Questions tagged [real-numbers]

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

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3answers
31 views

Problem proving that there exists an irrational number between any two real numbers

I want to show that: for $a,b \in \mathbb R$ with $a<b, \exists x \in (a , b)$ such that $x \notin \mathbb Q$ . I start with $0< \frac 1{\sqrt 2} < 1 \Rightarrow 0< \frac{ b-a}{\sqrt 2} &...
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1answer
33 views

How to factorize $P=(X^2-4X+1)^2+(3X-5)^2$ in $\mathbb R[X]$?

First of all, I searched for roots. I knew that $\exists x\in\mathbb R,\, P(x)=0 \iff \exists x\in\mathbb R,\, x^2-4x+1 = 0 \text{ and } 3x-5 = 0$. However, It is really easy to say that there is no ...
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1answer
18 views

Finding the amplitude of the difference between two sinusoids

I have the following function: $$y(t)=a\sin(wt+p)-b\sin(wt+p+\frac{2\pi}{3})\tag1$$ Question: what is the amplitude of the function $y(t)$ in terms of $a,b,w$ and $p$? I thought I can find: $$\frac{\...
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1answer
35 views

Prove there exists a real number $x>0$ such that $\sin(x)=0.5x$ [closed]

I have to prove: there exists a real number $x>0$ such that $\sin(x)=\frac{1}{2}x$. I have no idea how to prove this, can someone help me?
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1answer
86 views

$ \lim_{\epsilon \to 0^{+}} \int_{a+\epsilon}^{b-\epsilon} f(x) dx = \int_{a}^{b} f(x) dx $

Prove that$$ \lim_{\epsilon \to 0^{+}} \int_{a+\epsilon}^{b-\epsilon} f(x) dx = \int_{a}^{b} f(x) dx $$ Given that $f(x)$ is continuous in $(a+\epsilon,b-\epsilon) $ but is infinite at $a$ and $b$. ...
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2answers
47 views

Basic confusion in real analysis

I have just started real analysis (first time) and self studying from book Mathematical analysis by Apostol. In the first chapter of real and complex number system, it is written that We assume there ...
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1answer
44 views

The fundamental aspects of the square root [closed]

When I was in High School learning algebra we came upon solving for roots. When doing this for a quadratic you sometimes end up having square roots in your answer. Due to uncertainty we cannot ...
2
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1answer
57 views

Something similar to Kronecker's Theorem

I would like to prove (or have a reference to) the following: Given $n$ real numbers $a_1,\ldots,a_n>0$ and $\varepsilon>0$, there exist $k,k_1,\ldots,k_n\in\mathbb N\setminus\{0\}$ such that $|...
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1answer
17 views

Whether the following is a Dedekind cut?

I am able to verify that $$ T= \{t \in \mathbb{Q} : t^2\lt{2} \text{ or } t\lt{0} \} $$ is a Dedekind cut. However i have some confusion in verifying if $$ U= \{t \in \mathbb{Q} : t^2\le{2} \text{ ...
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0answers
10 views

Question about intersection of an open interval and the complement of a countable set

Let $A \subset \mathbb{R}$ a countable set, and $X = \mathbb{R} \setminus A$. Then we have to prove that For all open interval $(a,b)$, the set $(a,b) \cap X$ is uncountable; $X$ is dense in $\mathbb{...
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1answer
53 views

Prove that addition under “real numbers” is well defined.

Problem: If we denote real numbers as Cauchy sequences and: $$[\{a_i\}]+[\{b_i\}] = [\{a_i+b_i\}] ; i∈ N$$ Show that "$+$" is well defined under real numbers. My try: Assume that: $$ \begin{...
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2answers
189 views

Why cant we apply the square root of a negative number to equate more imaginary numbers?

In mathematics, polynomials like $x^2-1$ would have a clear solution of $x=\pm 1$. However, without complex numbers could you solve $x^2+1$? No, there would be no possible solution without adding a ...
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1answer
57 views

Is a an integer to the power of an irrational number irrational? [closed]

This seemed like a very obvious thing. I was playing around with exponents and logarithms, and I just started wondering, "if $a$$\in$$\mathbb Z$, and $r$$\in$$\mathbb R \setminus \mathbb{Q}$, is $...
4
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0answers
40 views

Creating a complete number diagram once and for all

Background I have been recently searching for a good number diagram I could save for future reference. I was looking for a picture that would show different types or real numbers, and also ...
2
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0answers
54 views

Given a real number what is the minimum number of times it can be tetrated to get an intiger?

I really enjoyed this video on the possibility that $\pi^{\pi^{\pi^\pi}}$ is an integer, but i thought that it was a case of a more interesting general problem. Given a real $x$ and an integer $y$, ...
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0answers
33 views

Let $x$ be an irrational number. Is the set $\ \{nx \mod 1: n \in \mathbb{N} \}\ $ dense in $[0,1]\ ?$ [duplicate]

Let $x$ be an irrational number. Is the set $\ \{nx \mod 1: n \in \mathbb{N} \}\ $ dense in $[0,1]\ ?$ Obviously if $x$ were rational, the set would be finite and therefore would have no limit points, ...
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1answer
27 views

What operations do I do in an inequality to raise a number to the variable?

