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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3answers
36 views

What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$

We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R ...
1
vote
1answer
18 views

Constant raised to the power of an even or odd function

Suppose that $a$ is a positive real number, that $f(x)$ is an even function and that $g(x)$ is an odd function. Would $a^{f(x)}$ be an even or odd function? And would $a^{g(x)}$ be an even or odd ...
1
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0answers
19 views

Constraining mathematics to a subset of $\mathbb{R}$

Let's imagine we're only using rational numbers for everything in mathematics. Problems arise quite soon when you try to calculate diagonals of squares or perhaps roots of something like $f(x)=x^2-2$. ...
2
votes
1answer
27 views

Series and comparison test

If $a_n>0$ and $\sum a_n$ diverges, what can be said about $\displaystyle \sum \frac{a_n}{1+na_n}$? I cannot prove that it is convergent or divergent. I think it is convergent for some examples ...
0
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2answers
49 views
1
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3answers
32 views

Use a function to represent positive real numbers?

Is it correct to define the positive real numbers as $\{f(x) = x^2\mid x \in \mathbb R\}$?
0
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0answers
23 views

How to find the general term of this sequence?

I would like to know if there is a way to find an explicit closed-form expression for the general term of the sequence (in $\mathbb{R}$) defined by $$ \begin{cases} a_0=1\\ a_{n+1}=1+\frac{1}{a_n}\ ...
1
vote
3answers
35 views

Calculate value of a real number, considering “n” as a natural number

How could I calculate the value of the real number: $$ (1 +i \sqrt{3})^n + (1 - i \sqrt{3})^n $$ ...considering $n$ as a natural number and $i$ as the imaginary unit.
0
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2answers
57 views

Show that the following statement is a theorem.

Suppose h is not a counting number and h is greater than 1, then there is a counting number n such that h is between n and n + 1. I am working through "Creative Mathematics" by H.S. Wall. The book ...
1
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2answers
35 views

Real numbers: powers inequality

Trying to prove the following inequality from textbook. Let $x>1$ be a real number, and let $q>0$ be a rational number. Then $x^q>1$. Let $p<q$ be a rationals numbers, and let $x>1$ ...
2
votes
3answers
78 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of R and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
1
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1answer
17 views

Proof from equality

suppose this equality holds: x, y, z are real 15(x + y + z) = 12(xy + yz + xz) = 10(x^2 + y^2 + z^2 ) and at least one variables isn't zero. I need to proof that x + y + z = 4 and find the ...
0
votes
2answers
33 views

First number in an open interval?

What is the first number in an open interval? For example, if I have the open interval (0, 1), what is the lowest number in that interval? Does this question even make sense with real numbers?
0
votes
1answer
21 views

How to prove these facts about integers using this definition?

A subset $A$ of $\mathbb{R}$, the set of real numbers, is said to be inductive if $1 \in A$ and if the statement $x \in A$ implies the statement $x+1 \in A$. The $Z_+$ of positive integers is ...
0
votes
2answers
37 views

Solving an unusual equation

I need to find a real number $n$ such that $n > 1$ and: $$ \sum_{k=1}^\infty \frac{2^k}{n^k} = \frac{n-1}{n} $$ Ideally, I'd find the minimum such $n$ (if more than one exists), but really, any ...
4
votes
1answer
146 views

Is the range of an injective function dense somewhere?

Consider an injective function $\,\,f:[0,1]\rightarrow[0,1].\,$ Then is it true that there always exists some non-empty open subinterval of $[0,1]$, such that $f([0,1])$ is dense in that subinterval? ...
0
votes
2answers
33 views

Simplifying fractional exponents

I am very confused about the following: whenever I put in into wolfram alpha the answer it gives me is "indeterminate", is it not possible to simplify fractional exponents or something? if the ...
24
votes
9answers
3k views

Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?

Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
0
votes
1answer
39 views

I think I was wrong: The supremum of a set whose elements squared less than a positive real number is square root of this number?

