of real numbers. But, it is interesting to know ...
M.H.Hooshmand's user avatar
0 votes
1 answer
88 views

are there 2 or more irrational numbers between any 2 rationals?

… in general, but also related to a calculus problem I have before me which is about continuity. The question regards continuity wrt the function $$ f(x) = \begin{cases} x, x \in \mathbb{Q} \\ 0, ...
El Jfe's user avatar
  • 41
1 vote
1 answer
105 views

Since $0.\bar9=1$, is $0.\bar9$ an Integer?

I've seen the following proof for $0.\bar9 = 1$: $0.\bar9$ can be represented as ($0.9+0.09+0.009+\dots$) which is effectively $\sum_1^n 0.9(0.1)^{n-1}$. Taking the limit as $n$ approaches infinity ...
Grey's user avatar
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0 answers
39 views

An Algebra Question with 3 Variables and 3 Conditions to be met for Real Answers.

The question is as follows: Solve for real numbers: $$\sqrt[3]{\left(1+\sqrt x\right)\left(1+2\sqrt x\right)\left(1+4\sqrt x\right)}=1+2\sqrt y$$ $$\sqrt[3]{\left(1+\sqrt y\right)\left(1+2\sqrt y\...
VihaMA's user avatar
  • 1
0 votes
1 answer
19 views

How to more formally prove this inequality

This is a simple problem I came up with while doing another problem: Given: $n < (n + \frac{1}{2}) < y < (n + 1)$ Prove: $|y - n| > |y - (n + 1)|$ So how I proved it was simply using the ...
Bob Marley's user avatar
1 vote
2 answers
92 views

why is $L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\}$ not the set of all Dedekind cuts?

Let the set $L$ be definded as $$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$ where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$. Clearly, every member of $L$ is neither an empty set ...
Mohamed Mostafa's user avatar
5 votes
2 answers
396 views

Prenex Normal Form of a Simple Proposition Reads Strangely.

I was trying to convert a simple (true) proposition concerning the real numbers to Prenex Normal Form but arrived at a logical statement that didn't appear equivalent to what I started with. The ...
eliza1024's user avatar
-1 votes
1 answer
31 views

Consider intervals of real numbers (a,b) and (c,d) with a<c. Is their intersection open? [closed]

Consider intervals of real numbers (a,b) and (c,d) with a<c. Then: A) The intervals are disjoint B) Their intersection is not empty C) Their intersection is closed D) Their intersection is open It ...
Samu's user avatar
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2 answers
54 views

Determine the values of the real parameter $m$, so that the equation $x^4-2x^3+2mx-m^2=0$ admits four different real solutions.

the problem Determine the values of the real parameter $m$, so that the equation $x^4-2x^3+2mx-m^2=0$ admits four different real solutions. the idea I tried applying the quadric formula and got that $...
IONELA BUCIU's user avatar
  • 1,115
-1 votes
1 answer
42 views

Determine the real numbers a and b for which the relation $a+b+246=22*(\sqrt{a+1024}+\sqrt{b-1020})$ occurs [closed]

the problem Determine the real numbers a and b for which the relation $a+b+246=22*(\sqrt{a+1024}+\sqrt{b-1020})$ occurs my idea The problem would have been so easy if the numbers would be rational... ...
IONELA BUCIU's user avatar
  • 1,115
1 vote
2 answers
65 views

how to prove $\forall a_n \in \mathbb{R}, n\in \mathbb{N} \exists x \in \mathbb{R} : \sum_{k=1}^n a_k \left( x^k-\frac{1}{k+1} \right)=0 $?

I tried to prove that $\forall a_n \in \mathbb{R} , n\in \mathbb{N}$ then there exist a real root $x$ such that $$ \sum_{k=1}^n a_k \left( x^k-\frac{1}{k+1} \right)=0 $$ for example if $n=2$ and $a_1=...
Faoler's user avatar
  • 1,329
6 votes
3 answers
756 views

Where is the mistake in the argument in favor of the (erroneous) claim "every Dedekind cut is a rational cut"?

