For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

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-1
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0answers
25 views

How to solve this complex-valued exponential signal ? Help !!! [on hold]

Consider the complex-valued exponential signal $$x(t)=Ae^{αt+jωt}, α>0$$ Evaluate the real and imaginary components of $x(t)$.
0
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0answers
10 views

Show that I is a non-empty interval if every continuous function has an interval as its image

Assuming $I$ is a non-empty interval of real numbers I want to show that for any continuous function $f$ that $f(I)$ is also an interval. So given, say, $y_{1}, y_{2}$ in $f(E)$ and without loss of ...
1
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0answers
30 views

An algorithmic approach to constructing the real numbers

To specify a real number, we can describe a rule which, given any rational number, tells you whether it's Too Big or Too Small. The rule should be self-consistent, in the sense that if $a$ is Too Big ...
0
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0answers
48 views

Proving that some extremal must be a minimizer.

I've been given the following problem; Let $x = x(t) : [t_0,t_1] \rightarrow \Re$ be a curve in $C^2$ with boundary conditions $x(t_0) = x_0$ and $x(t_1) = x_1$. Consider a function; $$J[x] = ...
2
votes
2answers
44 views

completeness property of real numbers

I am studying about how a real number is defined by its properties. The three type of properties that make the real numbers what they are. Algebraic properties i.e, the axioms of addition, ...
0
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2answers
39 views

Prove the completeness of the real numbers

It confused me for a while. Here is the question. Prove that given any Cauchy sequence of reals, there exists a Cauchy sequence of rationals that converges to the same value.
0
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1answer
24 views

How to show $\log a\le n(\sqrt[n]{a}-1) \le \sqrt[n]{a}\log a$

Let $ b=\sqrt[n]{a}$. How to show: $\log a\le n(\sqrt[n]{a}-1) \le \sqrt[n]{a}\log a$? Thank you ;)
0
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0answers
19 views

Automorphism of $(\mathbb{R}, <)$ that takes a finite set of reals to a set of naturals

I was working on another of Enderton's logic book exercises (specifically, exercise 20 (b) from section 2.2, p. 102), and, as part of the exercise, he asks us to show that, for any finite set of real ...
0
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0answers
21 views

Defining multiplication of monotone increasing seuqences to be strictly monotone increasing.

I am constructing the reals as the set of equivalence classes of monotone increasing sequences of rationals. However, one problem I have is defining multiplication such that the product of two ...
0
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0answers
10 views

What is the process of finding supremum/infimum of a continuous set?

For example given a set, for constant $k\in \mathbb{N}$ $$ \{ (1 + x^2)^{-k} \ | \ x\in [-1,0) \cup (0,1] \} $$ The supremum is $1$ ( the limit as $x$ approaches $0$), but how does one derive this ...
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0answers
32 views

Question concerning defining a particular class of functions

I have a multiset of real numbers $X \subseteq \mathbb{R} $ and I want to create a class of injective function to map the elements of $X$ to the unit interval(so basically a normalization). However ...
0
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1answer
27 views

Given $x \in \mathbb{R}$ and $N \in \mathbb{N}, N>1,$ find integers $h,k$ with $0 < k \leq N $ that satisfy $\lvert{kx - h\rvert} < 1/N.$

This is a problem from Apostol's $\textit{Mathematical Analysis}$. He provides a hint: Consider the $N+1$ numbers $tx-[tx]$ for $t=0,1,\ldots,N$ and show that some pair differs by at most $1/N$, where ...
6
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1answer
55 views

Are there numbers that we can't get with a usual compass and ruler, but can get with 3D compass and ruler?

If we have unit segment, we can use a compass and ruler to make segments whose length represents many numbers (all rational, sqrt(2)), but there are "unreachable" ...
0
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0answers
29 views

Rosenlicht Chapter 1 Problem 11 Unbounded Set

I've been working on this problem for over an hour. It is from Maxwell Rosenlicht's text book "Introduction to Analysis" in the first chapter before limits are discussed. I will state the claim, show ...
2
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1answer
51 views

$|\{0,1\}^\infty| = |\mathbb{R}|$?