I think I was wrong... Can you find it out and teach me how to prove (the title)? If $c>0$ a real number. Define $E=\{x\in\mathbb R\mid x^2<c\}$. $E$ is nonempty because $0\in E$ (positivity ...
0
votes
1answer
26 views

Simplify the algebraic expression

Can someone please explain to me how the algebraic expression in the picture is simplified. To be more specific, how (1) becomes (2). $3x^2(6x-4)^4 + x^3(6\times 4\times (6x-4)^3)$ ...
1
vote
4answers
55 views

Prove or disprove the rationality of $ x^y $

Prove or disprove: "If $x$ is a rational number, and $y$ is an irrational number then $x^y$ is irrational" I am stuck with this, these are my steps. let $x=2$ and $y=\sqrt{2}$ ...
0
votes
2answers
36 views

Find a subset of the real numbers

I have to find an open and dense subset of the real numbers with arbitrarily small measure. Since the set of the rational numbers is dense, could we use a subset of the rationals?? How could I find ...
6
votes
4answers
85 views

Is $[a, a)$ equal to $\{a\}$ or $\varnothing$?

Let us define the set $[a,b) = \{ x \in \mathbb{R}: a\le x <b\}$ Is $[a, a)$ equal to $\{a\}$ or $\varnothing$?
1
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2answers
30 views

Rudin - Exercise 12, Cap. 2 Principles of Mathematical Analysis

Let $\Bbb{K}\subset\Bbb{R}$ consist of $0$ and the numbers $\frac{1}{n}$, for $n=1,2,3,\dots$. Prove that $\Bbb{K}$ is compact directly from the definition (without using the Heine-Borel theorem).
1
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1answer
31 views

For $f(x, y) = x-y$, is $f(K \times K)$ closed if $K$ is closed?

$f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y) = x-y$. For $K \subset \mathbb{R}$ closed is $f(K\times K)$ closed? For the closed interval this is straight forwardly true ...
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votes
1answer
50 views

Identify a countable union of nested intervals using the Archimedean principle [closed]

$\displaystyle\bigcup_{n=2}^\infty \left[\frac1n,3-\frac 2n\right]=(0,3)$. I can't prove using limits. I have to use the Archimedean principle and I don't know how to go about doing that..
2
votes
2answers
116 views

Is the function $\,f(x, y) = x-y\,$ closed?

Is $f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$, such that $f(x,y)=x-y$ closed?
4
votes
2answers
44 views

Show that if $a,b \in \Bbb R$ then [duplicate]

$\max\{a,b\} = \frac12(a+b+|a-b|)$ and $\min\{a,b\} = \frac12(a+b-|a-b|)$ how would you go about solving this? I started with suppose $a \leq b$ Also, show min{a,b,c} = min{min{a,b},c}. How would ...
0
votes
0answers
27 views

Determine and sketch the pairs $(x,y)$ in $\mathbb{R} \times \mathbb{R}$ that satisfy some inequality

a) $|x| \leq |y|$ Continue my explanation below... If $y \geq 0$, then $-y \leq x \leq y$ and we get the region in the upper half-plane on or between the lines $y = x$ and $y = -x$
0
votes
1answer
88 views

Problem in Basic Real Analysis

I want to prove that if $$E_{b,y} = \{ r\in \Bbb Q |\; b^r\leq y \}$$ for $b\in \Bbb R^{>1}$ and $y\in \Bbb R^{>0}$ then, $\sup E_{b,y}$ is the unique solution to the equation $$b^x = y$$ I ...
2
votes
1answer
26 views

$A_\sqrt{2}$ isn't a Dedekind cut?

In a problem we consider a cut of $\mathbb{Q}$ a subset $A\subset\mathbb{Q}$ that fulfills: $A\neq\emptyset$ $\forall (q,q')\in A\times\mathbb{Q},\,q'<q\Rightarrow q'\in A$ $\forall q\in ...
2
votes
3answers
97 views

$\mathbf{Q}$ basis of $\mathbf{R}$.

Could someone give me an explicit basis of $\mathbf{R}$ as a vector space over $\mathbf{Q}$? I no some linearly independent subset, namely $1,e,e^2,\ldots$ but this seems to be a deep result already ...
0
votes
0answers
24 views

Checking this function for differentiability

$f(x) = |x|\sin x + |{x-\pi}|\cos x$ for $x \in \mathbb{R}$ Is the above function differentiable at $x=0$ ? At $x=\pi$ ?
-8
votes
7answers
266 views

How is greater than defined for real numbers?

