For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag. Do not use this tag for questions on solving equations.

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28 views

Can a change of basis modify irrationality/transcendance?

Fix a real number $x$. We can consider its binary expansion, for instance $x = (0.01101001100101101001011\ldots)_2$. Now we consider the real number $y = (0.01101001100101101001011\ldots)_{10}$ : we ...
0
votes
2answers
44 views

How to show that $e^{-\frac{1}{x}}<x^n$ within $0<x<\delta$?

I would just like to show that, given a positive integer $n$, it is possible to find a positive real number $\delta$ such that $$e^{-\frac{1}{x}}<x^n,~~~~0<x<\delta$$ For various values of $...
0
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1answer
21 views

Pugh's exercise on Dedekind cuts addition

I am trying to solve the following exercise: Let $x=A|B$ and $x'=A'|B'$ be cuts in $\mathbb{Q}$. Show that although $B+B'$ is disjoint from $A+A'$, it may happen in degenerate cases that $\mathbb{Q}$ ...
-1
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0answers
34 views

What would be the fastest analysis to determine what type of number 707 is. [on hold]

Bare in mind that I have about 40-60 seconds to solve this with a calculator. This type of question is not to frequent but I does come in upper level tests: What type of number is 707? a) Evil b) ...
3
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1answer
45 views

$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $ A^c = $ those element of the universe that are not in A. $ \Bbb R =$ ...
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0answers
29 views

proving the axiom of addition and multiplication for real numbers $a$,$b$ and $c$

I have learnt about the axioms of multiplication and a few of addition of real numbers but I still have problems with proving the uniqueness of the equalities (i) $a+b+c=a+c+b=b+a+c=b+c+a=c+a+b=c+b+...
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0answers
30 views

P(P(N)) is equinumerous to the set of real functions

I need to show that (the power set of power set of N) P(P(N)) is equinumerous to the set of real functions. I know that P(N) is equinumerous to R, thus it is equivalent to show that $\{0,1\}^R $ is ...
1
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1answer
51 views

Proving that $\sin(1/x)$ is not continuous at 0

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ My proof goes like this: for the function to be continuous ...
9
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2answers
828 views

How Can You Define Successors Over The reals?

I've been going through Introduction To Modern Set Theory by Judith Roitman, and am confused by her exposition of well orderings. She gives the following proof that every element $x$ of a well-ordered ...
1
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0answers
70 views

Brief overview of the foundations of math?

I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don'...
1
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1answer
39 views

Is the density of $\mathbb{Q}$ in $\mathbb{R}$ equivalent to the Least Upper Bound Property of $\mathbb{R}$?

Consider the following two assertions: (Least Upper Bound Property) If $S$ is a subset of $\mathbb{R}$ which has an upper bound, then $S$ has a least upper bound. (Density of $\mathbb{Q}$ in $\...
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2answers
36 views

How to separate real and imaginary parts of an expression

i am taking a course on complex numbers and I need to know how to separate the real and imaginary parts of a trigonometric expression like 1) $$\cos^{-1}(ix)$$ 2) $$\sin^{-1}(e^{i\theta})$$
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3answers
34 views

Show the countability of the non-zero elements when sum of elements is less than $\infty$ on the extended reals.

Show that for $(x_{\alpha})_{\alpha \in A}, x_{\alpha} \in [0, +\infty]$ where $\sum_{\alpha \in A}x_{\alpha} < \infty$ the number of non-zero elements is at most countable. Note $A$ can be ...
0
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0answers
16 views

finding the minimum and maximum values for $(q+r)$.

given positive integer $p$, $q$, and $r$ with $p=3^q\cdot2^r$ and $100<p<1000$. find the difference between maximum and minimum values for $(q+r)$. I did find the answers by hit and trial ...
2
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1answer
20 views

If $\frac{a}{b}\in \left[\frac{p-1}{q},\frac{p}{q}\right]$, is then $b\ge q$?

Let $x=\frac{a}{b}$ be a rational number (in its lowest terms) in $[0,1]$. Let $x\in \left[\frac{p-1}{q},\frac{p}{q}\right]$ for some positive integers $p,q$ with $p\le q$. Is it true that $b\ge ...
56
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3answers
4k views

Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots ...
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votes
1answer
52 views

Is this matrix going to be real or complex?