Let $\{0,1\}^\infty$ = {$(a_n)_{n \in \mathbb{N}}; a_n \in \{0,1\} \forall n \in \mathbb{N}$} Is there a bijection between $\{0,1\}^\infty$ and $\mathbb{R}$? I thought about something like this: If ...
2
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1answer
75 views

Are $+\infty$ & $-\infty $ elements of the real number line? [duplicate]

Can someone give an explanation/proof of whether these two numbers lie on the real number line?
0
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1answer
22 views

Trying to find base case in recursive argument that a sequence with only finitely many peak indices has a monotone subsequence.

I'm examining an argument in Fitzpatrick's Advanced Calculus that a sequence $\{a_n\}$ with only finitely many peak indices has a monotone subsequence. His argument is shown below, with a modification ...
2
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1answer
67 views

Natural and Real sets of numbers, which one is bigger than another?

From the years ago, it has always been this question in my mind which a teacher of high school talked about in a class but I never found it's correct answer. We have set of natural numbers ...
-1
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1answer
59 views

How is circle closed?

I have this thought that circle in 'real' is not a closed figure. We all know that 'pi' is irrational.And integers are nodes in a 'monstrous' line of real numbers. Irrational numbers are ...
2
votes
2answers
92 views

How to show that an infinite decimal is equal to a unique real number?

I don't understand how the proof above shows that two distinct real numbers correspond to different infinite decimal. All I got out of the explanation is given any two distinct real numbers $a$ and ...
1
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2answers
69 views

Bijection between an infinite set and its union of a countably infinite set

I have $A$ as an infinite set and $S$ as a countably infinite set, (so that means there exists a one-to-one correspondence between $S$ and $\mathbb{N}$). How do I show that there always exists a ...
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0answers
95 views

The topology generated by open intervals of rational numbers

Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$ Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of ...
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0answers
47 views

There exist a bijection between the Real numbers and points of a straight

Assuming that we are building our geometry on the axioms of Euclid/Hilbert, and using either the Dedeking or Cauchy construction of the reals, how can one prove this statement? I've looked up on the ...
0
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1answer
20 views

For every real number there are exactly two isometries of the real line that leave it fixed

I have some preliminary questions before I attempt this problem in my book. If $M$ is the metric space of all the real numbers and $x_0 \in M$, prove that there exist exactly two isometries of $M$ ...
4
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1answer
199 views

Replacing the “if $x ≤ y$, then $x + z ≤ y + z$” axiom in Reals.

How can I prove that we cannot (or maybe can) replace preservation of order under addition i.e. "If $x \leq y$, then $x + z \leq y + z$ with "if $0<x$ and $0<y$ , then $0<x+y$" in axioms ...
0
votes
2answers
79 views

How do you multiply infinite quantities?

Out of curiosity I was watching this video from njwildberger on youtube: https://www.youtube.com/watch?v=4DNlEq0ZrTo Where he says that you can't define associativity between irrational numbers ...
2
votes
2answers
40 views

Constructing $\mathbb{R}$ from $\mathbb{Q}$ using Cauchy Sequences.

Let $CS(\mathbb{Q})$ denote the set of all Cauchy sequences of rational numbers. Define an equivalence relation $\sim$ on $CS(\mathbb{Q})$ as follows: $$(x_n)\sim (y_n)$$ if $\lim_{n\to ...
0
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0answers
35 views

Question about the definiton of addition of Dedekind cuts

Given two Dedekind cuts alpha and beta, their addition is defined to be the set {a+b | a belongs to alpha and b belongs to beta}. But why does this define addition? Why does this deserve to be called ...
4
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6answers
129 views

Why is $0^0$ undefined when $x^x=1$ as $x$ approaches $0$?

This question was born in another post available here. I believe $0^0=1$, because $x^x$ is continuous as $x$ approaches $0$. Consider $\lim_{x \to 0}x^x$. Let ...
2
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2answers
68 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 ...
-3
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1answer
69 views

Someone can solve this limit? [closed]

$$f(x) = \frac{9-2\sqrt{\left\vert\,x\,\right\vert}}{3\sqrt{-x}}$$ $$\lim_{x\to-\infty} f(x) = l$$ I need the method of calculate l and solve this: (proving limit using epsilon-delta definition) ...
1
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0answers
46 views

Inverse element of “-”

What is meant by the inverse element of "-"? There is a statement in my book that says there exists an inverse element of "-" in $\mathbb{R}$ and I have to mark it true or false. I know that the ...
7
votes
5answers
631 views

Is the product of uniformly distributed numbers, uniformly distributed too?