A cut is a set $C$ such that: (a) $C\subseteq \mathbb Q $ (b) $C \neq \emptyset $ (c) $C \neq \mathbb {Q} $ (d) for all $a, c \in \mathbb Q $ , if $c\in C$ and $a\lt c$ , then $a\in C $ (e) for all $c\...
Vince Vickler's user avatar
3 votes
0 answers
107 views

Proof that for all nonzero real numbers $a$, $\frac{1}{a}$ is nonzero

I was wondering if someone could check my proof that "For all $a\in\mathbb{R}$, if $a\neq 0$ then $\frac{1}{a}\neq 0$". The definitions/assumptions I am basing the proof off of come from &...
user1320946's user avatar
0 votes
0 answers
32 views

Can there be a sensible unique subset of all definable real numbers? Why is it unable to serve as a replacement of the canonical set?

I've heard that reals is the least (well, I'll say it in intentionally vague way, so it will precisely reflect my current understanding) number-like set that is closed under an operation of taking ...
Andrew Zabavnikov's user avatar
0 votes
2 answers
58 views

Proving, using Dedekind cuts, that $C(0)$ is an additive identity for addition on cuts.

My source is Franck Ayres' Modern Algebra. The author states the fact under discussion (Chapter 7, "Real numbers") but does not provide a proof. My question is about the second part of the ...
Vince Vickler's user avatar
2 votes
5 answers
403 views

Clarification about Cantor's Diagonal argument compared to Natural Numbers

I'm not a mathematician but I am a software engineering student. From what I've understood so far, the Cantor diagonal argument proves that the real numbers are infinite and uncountable. My biggest ...
peachyoana's user avatar
0 votes
0 answers
31 views

Dedekind cuts : establishing the equivalence of 2 definitions of addition on positive cuts

My source is Ayres, *Modern Algebra", $1965$ ed , ch. $7$, § "additive inverses" , p.$68$. A cut in $\mathbb Q$ is defined as a non empty proper subset $C$ of the rationals such that (...
Vince Vickler's user avatar
9 votes
5 answers
2k views

Can a straight line be drawn through a single node on an infinite square grid without passing through any other nodes?

The problem is from an advanced 8th grade math curricula, and marked with a star: *The topic is "Real numbers" The plane is covered by an infinite square grid. Is it possible to draw a ...
curioushuman's user avatar
0 votes
1 answer
53 views

$-a=(-1)\cdot a,\forall a\in\mathbb{R}$ using axioms

I've been trying to prove $$-a=(-1)\cdot a$$ for every $a\in\mathbb{R}$ using only axioms, however, every demonstration I found use one of these two properties: $$0\cdot a=0,\quad\forall a\in\mathbb{R}...
mvfs314's user avatar
  • 2,075
0 votes
2 answers
55 views

Are there any theorems that use the uncountability of the reals in their proof?

Can we use the uncountability of the reals as a tool to prove any theorem? Can we use this to calculate anything? Suppose I was trying to convince a pragmatist that uncountability is useful.
Alex's user avatar
  • 460
0 votes
1 answer
36 views

Unable to come up with correct bounds for showing sequence convergence. Analysis I, Terence Tao, Theorem 6.1.19, part (g)

I was trying to prove the following (part (g) of the Theorem): Theorem 6.1.19 (Limit Laws). Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numbers, and let $x,y$...
Paul Ash's user avatar
  • 1,350
1 vote
0 answers
51 views

Measure theory, prove the countable additivity of measure

I am reading Cohn's Measure Theory, here is an exercise from Chapter 1, Section 2. Let ($X,\mathcal A,\mu$) be a measure space, and define $\mu^{\star}:\mathcal A\rightarrow[0,\infty)$ by \begin{...
Hao Shen's user avatar
1 vote
1 answer
95 views

Prove that for any rational number $t$ , there is a solution of the equation $ax^2+by^2=t$.