I hope that this is the right forum where to post this question (and not here). I have a Chi-Square Kernel Matrix (using the second version, which is positive-definite) ...
1
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4answers
77 views

Constructing $\mathbb{R}$ from $\mathbb{Z}$?

I have been told that the real number line $\mathbb{R}$ can be constructed from the cartesian product $\mathbb{Z} \times [0,1)$. How exactly is that true? Surely, the cartesian product $\mathbb{Z} \...
0
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1answer
25 views

Decimal expansion of an irrational number not ending in a particular sequence

Consider the set $N$ of natural number, $N$={1,2,3,4,5,6,7,8,9,10,11...} and consider the subsequence {$N_i$} ,$i \in N$ Each $N_i$ consists of elements in ascending sequence greater than $i$ $N_1$...
2
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0answers
45 views

Could Euclid have proven Dedekind's definition of real number multiplication?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
0
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0answers
27 views

Concave optimization on closed unit ball, using penalty function

Background: I want to solve an optimization problem like $$\begin{align*}\text{minimize }&f(x)\\ \text{subject to }&\|x\| \le 1.\end{align*}$$ where $x \in \mathbb{R}^d$, $\|\cdot\|$ is the $...
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0answers
89 views

Geometric basis for the real numbers

I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients. But I never actually think of real numbers in this way. I ...
0
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2answers
47 views

How to prove $a,b,c \in \mathbb{R} \mid a+b+c \geq abc \implies 3abc(a+b+c) \geq 3(abc)^2$?

I'm working through some proofs from Cvetkovski's "Inequalities", when I came across a more difficult one (for newbies like me). Given $a, b, c \in \mathbb{R} \mid a+b+c \geq abc$, how can we prove ...
6
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2answers
242 views

What benefits do real numbers bring to the theory of rational numbers?

Complex numbers make it easier to find real solutions of real polynomial equations. Algebraic topology makes it easier to prove theorems of (very) elementary topology (e.g. the invariance of domain ...
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2answers
46 views

Help determining if a finite subset of $\mathbb R$ is closed and bounded.

If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
11
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3answers
414 views

A topology on the set of lines?

Of course any set $X$ can have a topology, but are there more natural topologies, metrics or similar on the set of straight lines in $\mathbb R^2$?
0
votes
1answer
36 views

Are these statements true or false?

$\forall$ real $r \gt 0$, $\exists$ and natural number $M$ such that $\forall$ natural numbers $n>M$, $0 \lt \frac{1}{n} \lt r$ I think I understand this up until the last part $0 \lt \frac{1}{n} ...
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2answers
41 views

Solve the equation $2^{x^2 + x} + \log_2x=2^{x+1}$

Solve the equation $2^{x^2 + x} + \log_2x=2^{x+1}$ where $x$ is real. I tried to use derivatives, without success. It's obvious that $x=1$ is solution. Also, if $x \gt 1$ then $2^{x^2 + x} \gt 2^{...
3
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1answer
44 views

Find the smallest $n$ such that an integer lies between $nx$ and $ny$ for real numbers $x$ & $y$.

Say I'm given two real numbers as inputs: $x$ and $y$, with $x < y$. I want to find the smallest natural number n such that there's at least one integer between $nx$ and $ny$ (inclusive of $nx$ and ...
8
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1answer
234 views

Is the Fibonacci constant $0.11235813213455…$ a normal number?

Recall that a normal decimal number is an irrational number $\alpha \in \mathbb{R}$ such that each digit 0-9 appears with average frequency tending toward $\frac{1}{10}$, each pair of digits 00-99 ...
0
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1answer
30 views

Show that the closure operation has the following properties

Let $E_1$ and $E_2$ be subsets of $\mathbb{R}$. I need to show that the closure operation has the following properties: a)$(E_1 \subset E_2) \Rightarrow (closure(E_1) \subset closure(E_2))$ b)$...
4
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2answers
89 views

What properties of the real numbers are almost always true and there are no (or very few) known examples of?

What properties of the positive real numbers are almost always true and there are no (or very few) known examples of? Two that come to mind are numbers that are normal in every base and numbers ...
3
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4answers
96 views

How are sets “detached” from their structure?

