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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.

0
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3answers
32 views

Show set of all real number in ($0,1$) with base $10$ decimal expansion contains no $3$s or $7$s is uncountable

here is the question: Show set of all real number in ($0,1$) with base $10$ decimal expansion contains no $3$s or $7$s is uncountable My thoughts: to show it's uncountable, we should map it to an ...
-1
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1answer
33 views

Probability two numbers add such that the number of digits increase by at least 1?

Take two number $n_1, n_2 \in \mathbb{Z}$ where $n_1$ is known but $n_2$ is not, what is the probability that, $$ \frac{n_1}{10} \leq \frac{n_1 + n_2}{10} $$ Given that $p_1 < n_1, n_2 < p_2$ ...
1
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1answer
22 views

Prove that if $x =\frac{p}{q} \in (0, 1], q > 1$, then the period $P$ of repeating digits of $x$ is in fact less than or equal to $q − 1$.

Prove that if $x =\frac{p}{q} \in (0, 1]$ is a rational number, $q > 1$, then the period $P$ of repeating digits in the decimal representation of $x$ is in fact less than or equal to $q − 1$.
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0answers
33 views

Using the decimal representation of real numbers, construct an injection from $[0, 1] \times [0, 1]$ to $[0, 1]$. [duplicate]

Using the decimal representation of real numbers, construct an injection from $[0, 1] \times [0, 1]$ to $[0, 1]$. I don't fully understand how a 2D mapping 1D space can be one-to-one. The question ...
1
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2answers
34 views

How does the well-orderedness of the set of natural numbers follow assuming the inductive-set definition of natural numbers

Assume that $\mathbb R$ is an ordered field (i.e. $\mathbb R$ is a model of real numbers). We define the set of natural numbers $\mathbb N$ as the smallest inductive set containing $1_\mathbb R$ (...
5
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2answers
77 views

Does there exist a measurable subset $A \subset \mathbb{R}$, such that $\mu(A)$ is finite, but $\mu(\{a+b|a,b\in A\}) = \infty$?

Does there exist a measurable subset $A \subset \mathbb{R}$, such that $\mu(A)$ is finite, but $\mu(\{a+b|a,b\in A\}) = \infty$? Here $\mu$ stands for Lebesgue measure. If such subset exists, it can ...
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3answers
84 views

What is the problem with my “proof” that $\mathbb R$ is countable?

The problem I wanted to answer is Determine whether or not the set of irrationals, $\mathbb R \backslash \mathbb Q$, is countable. My attempt. For nicer notation, let $\mathcal I := \mathbb R \...
2
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0answers
72 views

“Recursive” Sequence Reaching Every Open Interval

Let $x_0$ denote the position of a particle at some point on a number line. From $x_0$, it can move to either the point $a-a^2+ax_0$ or to the point $x_0-ax_0-a+a^2$, for some fixed $0<a<1$. ...
-1
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1answer
54 views

Need to find the formula ($n$th term) of sequence [on hold]

I have been going over this sequence for weeks now and nothing. If anyone can help at least get a bit closer to the formula I would appreciate it. The sequence is as follows $1 , 5 , 49 , 820 , 21076$...
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1answer
22 views

Set with elements in any open interval

Suppose a set of real numbers has infimum b and superemum a. Further suppose that if x and y are within the set, then another real number between x and y is also within the set. Must it be true that ...
0
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2answers
48 views

Infinite numbers of decimals for a finite point in a line

Recently I started studying real analysis. In the beginning itself I was introduced to numbers which can't be represented as ratios of other natural numbers. But before studying them I had doubts ...
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0answers
9 views

Prove that in the cartesian plane, the set of all half open rectangles form a semi-ring.

The only axiom I am having trouble with is the one which involves writing the set difference as a finite union of disjoint elements in the semiring.. I can draw pictures and convince myself but I don'...
7
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1answer
47 views

Remove any number and the remaining numbers can be partitioned into two subsets of equal sum; prove all numbers are equal.