Let $a$ and $b$ be two non-zero rational numbers such that the equation $ax^2+by^2=0$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the ...
Ayan Bhowmik's user avatar
-1 votes
1 answer
31 views

Complex numbers : finding real & imaginary numbers and the magnitude

enter image description here My Thoughts are: Using geometric series $S_n= a(r^{n+1} -1) /(r-1) )$ for $a=1 , r=2i , n=26$ Now as I know that $i^2=-1$ , so $i^{27} = i^3 =-1$ $\implies (2i)^{27} = (2^...
user1315456's user avatar
0 votes
1 answer
22 views

Convergence of rapidly decaying series

Let $0<\lambda_1<\lambda_2<\dots$ be an increasing sequence of positive real numbers with $\lim_{k\to\infty}\lambda_k=\infty$. Let $a_1,a_2,\dots$ be a sequence such that for all $m>0$, $$...
geometricK's user avatar
  • 4,786
28 votes
1 answer
2k views

Can a non-trivial continuous function "undo" the discontinuities of another function?

Apologies for the unclear title, I have no idea if the property I'm looking for has a better name. I'm wondering if there exists a pair of functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ such ...
TheAmazingKitchen's user avatar
2 votes
1 answer
92 views

How do I write this Theorem with quantifiers?

Here is the theorem from Steven Abbot's Understanding Analysis. Theorem. Two real numbers $a$ and $b$ are equal if and only if for every real number $\epsilon > 0$ it follows that $|a - b| < \...
Dr. J's user avatar
  • 73
6 votes
1 answer
290 views

How to construct a nonzero real number between two given nonzero real numbers?

Statement: Let $$X=$$ $$\{(a,b) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}\setminus \{0\}:a<b\}$$ There exists a function $f:X \rightarrow \mathbb{R} \setminus \{0\}$ such that for all $(a,b) ...
Mohammad tahmasbi zade's user avatar
5 votes
2 answers
225 views

How can a subset of reals not exist?

Let's take a Vitali set in a model of ZFC, then map its elements to the corresponding reals in the Solovay model and consider them as a set. We get a Vitali set in the Solovay model while it shouldn't ...
Maxim's user avatar
  • 329
3 votes
0 answers
222 views

How to use Finite Decimal Continuity to Define Real Number Arithmetic?

Here is some context for my question: In "Vector Calculus, Linear Algebra and Differential Forms" by Hubbard & Hubbard, in Appendix A.1 on real number arithmetic they construct what we ...
John Doe's user avatar
0 votes
1 answer
66 views

How to generalise the argument in Chap. 1 in Baby Rudin to show that these sets $A$ and $B$ have no largest and smallest elements, respectively?

Let $n$ be a positive integer that is NOT a perfect square, and let the sets $A$ and $B$ be defined as follows: $$ A := \left\{ p \in \mathbb{Q} \colon p > 0, p^2 < n \right\} $$ and $$ B := \...
Saaqib Mahmood's user avatar
-3 votes
1 answer
62 views

How to find the digits a, b and c? [closed]

I am trying to solve the following mathematical expression: $\frac{1}{a+b+c}=0,abc$ Where a, b and c represent digits. My goal is to find the values ​​of the digits a, b and c. I have tried to ...
golub001's user avatar
0 votes
1 answer
51 views

Why is compactness necessary for IVT

The version of IVT I am working with is as follows: Let $f\colon [a,b] \to \mathbb{R}$ be a continuous function from a compact subset of the real line, and suppose that $f(a) \lt 0$ and $f(b) \gt 0$ . ...
Micah Zarin's user avatar
3 votes
1 answer
82 views

Historically, when have the the real numbers been constructed via the "positive" (non-negative) reals first, and then usual real numbers second?

There has been something that has been bugging me for the longest time, at least since grad school. In the teaching of mathematics, during the construction of the "usual" real numbers, why ...
Rex Butler's user avatar
  • 1,622
1 vote
1 answer
44 views

What does Artin mean by "real numbers are the *only* ones needed for the usual for the usual algebraic operations?"