This question is best asked with an example. Consider the real numbers. However we construct the real numbers, the "final product" so to speak, is not just a set, but it is a complete ordered field. ...
0
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1answer
29 views

Find the Limit point of this exercises

Good morning, i'm working in this exercise and i solve this, but, i don't know it's fine, please how you can find the limit point? 1) $\left\{ 1-\frac{1}{n}\::\:n=1,2,3...\right\}$ Well, i say the ...
2
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1answer
88 views

Value of the integral $\int_{0}^{1}x^{n}(1-x)^{n}dx$

If $A=\int_{0}^{1}x^{n}(1-x)^{n}dx$ then which of the following is/are true? $1.$ $A$ is not a rational number. $2. 0<A\leq 4^{-n}.$ $3.$ A is a natural number. $4.$ $A^{-1}$ is a natural ...
2
votes
1answer
30 views

Confusion in finding left and right hand limits [duplicate]

Let $f:\mathbb R$→$\mathbb R$ defined as - $f(x)=0$, if $x$ is irrational or $x=0$ and $f(x)=1/q$, if $x=p/q$, $p\in$$\mathbb Z$ ,$q\in$$\mathbb N$, $(p,q)=1$. What are the points of continuity of $...
0
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1answer
47 views

Nearest neighbor of an irrational number

I am confused in my thoughts about the irrational numbers in real line. My confusion is: If $x\in$$\mathbb R$$-\mathbb Q$ then for $\epsilon>0$ as small as you please, the element ($x+\epsilon$) ...
5
votes
1answer
59 views

For an arbitrary uncountable set of irrational numbers, can I always construct a sequence from them that converge in the rationals?

Suppose you have a set $S$ of uncountably many irrational numbers. Can you construct a sequence of $S$ that converges to a rational number? What I have tried: Since $S$ is uncountable, the inf of ...
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1answer
88 views

Is $\pi = 3.14159…$ first-order definable in the reals?

Given first-order logic with equality and the real field $\mathbb{R} = (R, 0, 1, <, +, \cdot)$, is $\pi$ first-order definable? By first-order definable, I mean a sentence of the form $\exists x \;...
3
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6answers
708 views

Real and Non-real Numbers; Value of Zero? [closed]

There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-...
3
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4answers
37 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
0
votes
2answers
95 views

Why we square while doing the proof of √2 is irrational? [closed]

When we prove that $\sqrt 2$ is irrational by the method of contradiction, we assume $\sqrt 2$ is a rational number: $\sqrt 2 = a/b$ Squaring both sides, $2 = a^2/b^2$. Here is my question: is ...
1
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2answers
73 views

$\aleph_1$ and $\omega_1$, what are they?

Sorry for my ignorant question but.. I understand that some sources says that $\aleph_1$ is the cardinality of the real numbers (ℝ) because In set theory $$\mathfrak{c} = 2^{\aleph_0} $$ and the ...
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0answers
96 views

An equivalence relation on Dedekind cuts

The definition of a cut $(A,B)$ is: $$ \mathbb{Q} = A \cup B \\ A \ne \emptyset ,B \ne \emptyset \\ \forall a \in A,\forall b \in B,a < b \\ $$ Define a relation on the set of all cuts of $...
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2answers
40 views

Listing real numbers as countable like listing rational numbers [closed]

like proving the set of positive rational numbers are countable, where we list the rationals as the following list, why can't we represent real numbers like the same? If positive Rational numbers (p/...
4
votes
4answers
217 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
1
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1answer
78 views

approximate irrational numbers by rational numbers

I want to prove this below: (1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$...
0
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1answer
35 views

A bounded interval covered by finite open intervals

If a bounded closed interval $[a,b]$ is covered by finite open intervals $\bigcup\limits_{j = 1}^n {({c_j},{d_j})} $, I want to prove $b - a < \sum\limits_{j = 1}^n {({d_j} - {c_j})} $. It seems ...
0
votes
0answers
22 views

Poll Ranking Formula

I am having a hard time making a ranking formula for a project I am working on. Here's a practical approach and I would really appreciate any help. Lets say we have $1000 to share among 5 people,...
0
votes
1answer
39 views

Countable Set, the numbers rational

Good morning, i want to try solving this exercise: Prove $\mathbb{Q}$ is countable set. I make this: $f:\mathbb{N}\rightarrow\mathbb{Q}$ Be q $∈$ $\mathbb{Q}$, if $q>0$ then: $f\left(n\right)=\...