Supposed I have a list of $n$ real numbers, where $n$ is odd. The list is constructed such that I can remove any arbitrary number from the list, and the remaining numbers can be partitioned into two ...
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0answers
33 views

Does the completeness of $\mathbb{R}$ mean the metric space $(\mathbb{R},d_E)$ over the field $\mathbb{R}$ complete?

As far as I know, completeness is a property of topological vector space whose topology induced by a metric. While $\mathbb{R}$ is a field, generally speaking, a set, rather than a space. But ...
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1answer
34 views

How to prove the following statement from the solution manual of Convex Optimization by Boyd & Vandenberghe?

This question is related to Problem 3.51 of the book. Consider a polynomial $p(x)$ which has $n$ roots i.e. $$p(x)=(x-s_1)(x-s_2)\cdots(x-s_n)$$ where we assume w.l.o.g. that $s_1\leq s_2\cdots \leq ...
1
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1answer
54 views
+150

Abstracting Magnitude Measurement Systems (i.e. subsets of ${\mathbb R}^{\ge 0}$) via Archimedean Semirings.

I did some googling but could not find any easily accessible theory so I am going to lay out my ideas and ask if they hold water. Definition: A PM-Semiring $M$ satisfies the following six axioms: (1)...
10
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1answer
89 views

Are “most” sets in $\mathbb R$ neither open nor closed?

It seems intuitive to believe that most subsets of $\mathbb R$ are neither open nor closed. For instance, if we consider the collection of all (open, closed, half-closed/open) intervals, then one ...
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2answers
63 views

A question about Borel sets

I have a question regarding the Borel sequence of real sets. Given the collection $\cal{F}_\sigma$ as sets that can be expressed as the countable union of closed sets and $\cal{G}_\delta$ as the ...
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0answers
53 views

Given that there are real constants $a, b, c, d$ such that $\lambda x^2 +2xy + y^2 =(ax+by)^2 +(cx+dy)^2$ holds for all $x,y \in \mathbb{R}$. [closed]

Given that there are real constants $a, b, c, d$ such that the identity $\lambda x^2 +2xy + y^2 =(ax+by)^2 +(cx+dy)^2$ holds for all $x,y \in \mathbb{R}$. This implies (i) $\lambda =-5$ (ii) $\lambda \...
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1answer
36 views

Spivak Ch. 1 Q20

Prove that if $\ |x-x_0| < ε/2$ and $|y-y_0| < ε/2 $ then $|(x+y) - (x_0 + y_0)| < ε$ $|(x-y) - (x_0 - y_0)| < ε$ So far I've just gotten $|x-x_0| - |y-y_0| < ε$ How do I deal ...
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1answer
34 views

Real numbers axioms proof, is this correct?

What do you think is my proof correct? ...
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0answers
36 views

Maths problem fractions [closed]

How is 8/27 + 19/27 not 27/27 can anyone tell me what I’m doing wrong thanks! Idk know how to possibly do this lol
1
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1answer
53 views

Are there countably many numbers than can be described?

I can explain to you every natural number (in theory) in the sense, that I could describe it and you would know exactly which number I'm talking about, e.g. by writing it down, this can be done in a ...
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1answer
26 views

Equality of real numbers : transitivity holds or not with this definition ??

I was reading this book called "The Taylor series" by Paul Dienes. He wrote and I quote, "We say that the positive real number $a=a_0.a_1a_2a_3a_4... $ Precedes $b=b_0.b_1b_2b_3.. $ if $a_0<b_0$ ...
0
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0answers
50 views

Limitations of Marc-32 machine numbers

I am in doubt whether I understand Marc-32 machine numbers limitations correctly, however, before I try to explain my questions, I will just shortly explain how one find Marc-32 numbers (to ensure ...
0
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0answers
42 views

Solving $\left\{\frac{1}{1-|a|^b}\right\}^{1/b} a= k$

I seek real-valued solution the following equation for $a$: $$\left\{\frac{1}{1-|a|^b}\right\}^{1/b} a= k, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$ here $k$ and $0<b\leq2$ are given. I tried solving ...
3
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1answer
56 views

How to prove $\sqrt{2}\in \Bbb{R}$ with Dedekind cuts?