Suppose we have $f(x)=x^2+2x$ and an interval of $[0,1]$. What I normally do to find a range in that interval is: Ok. that's the easy part. I need do the same, but with $f(x)=3^-x$ Here is my ...
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0answers
31 views

How many times digit 9 appears between 1 to 10000 [closed]

While writing numbers from 1 to 10,000 how many times the digit 9 will be written ?
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0answers
129 views

Roth's theorem: contradiction?

Roth's theorem says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions to this: $$\displaystyle \left|\alpha-\frac pq\right|<\frac 1{q^{2+\epsilon}}...
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1answer
29 views

How can I prove this question?

We know that it is not proved that $e^e$ is transcendental, so neither is the number that $e^{e\sqrt{2}}$. My question is, if one turns out to be, how can it be proved that the other is? Because there ...
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1answer
20 views

Can we multiply a matrix by an imaginary or complex number not purely real?

I am currently reading the topic of scalar multiplication of matrices. It always says that if k is a scalar then k times matrix = matrix with all the elements multiplied by k. My question is what is ...
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0answers
12 views

Can you decompose a real number $a = \sum_{i \in S} n_i Q b^i$?

Can you represent any real (or at least rational) number $a$ as the sum $$a = \sum_{i \in S} n_i Q b^i$$ where $S$ is a set of integers (including those less than 0), $Q$ is a real number greater than ...
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1answer
32 views

How to solve a recursive sequence that is oscillating to a limit point

In class we only learned the "tools" to prove that a sequence that is monotonically increasing or decreasing converges. How to prove that a sequence converges if it is given in recursive ...
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2answers
32 views

Proving a set X is dense in [0,1] equivalence relation [duplicate]

Let the relation in $\mathbb{R}: x \equiv y \ \mbox{mod} \ \mathbb{Z}$, when $x-y \in \mathbb{Z}$. For each $n \in \mathbb{N}$, let $x_n \in [0,1)$ such that $x_n \equiv \sqrt{n} \ \mbox{mod} \ \...
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1answer
48 views

Does $[0,+ \infty)$ counts for a closed interval in just real number system?

This is different from my previous question which asked about the situation in extended real number system. So, we are discussing the situation when $X=\mathbb{R}$. Without the elements $\{-\infty, +\...
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1answer
37 views

Does $[0,+ \infty)$ counts for a closed interval in extended real number system?

In Topology by Munkres, page 84, chapter 2, section 14 the order topology: So, we are discussing the situation when $X=\mathbb{R}\cup \{+\infty, -\infty\}$. Then $[0,+\infty):= \{t\in X: 0 \leq t &...
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1answer
43 views

Is there a way to represent the relation $<$ on the real numbers without using multiplication?

I know for $x,y\in\mathbb R$ we have $x<y$ iff there exists a $z\in\mathbb R$, so that $y=x+z^2$. Is there a similar way to „prove“ $x<y$ using only $+$ and $=$? And if not, is there a more or ...
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3answers
23 views

I am trying to prove the following theorem: Let $x$ and $y$ be real numbers. Show that if $x \ne y$ and $x,y \geq 0$, then $x^2 \ne y^2$.

I am trying to prove the following theorem: Let $x$ and $y$ be real numbers. Show that if $x \ne y$ and $x,y \geq 0$, then $x^2 \ne y^2$. $Proof.$ The contrapositive of the statement is: If $x^2 = ...
2
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1answer
68 views

Completeness Axiom of $\mathbb{R}$.

I use the following as the axiom of completeness of the reals $\mathbb{R}$: $$\forall X,Y\in \mathcal{P}(\mathbb{R})\backslash\{\emptyset\}: (\forall x\in X\quad\forall y\in Y: x\leq y) \implies \...
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0answers
122 views

$x^{x^{1/x}}$ has the same first $\lfloor x \rfloor$ decimals as the number $3\sqrt{\lfloor x\rfloor}$ when $x=\pi$ and $x=e$. Why? Are there others? [closed]

Suppose we are in year $0$ (I mean no computers), and someone gave us this question: (edited) The real number $\pi^{\pi^{1/\pi}}$ has the same first $\lfloor \pi \rfloor$ decimals as the number $3\...
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2answers
35 views

Proving uncountable sum as series using nets.

I saw this question: The sum of an uncountable number of positive numbers Asking about a proof of the following: Let $A = \{a_i\}_{i\in I}$ be a set of positive numbers. If the uncountable sum ...
6
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2answers
128 views

Can you expand induction proofs to the real numbers?

