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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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12 views

What is the use of extension of the real and complex field?

What is the use of the extension of the real and complex field to extended real and complex field (including $\infty$)? If the extended real and complex sets are no more field then what is the use of ...
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3answers
20 views

Why is a < 0 the only solution to the following inequality?

I have been given the following equation, semi-derived from the quadratic equation: $\frac{+\sqrt{b^2-a}}{a}<\frac{-\sqrt{b^2-a}}{a}$ I need to prove that ${a}<0$ is a possible real solution ...
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2answers
28 views

Prove $\forall x,y \in \mathbb{R} \ $ $x^2+y^2 \geq x^2-y^2$

I know this may sound obvious, but I was wondering if both $x, y$ are real numbers, then why is it that $$x^2+y^2\geq x^2-y^2.$$
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1answer
28 views

Exercise regarding density of rational numbers in $\mathbb{R}$ [on hold]

Let $a,b\in\mathbb{R}$ be such that $a<b$. Show that $$ \exists\ n\in\mathbb{N}\ \ \{p/2^n : p\in\mathbb{Z}\} \cap (a,b) ≠ \emptyset. $$ I suppose two cases : $a>0$ and $a<0$. First, ...
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1answer
93 views

Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)

I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body ...
2
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1answer
40 views

Group automorphism of multiplicative group of real number field

Let $\mathbb{R}$ be the real number field and $\mathbb{R}^{\times}$ be the multiplicative group of it. $\mathrm{Aut}(\mathbb{R}^{\times})$ denotes the group automorphism of $\mathbb{R}^{\times}$. [...
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2answers
51 views

If $x$ and $y$ are irrational, then $x^y$ is irrational

I thought it was true, however my textbook claims it to be false. I need a counter example but I can't really think of one.
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0answers
55 views

Why is it necessary to show subsequence convergence in the extreme value theorem?

I’m probably making this more complicated than it needs to be, but I’m trying to figure out why it is necessary to prove the convergence of a subsequence in proving the extreme value theorem (EVT). I ...
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0answers
44 views

Can one sample rational number from $[0, 1]$ in $\Bbb R$?

I learned from real analysis course that when we devide [0,1] into irrational part $I$ and rational part $R$, we have $m(R) = 0, m(I) = 1$. So my question is, can I say we have zero probability that ...
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0answers
37 views

Can linear combinations of real numbers with integer coefficients produce every real number?

Can there exist a finite set $S$ such that the set $$S' = \{ \sum a_i \beta_i, a_i \in \mathbb Z, \beta_i \in S\}$$ Is the set of all real numbers? If no, prove the same
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1answer
47 views

Jordan's theorem implication

In class we proved we can write a function in bounded variation as the difference of two non-decreasing functions. (Jordan's theorem) My prof then said this suggests that this could be a good source ...
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3answers
273 views

Proving $∀ε>0:x<y+\epsilon ⇒ x<y$

If $x, y \in \Bbb R$ and $x<y+\epsilon$ for every $\epsilon >0$, then $x<y$. Okay so I went about this by proving the contrapositive. Proof: Let $x,y\in\Bbb R$ and let $\epsilon>0$. ...
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3answers
1k views

Do cubics always have one real root? [closed]

I've seen a few conflicting pieces of information online. So far, I know that with real coefficients there will always be one real root. But how about with complex coefficients? At very least ...
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0answers
44 views

What are the elements in set $b: S_b =\{(x,y):(x,y)\in \mathbb{R}^2 \text { and } y=x/4 \text{ and } y=4/x \}$

I am completely confused as to how to answer this. I can answer other questions in this topic but this one is confusing me. Help please? What are the elements in the set: $S_b=\{(x,y):(x,y) \in \...
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1answer
50 views

Do real numbers include all kinds of numbers as long as they are on the number line? [duplicate]

Non-computables, hyperreals, surreal, and even more obscure numbers exist between even just, e.g., 0 and 1. Does that mean they are all real numbers? Is a number real as long as it is on the number ...
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1answer
25 views

Let $h:R\to R$ be continuous on$ $R satisfying $h(m/2^{n}) =0$ for all $ m \in Z $, $ n \in N $. Show that $h(x)=0$ for all $ x \in R $.