Problem statement: Prove that $\sqrt{2}\in\Bbb{R}$ by showing $x\cdot x=2$ where $x=A\vert B$ is the cut in $\Bbb{Q}$ with $A=\{r\in\Bbb{Q}\quad : \quad r\leq 0\quad \lor \quad r^2\lt 2\}$. Denote the ...
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2answers
52 views

Need help finding the formula (nth) term of the following sequence [closed]

So my friend gave me this sequence $\frac 32$ $\quad$ $ \frac 54$ $\quad$ $\frac {21}{16}$ $\quad$ $\frac {45}{32}$ Each of these numbers corresponds to n = 2 , n = 4 , n = 6 ... so to even n ...
5
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4answers
81 views

Does there always exist a strictly increasing function from $\mathbb{R}$ to $\mathbb{R}$ less a countably infinite set?

Given some countably infinite set $Z$, can one always construct a strictly increasing function whose domain is all of $\mathbb{R}$, and whose codomain is $\mathbb{R} \setminus Z$? Specifically, I mean ...
2
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0answers
82 views
+50

Calculating the boundary of a specific subset of real numbers

Consider the subsets {$J_m| m=0,1,2,...$} of $\mathbb R$ defined by $$J_m:= \bigcup_{s=0}^{m-1} \bigcup_{n\in \mathbb Z} \left(\frac{3^m(2n+1)+2\times 3^{s}+1}{3^m},\frac{3^m(2n+1)+2\times 3^{s}+2}{3^...
0
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1answer
27 views

Property of real exponent

I'm trying to prove that even for real exponent we have that $a^{x_1+x_2}=a^{x_1}a^{x_2}$, for every $x_1,x_2\in\mathbb{R}$ and $a>0$. In other words, I have to show this: $$\left(\lim_{\mathbb{Q}...
0
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1answer
29 views

How to explain that the the following points are optimal in a region?

I have a region defined by following inequalities $$y\leq \min(a,b)\tag{1}$$ $$x\leq\frac{cd}{c+d-1}\tag{2}$$ $$\frac{c+d-1}{c}x+\frac{d}{\min(a,e)}y\leq d\tag{3}$$where $a,b,c,d,e$ are positive ...
2
votes
1answer
58 views

Do unexpressible numbers exist? [closed]

I just learned about the difference between transcendental numbers and irrational numbers (I guess I had been mis-educated into thinking they were the same thing) and it made me wonder if there is ...
1
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1answer
81 views

Are random numbers transcendental?

Do I see it right that a randomly chosen number $a$ of the form $ a = 0.abc...xyz$ with random digits $a,b,c,x,y,z$ an approximation of a transcendental real number is? It is an approximation of a ...
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2answers
27 views

LUB (if it exists) of a complete set belongs to that set: Validity

By LUB I mean the least upper bound of the set. And the definition of complete set I am using is that every Cauchy sequence in that set must converge in that set. So by these two assumptions. I cannot ...
0
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0answers
45 views

Connectedness on the real line

This is rather long proof of equivalence of the definitions of connectedness on the real line. Because I am self-learning, I would really like to know if my proof contains any flows or logical ...
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3answers
36 views

If $x\in [0,1)$ then there exists $n\in \mathbb N$, $x\leq 1-\dfrac{1}{n}$ ?

I'm trying to show that $\bigcup_{n\in \mathbb N^*}[0,1-\dfrac{1}{n}]=[0,1)$ and I'm stuck at the following step: If $x\in [0,1)$ how to justify that there exists $n\in \mathbb N$ such that $x\leq 1-...
2
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1answer
50 views

Does there exist an interval on the reals, however small, in which every number is irrational? [duplicate]

Since the irrationals are uncountably infinite, I've always imagined them "filling the space" between the rational numbers, but does that make sense? Or is it the case that any two irrational numbers ...
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10answers
125 views

Square root of a squared number: $\sqrt{a^2} = a \in \mathbb R$? [duplicate]

True or False $$a\in \mathbb R\implies \sqrt{a^2} = a$$ a positive or negative, will always be equal to a, so for me it is true, but the teacher says that the expression is false, but I can not ...
1
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3answers
54 views

Why is this not a choice function from subsets of $\Bbb R$?

