Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
2,284 questions
0
votes
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
votes
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
vote
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 $q − 1$.
-1
votes
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
vote
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
votes
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 ...
0
votes
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
votes
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
votes
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
$1 , 5 , 49 , 820 , 21076$...
0
votes
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
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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
vote
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
votes
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 ...
0
votes
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 ...
-3
votes
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 \...
0
votes
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 ...
0
votes
1answer
34 views
-3
votes
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
vote
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 ...
0
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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 $$J_m:= \bigcup_{s=0}^{m-1} \bigcup_{n\in \mathbb Z} \left(\frac{3^m(2n+1)+2\times 3^{s}+1}{3^m},\frac{3^m(2n+1)+2\times 3^{s}+2}{3^...
0
votes
1answer
27 views
Property of real exponent
I'm trying to prove that even for real exponent we have that $a^{x_1+x_2}=a^{x_1}a^{x_2}$, for every $x_1,x_2\in\mathbb{R}$ and $a>0$.
In other words, I have to show this:
$$\left(\lim_{\mathbb{Q}...
0
votes
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
vote
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
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 ...
-1
votes
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
vote
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
vote
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
vote
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 ...
-1
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 ...