Let $h:R\to R$ be continuous on $R$ satisfying $h(m/2^{n}) =0$ for all $ m \in Z $, $ n \in N $ .Show that $h(x)=0$ for all $ x \in R $. I think sequential criterion of continuity .but how to show $(...
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2answers
40 views

Mapping $\mathbb{R}^n$ “close” to integer points

For $\epsilon>0$ and a given $n$, define $$S_\epsilon=\bigcup_{x\in\mathbb{Z}^n}B_\epsilon(x)\subset\mathbb{R}^n$$ where $B_\epsilon(x)$ is the open ball of radius $\epsilon$ around the point $x$ ...
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2answers
10 views

Partitioning the real line

I was looking for an equivalence relation between the points of the real line such that the equivalence classes are $1$ unit long segments. I found two so far 1) Let $x_1,x_2\in\mathbb{R}$ and $$ ...
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2answers
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Reference on polynomial equation that demonstrates the history of complex numbers

Pardon my ignorance and lack of thorough understanding, but I am missing a piece of the puzzle when it comes to complex numbers and can't seem to find an answer. I have been trying to understand ...
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1answer
19 views

$BV[a,b] \subset B[a,b]$

I am a little confused about this statement. Can't we take $\tan(x)$ where $x \in [0,\pi/2]$, then it is monotonic hence of bounded variation but obviously it is not bounded as a counterexample? In ...
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1answer
37 views

Density of $\mathbb{Q}$ in $\mathbb{R}$

Let $X$ be a topological space and $Y\subseteq X$ a subset. $Y$ is said to be dense in $X$ if $\overline{Y}=X$ (where $\overline{Y}$ denotes the closure of $Y$). Now consider $X=\mathbb{R}$ (with ...
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2answers
84 views

Why would a facility use room numbers $3233$, $53$, $61$, $\infty$, $10^{100}$, $1729$, $4$, $1.6180$, $3.1415$, $1.3333$, $\sqrt{-1}$, $0$, $1$?

I was part of TCS Ignite training after my BSc graduation from July 2007 to October 2007. The training space had a numbering system for its rooms such as $$3233,\; 53,\; 61, \;\infty, \; 10^{100}, \; ...
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0answers
33 views

Least Squares Normal Equations in Explicit Form

I am struggling with the following least squares problem: Find the minimiser x* $\in \mathbb{R}^{m}$ of $$F(\textbf{x})=||\textbf{b}-A\textbf{x}||_2$$ where $A \in \mathbb{R}^{(m+1) \times m}$, $...
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3answers
60 views

Proof hint, prime numbers [duplicate]

I am trying to prove this: Prove that a positive integer $n$ is a prime number, if no prime $p$ less than or equal to $\sqrt n$ divides $n$. This is how I thought of starting: Let us assume the ...
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2answers
102 views

Showing that there exists a positive integer $t$ such that $5^t\equiv -3\pmod {2^{n+4}}$ [closed]

When I deal with number theory, I encounter a problem that seems to be easy but I can't prove. let $n$ be a positive integer, there exists a positive integer $t$ such that $$5^t\equiv -3\pmod {2^...
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1answer
45 views

Countability of Sets with rational and real numbers [closed]

Determine whether it is finite, countably infinite, or uncountably infinite. Justify $$\Big\{\Big(\frac{m}{2}, \frac{n}{3}\Big) \in \mathbb{R}^2 \mid m,n \in \mathbb{Z}\Big\}$$ The set is ...
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Is it possible to construct a four dimensional coordinate frame by changing the notion numbers that are used to express the angles between the axes?

Constructing a four dimensional coordinate system or frame is so difficult that it cannot even be imagined . One of the fundamental problems in doing so is that the angles between the four axes of the ...
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2answers
31 views

Square of hyperreal numbers

If I were to take the square of a hyperreal number that approaches 0 such as differential distance, how would I prove that it is 0?
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1answer
43 views

How to represent an arbitrary real number in $[0,1)$

Let $\{a_k\}_{k\geq0}$ such that each $a_k \geq 1$. Show that every real number $x\in[0,1)$ can be represented as $$x=\sum_{k=1}^\infty\frac{x_k}{a_1a_2…a_k}$$,where $x_k\in\{0,1,...,a_k-1\}$
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1answer
36 views

Does the following Infinite product converge to anything?

The infinite product in question is $$\prod_{n=1}^{\infty}(1-\frac{x}{n\pi})$$ I see that this product is similar to that of $$\frac{sin(x)}{x}= \prod_{n=1}^{\infty}(1-\frac{x^2}{n^2\pi^2})$$ ...
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5answers
65 views

What is the number of pairs of $(a,b)$ of positive real numbers satisfying $a^4+b^4<1$ and $a^2+b^2>1$?