I have an understanding of real numbers. For example, I can imagine real numbers as points on the line. The point which is more to the right represents bigger number. Or if I have decimal ...
1
vote
2answers
26 views

rational number plane vector space or not?

Two questions: 1. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{Q}$? 2. Is $\mathbb{Q}^2$ a vector space over the field $\mathbb{R}$? My answer to the first question is yes. Because the ...
0
votes
0answers
25 views

What are alphamatic numbers?

I came across a word called alphamatic numbers today and I searched the internet for it but couldn't find what it is. I know it's something to do with English language.
0
votes
2answers
37 views

Does the mean of two positive numbers obey these basic inequalities?

Suppose $a,b$ are positive real numbers. Does it follow that $$ a < \frac{a+b}{2} < b $$ provided $a < b$?
3
votes
1answer
69 views

Unconstrained optimal control - $J = \int_0^{t_1} (x^2 + ux + \frac{1}{2} u^2) dt$

I've been given the following problem to solve, and I'm having a lot of difficulty in understanding what I can do. The system $\dot x = x + u$, where $u = u(t)$ is not subject to any constraint, ...
3
votes
1answer
93 views

Undefinable Real Numbers

Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question Part ...
1
vote
2answers
27 views

How to prove: If $x \geq 0 $ and $x \leq \epsilon$, for all $\epsilon > 0$, then $x = 0$?

I am trying to prove this problem for my homework. I am having some difficulty with this, because we are just supposed to use several ordered field axioms, the four order axioms, and several basic ...
2
votes
1answer
81 views

Prove that $\mathbb{Q}\!\smallsetminus\!\mathbb{Z}$ is dense in $\mathbb{R}$

Can someone just tell me if this is a correct way to prove it. let $(a,b)$ be a nonempty open interval in $\mathbb{R}$. Then by density of $\mathbb{Q}$ in $\mathbb{R}$ there exists $q\in \mathbb{Q}$ ...
4
votes
5answers
408 views

Quick and painless definition of the set of real numbers

I am looking for a simple way to describe the underlying set of the real numbers without getting into cauchy sequences or dedekind cuts. Furthermore, I want the description to not rely on some notion ...
-1
votes
3answers
53 views

Showing $|\mathbb{R}| \leq$ cardinality of the set of open sets in $\mathbb{R} $

Let $(O_\lambda)_{\lambda \in A}$ be the family of open sets in $\mathbb{R}$, we define $O_i= (-i, i)$. Then {$O_i$}$_{i=1}^\infty$ is an open cover for $\mathbb{R}$. Hence we have $\mathbb{R} ...
1
vote
2answers
42 views

Prove that $AB-BA=I_2$ cannot hold whatever the real $2\times 2$ matries $A, B$ [duplicate]

How can I prove that $AB-BA=I_2$ cannot hold whatever the real $2\times 2$ matries $A, B$. Suppose $A=\pmatrix{ a_{11} & a_{12}\\ a_{21} & a_{22}}$ and $B=\pmatrix{b_{11} & b_{12} \\ ...
-1
votes
4answers
58 views

Proof about real numbers

Question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 28. Exercise 1. If $x$ and $y$ are arbitrary real numbers with $x<y$, prove that there is ...
1
vote
4answers
63 views

Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
0
votes
2answers
24 views

finite padic number uniqueness

suppose $\sum\limits_{i=0}^{n}a_ip^i=0$ where $a_i\in \{0,1,-1,\dots,(p-1),-(p-1)\}$ and $p\geq 2, p\in N$, how to prove that $a_0=a_1=\dots=a_n=0$?
0
votes
0answers
29 views

cantor staircase function uniform distribution on cantor set

suppose Cantor staircase function $F$ is extended to have $F(a)=0$ for $a<0$ and $F(a)=1$ for $a>1$. Then how can one show that $F$ is the cumulative distribution function of the uniform ...
0
votes
1answer
53 views

Monotone increasing sequence of rationals with an irrational limit

I am trying to use rationals in order to approximate irrationals. Is it possible to construct a monotonically increasing sequence of rationals the limit of which is an irrational? If so, how?
1
vote
4answers
80 views

locating any point on a real number line

So my question is really simple (and may be a bit naive): The claim is, I can locate any point in a 2D-plane by recursively applying the following method (possibly infinite number of times): For ...