My question is simple, I think. If we took two random natural numbers $a$ and $b$ uniformly distributed in a specific range $[c,d]$, is $ab$ a uniformly distributed too? What if $a$ and $b$ are not ...
2
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3answers
67 views

Integration of a real powered rational expression

Peace be upon you, I've encountered this pretty integral \begin{align*} \int_0^1&\frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}dx,\\ \\ &\alpha,\beta\in\Re^+ \end{align*} It seems much simpler ...
0
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2answers
53 views

Equality of Real Numbers

Is the following statement provable from the axioms of $\mathbb{R}$? If $\forall \epsilon>0$, $|r-s|\leq \epsilon$, then $r=s$.
30
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9answers
4k views

Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...
1
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0answers
59 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
1
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2answers
87 views

Measure of set of rational numbers

I find it difficult to understand why the 'size' of the set of rational numbers in an interval such as [0,1] is zero. I know that there are way more irrational numbers than rational numbers such that ...
2
votes
1answer
54 views

In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable?

I was playing with Baire's Theorem, and seemed to deduce the following: In a complete metric space $X$ that has no isolated points, any countable intersection of open dense sets is uncountable. ...
5
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2answers
330 views

Why is this argument for the reals being countable wrong? [duplicate]

After watching a proof of the set of computer programs being countable, I thought of the following argument: Consider the sets of real numbers (of the form $0.xxx...$) with $0$, $1$, $2$, $3$, $...$ ...
2
votes
2answers
49 views

Proving some statements only by the definition of Real numbers.

Let $f \colon \ [0.1] \to \mathbb R$ is monotonically increasing function and $f(0)>0$ and $f(x)\neq x $ for all $x\in [0,1]$. $$A=\{x\in [0,1] : f(x)>x \}$$ We know: every non-empty subset ...
3
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3answers
74 views

Can every uncountable subset of $\mathbb R$ be split up this way?

For me this question is like a fish that anytime when I (seem to) catch it, manages to slip out of my hands again. If $U$ is an uncountable subset of $\mathbb R$ then can it be shown that some ...
0
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1answer
48 views

Duality of zero and infinity [closed]

The duality of zero and infinity is striking: e.g. Some lengths are more zero or infinite than others, but no length is zero or infinite. If a is more infinite than b, 1/a is more zero than 1/b. ...
9
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5answers
374 views

How prove $ \sqrt{2}+\sqrt{3}>\pi$?

How prove that $ \sqrt{2}+\sqrt{3}>\pi$? Maybe some easy way?
0
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1answer
28 views

Is Champernowne's constant Liouville?

By looking at extreme spikes of Champernowne's constant and how well it's approximated by some rational numbers I think it's reasonable to think that this is a Liouville number. However, no source I ...
2
votes
2answers
36 views

Is it still possible for their to exist a well-ordering on the reals such that there is always a least next element?

Let $a \leq b$ be a well-ordering on $\Bbb{R}$. And suppose for any $a_n \in \Bbb{R}$ there exists another element $a_{n+1} \in \Bbb{R}$ such that under the given ordering $(a_n, a_{n+1})$ is empty. ...
2
votes
1answer
65 views

Characterization of dense open subsets of the real numbers

Does the complement of every dense open subset of the real numbers have Lebesgue measure $0$? This is certainly not a characterization of dense open subsets of reals, since the complement of the ...
0
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0answers
26 views

Distribution of $\lfloor n^{\log{n}} \rfloor$ modulo $q$.

Let $q$ be an arbitrary integer. I want to investigate the distribution of the set $\mathcal{S} = \{\lfloor n^{\log{n}} \rfloor : n \in \mathbb{N}\}$. After a few explicit computations with SAGE, it ...
60
votes
21answers
12k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
2
votes
1answer
91 views

Maximal model for $\Bbb R$?

I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...