The number of pairs of $(a,b)$ of positive real numbers satisfying $a^4+b^4<1$ and $a^2+b^2>1$ is - $(i)$0 $(ii)$1 $(iii)$2 $(iv)$ more than 2 Solution:We have $a,b>0$,...
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1answer
44 views

On the existence of an algebraically closed field containing other fields

This question arose while I was reading a paper I found in the web. It might be very simple, but I don't know the answer. Let $\mathbb{R}$ be the set of real numbers and $\mathbb{Q}_p$ the set of all $...
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0answers
64 views

Discrete math proof help. Let X be a real number, prove the following

$\left\lfloor3x \right\rfloor = \left\lfloor x \right\rfloor + \left\lfloor x + \frac{1}{3} \right\rfloor + \left\lfloor x + \frac{2}{3}\right\rfloor$ I get stuck a quarter way in, can someone offer ...
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1answer
17 views

Continuity of the reals in terms of decimal expansions

I was wondering about how we could prove the completeness of $\mathbb R$ when this set is defined to be the set of all decimal expressions of the form : $$\underbrace{-}_{\text{sign}}\underbrace{317}_{...
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1answer
27 views

An elementary problem concerning real nuumbers

Take any $y\in \mathbb{R}$, $r>0$ and let $I_{y,r}=(y,y+r)$. Consider the set $A=\bigcup\limits_{i=1}^\infty (i^2,i^2+1)\cup (-i^2,-i^2+1)$, how to explicitly write the set $A\cap I_{y,r}$ as a ...
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1answer
54 views

Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$

Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$. Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
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Calculate: $\lim_{m\rightarrow\infty} (1+\frac{1}{m})^m$ [duplicate]

I am trying to calculate. $\lim_{m\rightarrow\infty} (1+\frac{1}{m})^m$
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3answers
287 views

Convert numbers between 0 and infinity to numbers between 0.0 and 1.0

Can we arrange all numbers x; such that x lies between 0 and ...
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0answers
37 views

What is intuition behind uniform continuity?

What is intuition behind uniform continuity ? Actually I want to know the geometrical approach .If we discuss uniform continuity on $(- \infty ,\infty ) ,(-\infty,-1),(0,\infty)$ and $[0,\infty] $ of $...
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2answers
66 views

Direct proof of Archimedean Property (not by contradiction)

I looked at the proof of Archimedean Property in several places and, in all of them, it is proven using the following structure (proof by contradiction), without much variation: If $\space x \in \...
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0answers
28 views

Show that for every real number x, there exists an integer n so that $\ \left|n\ -x\right|\ \le\ \frac{1}{2}$

I am very new to proofs and just wanted to make sure that mine isn't absolute nonsense. Here's the question and what I wrote: Show that for every real number x, there exists an integer n so that $...
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1answer
100 views

Prove that $\lfloor n\sqrt{2}\rfloor $ doesn't satisfy any linear recurrence with constant coefficients.

Prove that $\lfloor n\sqrt{2}\rfloor $ doesn't satisfy any linear recurrence with constant coefficients. Using one of the formula for floor function, we can write: $$ \lfloor n\sqrt{2}\rfloor =\frac{-...
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2answers
28 views

What is the general term of this sequence of integers?

Consider the following sequence: 0, 1, -1, 2, -2, 3, -3, 4, -4 ... What is the general term of this sequence? Also, can its general term be expressed without using any other functions, such as floor ...
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0answers
28 views

symmetric matrix eigenvalues real (by induction)

Let's say we have an $n\times n$ symmetric matrix $A \in M_n(\Bbb R)$ and function $f(\lambda)=\det(A-\lambda I)$, where $I$ is the identity matrix. If we have some number $\lambda_0$ that is a ...
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2answers
77 views

Let $x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{…}}}}$; then the value of $(2x-1)^2$ equals…

Let $$x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{...}}}};$$ then the value of $(2x-1)^2$ equals... I don't how to start this question. Please help.
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0answers
32 views

Bijection between points on a line and the real numbers

I need some help. Quoted from Elliott Mendelson's book, "Real numbers will have to be defined in such a way that, not only are the ordinary arithmetic operations of addition, subtraction, ...
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0answers
17 views

Why the universe of neural network is a ring K?

I wish to find an algebraic or category theory approach to describe neural network in particular to have use algebraic method to extract the concept of 'synaptic weight'. I find this for the moment, ...
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2answers
41 views

On some inequality about positive numbers [closed]

I was wondering whether you can help me with the following question: Let $a,b\geq0$. Is there some $c\geq 1$ such that \begin{equation} (ab)^c\left|(a^4+1)^{\frac{-3}{4}}-(b^4+1)^{\frac{-3}{4}}\...
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1answer
86 views

Is there a sum of an uncountable set of Real numbers? [duplicate]

The addition of Real numbers is commutative, so instead of saying we can find the sum of a sequence $\{a_1,...,a_n\}$ of real numbers that are pairwise not equal, we can say that there is a sum of a ...
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2answers
51 views

Prove that if $a < b$ and $c < d$ then $a + c < b + d$

I'm having a tough time with this questions. Could someone perhaps give me a hint? I'm only allowed to use the axioms of real numbers: A1. $a + b = b + a$ A2. $a + (b + c) = (a+b) + c$ A3. $a + 0 = ...