Everyone knows the principle of induction, where you first prove a base case for some $n_0\in \mathbb{N}$, and than show that by assuming the case for an $n \in \mathbb{N}$ the $n+1$ case follows. ...
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1answer
15 views

Euclid's Division Lemma Extended to Negative Integers Conflict

My textbook states that Euclid's Division Lemma can be extended to all integers with the following information: Let a and b be any two integers with b ≠ 0. Then, there exist unique integers q and r ...
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0answers
19 views

Infimum of the union of 3 sets under some restriction

I was given the following to solve- At first I assumed that I would have to solve it by breaking up and showing that for each set (A, B and my arbitrary set D) that the infimum was as proven by also ...
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5answers
71 views

How to prove $\sqrt{x} - \sqrt{x-1}>\sqrt{x+1} - \sqrt{x}$ for $x\geq 1$?

Intuitively when $x$ gets bigger, $\sqrt{x+1}$ will get closer to $\sqrt{x}$, so their difference will get smaller. However, I just cannot get a proper proof.
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1answer
47 views

What is the difference between $\mathbb{R}^+$ and $\mathbb{R}^*$?

I know that both of them contain all positive numbers from $\mathbb{R}$ but one notation contains $0$ too. I don't know which one. Thanks in advance.
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1answer
23 views

Prove that for two real numbers $a,b$, if for any $e>0$ they can be bounded by $s,s'$: $s'\geq a\geq s,s'\geq b\geq s$ and $s'-s<e$, then $a=b$.

I don't understand the proof of lemma presented in the title. I found it in the book "Differential and integral calculus" by G. M. Fichtenholz, it's lemma 2, section 2.8, chapter 1. I don't ...
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2answers
34 views

Finding the values for which a series converges [closed]

I need to find the values of $x$ for which the sum $\sum_{j=1}^\infty jx^j$ converges.
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2answers
75 views

How would the mathematics relevant to physical theory be different if it didn't use real numbers?

Real numbers assume we can have infinite precision and some of the theory behind them uses infinite processes to establish certain proofs. A small band of mathematicians–eg ultrafinitists–disagree ...
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0answers
16 views

Does any nonempty open set in $\mathbb{R}^n$ contains a point with rational coordinates? [duplicate]

I would like to determine whether any nonempty open set in $\mathbb{R}^n$ contains a point with rational coordinates. The answer to this question seems to be confirmative because $\mathbb{Q}^n$ is ...
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2answers
64 views

Construct a set such that $a, b \in X \implies a + b \not\in X$

I'm looking for an uncountable set $X \subset \mathbb{R}_{\geq 0}$ such that for all $a, b \in X$, $a \neq b \implies a + b \not\in X$. Two points about this question: first of all, I'm not sure ...
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1answer
31 views

Proving if a function is continuous and not one-one then it has many such points.

Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
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1answer
35 views

Showing an inequality in the real positive numbers

I want to show that $\forall c>0$: $$0<\left(1-\frac{(wl)^2}{r^2+(wl)^2}\right)^2+\left(\frac{wrl}{r^2+(wl)^2}\right)^2<(1-w^2lc)^2+(wrc)^2\tag1$$ Where all variables are bigger than zero. ...
4
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1answer
148 views

$\lim_{n \to \infty} (u_{n+1}-u_n)$ does not converge then $\lim_{n \to \infty} u_n$ does not converge

Prove that if $\lim_{n \to \infty} (u_{n+1}-u_n)$ does not converge then $\lim_{n \to \infty} u_n$ does not converge. My try- Contrapositive statement - If $\lim_{n \to \infty} u_n$ converges then $\...
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0answers
30 views

Orthonormal basis for $\mathbb{R}^3$ and its determinant

Given two vectors $u = \frac{1}{\sqrt{3}}*(1, -1, 1)$ and $v = \frac{1}{\sqrt{6}}*(1, -1, -2)$. How would I go about finding a third vector $w$ that is orthonormal to the other two and linearly ...
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0answers
15 views

Proof for indexed family of sets and intervals in real analysis [duplicate]

I am asked to either prove or disprove the following statement. $\cap_{\textrm{x} \in (1, \infty)}$ (-x, x+1] = $\left[-1,2\right]$ I attempted to show that each side was a subset of each other ...
3
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1answer
77 views

Irrationality of an “Euler-like” number

Let $(a_n)_{n=0}^{\infty}$ be a sequence of zeroes and ones such that $a_n=1$ for infinitely many $n$. Let $\displaystyle x:=\sum_{n=0}^{\infty} \frac{a_n}{n!} .$ Is $x$ irrational? I believe it is, ...
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0answers
38 views

“Integer division” of reals

My claim: Given $x,y \in \mathbb{R}$, $y\neq0$ there exist unique $q \in \mathbb{Z}, r \in \mathbb{R}$ such that $$x = yq +r, \space\space\space 0\leq r< |y| $$ Sketch of proof: Prove it first for ...
0
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2answers
62 views

Proof Concerning Indexed family of Sets and Intervals

I have been stuck on the following problem for some time. I know to prove two things are equal, you use the proof where you show they are both subsets. I think I was able to show that the set on the ...
3
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1answer
51 views

Solve the equation $x=1-5(1-5x^2)^2$

Solve the equation $$x=1-5(1-5x^2)^2$$ ###My work Let $f(x)=1-5x^2$. Then we have tha equation $f(f(x))=x$. But in this case we don't use the equation $f(x)=x$ because $f(x)$ is not monotonic function

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