A definition of the Axiom of Choice mentions that: [...] no choice function is known for the collection of all non-empty subsets of the real numbers --Wikipedia However, for all non-empty ...
1
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2answers
59 views

$f(x)=x^3$ is convex nor concave without to derivate

Drawing the graph of a function is not intuitive. For example, the function $f(x)=x^2$ is special, we can say that given two points of the function the straight line that join those points is above ...
1
vote
1answer
77 views

Prove $\frac{b-a+1}{ab} \leq \sum_{i=a}^b \frac{1}{i^2} \leq \frac{b-a+1}{(a-1)b}$ [closed]

Given $b>a>1; \ a,b \in Z^+$. Prove $\frac{b-a+1}{ab} \leq \sum_{i=a}^b \frac{1}{i^2} \leq \frac{b-a+1}{(a-1)b}. \ $ I mostly care about the second inequality. Thank you!
1
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2answers
42 views

Is $2^n$ always dominate $n^k$, where $k$ is some fixed natural number

Is $2^n$ always dominate $n^k$, where $k$ is some fixed natural number. I have hard time thinking about this? My friend say yes but we are not able to come up with any proper proof and it also feels ...
4
votes
1answer
65 views

Finding minimum value of $\mu$ in cubic $x^3-\lambda x^2+\mu x-6=0$

If $\lambda,\mu$ are the real number such that, $x^3-\lambda x^2+\mu x-6=0$ has its real roots and positive, then the minimum value of $\mu$ is? My attempts: As it has real and positive roots its ...
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votes
1answer
36 views

A little question on rationals and open sets in $\mathbb{R}$.

Let $A$ be an open set of real numbers, and $D$ be the set of rationals in $A$. For every $d\in D$, let $J(d)$ be an (arbitrary) open interval such that $d \in J(d) \subseteq A$. Is it true that $\...
2
votes
3answers
52 views

$x \neq 0$ implies that $\frac{1}{\frac{1}{x}} = x$.

I'm having trouble proving this theorem from Rudin: that if $x \neq 0$ then $\frac{1}{\frac{1}{x}} = x$. Rudin seems to solve this by referring to an earlier result that $xy = xz$ for $x \neq 0$ ...
2
votes
1answer
44 views

Can any computable real number be represented as the sum of some integer plus some rational number times some other computable real number?

It's known that imaginary numbers can be expressed as vectors such as $\alpha + \beta i$. My question is: Can you represent a computable real number with a similar algebraic definition? For example ...
0
votes
5answers
60 views

How would you prove that $\mathbb{Q} - B$, where $B$ is a finite subset of $\mathbb{Q}$, is dense in $\mathbb{R}$?

I had a problem in my book asking to prove if various sets were dense in $\mathbb{R}$. This set I was working with ended up being equivalent to $\mathbb{Q} - \{0\}$ , and I realized $\mathbb{Q}$ ...
0
votes
3answers
50 views

$5$ real numbers tied by two equations: understanding a proof of the related problem.

If $a,b,c,d,e$ are real numbers such that $$ \left\{ \begin{array}{lcl} \phantom{0}40-e &=& a+b+c+d, \\ 400-e^2 &=& a^2+b^2+c^2+d^2, \end{array} \...
2
votes
2answers
55 views

Binomial expansion lower bound $A^n + B^n \le (A+B)^n$ for non-integer $n$

By the Binomial expansion for integer powers, $$ (A+B)^n = \sum_{k=0}^n {n\choose{k}} A^{n-k} B^{k}$$ (I'm assuming $A,B\ge 0$) and so we get the easy estimate $A^n + B^n \le (A+B)^n$ for any ...