In page 81 of the 2nd edition Michael Artin's Algebra, he introduces fields and presents $\mathbb{R}$ as a familiar example, but goes on to say that "the fact that they are the only ones needed ...
Maqdounes cozbora's user avatar
0 votes
0 answers
29 views

Proving convergence of rational function from first principles

The full question and solutions are, here However, the bit I am confused on is how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here ...
Chen Wu's user avatar
  • 11
2 votes
1 answer
150 views

Is it true but unassertable that there are undefinable real numbers?

I know of Joel David Hamkins's analysis of the so-called "math tea argument", namely that there are undefinable real numbers. Supposedly, he debunked this argument by constructing a ...
user107952's user avatar
  • 20.9k
2 votes
3 answers
96 views

How do I understand that there is no infinitesimal element in $\mathbb{R}$?

I read a problem in my book that if there is an element $o>0$ in $\mathbb{R}$, such that for any $x>0$, we will have $o<x$, and we call $o$ as an infinitesimal element. Prove that there is no ...
Young's user avatar
  • 21
8 votes
4 answers
893 views

Is there a dimensional multiplication operation? [closed]

When expressing numbers with any unit, we know this. We can multiply and divide numbers with different types of units, but we cannot add or compare them. From Terry Tao's 2012 blog post "A ...
Bilgehan Yılmaz's user avatar
0 votes
1 answer
31 views

Appropriate model to represent negative numbers

Negative numbers can be introduced by means of temperature, but it does not make sense to multiply two negative temperatures. Moreover, it is even objectionable to say 20°C is twice as hot as 10°C. A ...
apprenant's user avatar
  • 746
16 votes
1 answer
165 views

What is the "higher cohomology" version of the Eudoxus reals?

The "Eudoxus reals" are one way to construct $\mathbb{R}$ directly from the integers. A full account is given by Arthan; here is the short version: A function $f: \mathbb{Z} \to \mathbb{Z}$ ...
user263190's user avatar
  • 1,247
1 vote
0 answers
24 views

Uncountable version Helly's theorem

I am trying to solve Exercise 7.4 of Imre Bárány's lecture note of Discrete mathematics. "Prove that Helly's theorem when family is not countable and every set in the family is compact" I ...
김현성's user avatar
0 votes
1 answer
40 views

Is this proof of the density of the rationals in the reals correct?

Theorem: For any 2 non-equal positive real numbers, there exists a rational number "between" them. $ x , y \in \mathbb{R} $, $ 0 \lt x \lt y \implies \exists q \in \mathbb{Q} | x \lt q \lt ...
Tim's user avatar
  • 53
2 votes
2 answers
178 views

Finding an irrational number between two given irrational numbers constructively

Statement: Let $$X=\{(a,b) \in \mathbb{R} \setminus \mathbb{Q} \times \mathbb{R} \setminus \mathbb{Q}:a<b\}$$ There exists a function $f:X \rightarrow \mathbb{R} \setminus \mathbb{Q}$ such that for ...
Mohammad tahmasbi zade's user avatar
-4 votes
2 answers
124 views

How can the reals be the set of all points on a number line when there exist non-constructible reals? [closed]

We are given the intuition that the reals form all the numbers on the numberline. However, this intuition wasn't working for me as the existence of non-constructible reals seems to me to imply that ...
Princess Mia's user avatar
  • 2,495
1 vote
0 answers
39 views

constructing $\mathbf R$ via binary search

Using the function $$f(x) = \begin{cases}\frac{x}{4} &\text{if }|x|\le 2\\[0.3em] \frac{|x| - 1}{x} & \text{otherwise}\end{cases}$$ one can map $\mathbf Q$ into the interval $[-1, 1]$ in a way ...
node196884's user avatar
2 votes
2 answers
64 views

Finding Maximum Value using AM-GM Inequality

Let us have a set of natural numbers $S=\{x_1,x_2,...,x_n\}$ where $n≥4$, $n$ is even, such that all $(x_i\in S)≥0$ and $\sum_{i=1}^nx_i=1$.Find the maximum value of $\sum_{i=1}^{n-1}(x_i*x_{i+1})$.My ...
20DPCO190 Amanul Haque's user avatar

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