Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
3,573
questions
0
votes
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} &...
3
votes
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 ...
-1
votes
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:
$$\frac{\...
-1
votes
1answer
35 views
Prove there exists a real number $x>0$ such that $\sin(x)=0.5x$ [closed]
I have to prove: there exists a real number $x>0$ such that $\sin(x)=\frac{1}{2}x$. I have no idea how to prove this, can someone help me?
0
votes
1answer
86 views
$ \lim_{\epsilon \to 0^{+}} \int_{a+\epsilon}^{b-\epsilon} f(x) dx = \int_{a}^{b} f(x) dx $
Prove that$$ \lim_{\epsilon \to 0^{+}} \int_{a+\epsilon}^{b-\epsilon} f(x) dx = \int_{a}^{b} f(x) dx $$ Given that $f(x)$ is continuous in $(a+\epsilon,b-\epsilon) $ but is infinite at $a$ and $b$.
...
0
votes
2answers
47 views
Basic confusion in real analysis
I have just started real analysis (first time) and self studying from book Mathematical analysis by Apostol.
In the first chapter of real and complex number system, it is written that
We assume there ...
0
votes
1answer
44 views
The fundamental aspects of the square root [closed]
When I was in High School learning algebra we came upon solving for roots. When doing this for a quadratic you sometimes end up having square roots in your answer. Due to uncertainty we cannot ...
2
votes
1answer
57 views
Something similar to Kronecker's Theorem
I would like to prove (or have a reference to) the following:
Given $n$ real numbers $a_1,\ldots,a_n>0$ and $\varepsilon>0$, there exist $k,k_1,\ldots,k_n\in\mathbb N\setminus\{0\}$ such that $|...
0
votes
1answer
17 views
Whether the following is a Dedekind cut?
I am able to verify that
$$ 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{ ...
1
vote
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{...
2
votes
1answer
53 views
Prove that addition under “real numbers” is well defined.
Problem:
If we denote real numbers as Cauchy sequences and:
$$[\{a_i\}]+[\{b_i\}] = [\{a_i+b_i\}] ; iā N$$
Show that "$+$" is well defined under real numbers.
My try:
Assume that:
$$
\begin{...
2
votes
2answers
189 views
Why cant we apply the square root of a negative number to equate more imaginary numbers?
In mathematics, polynomials like $x^2-1$ would have a clear solution of $x=\pm 1$. However, without complex numbers could you solve $x^2+1$? No, there would be no possible solution without adding a ...
-1
votes
1answer
57 views
Is a an integer to the power of an irrational number irrational? [closed]
This seemed like a very obvious thing. I was playing around with exponents and logarithms, and I just started wondering, "if $a$$\in$$\mathbb Z$, and $r$$\in$$\mathbb R \setminus \mathbb{Q}$, is $...
4
votes
0answers
40 views
Creating a complete number diagram once and for all
Background
I have been recently searching for a good number diagram I could save for future reference. I was looking for a picture that would show different types or real numbers, and also ...
2
votes
0answers
54 views
Given a real number what is the minimum number of times it can be tetrated to get an intiger?
I really enjoyed this video on the possibility that $\pi^{\pi^{\pi^\pi}}$ is an integer, but i thought that it was a case of a more interesting general problem.
Given a real $x$ and an integer $y$, ...
0
votes
0answers
33 views
Let $x$ be an irrational number. Is the set $\ \{nx \mod 1: n \in \mathbb{N} \}\ $ dense in $[0,1]\ ?$ [duplicate]
Let $x$ be an irrational number. Is the set $\ \{nx \mod 1: n \in \mathbb{N} \}\ $ dense in $[0,1]\ ?$
Obviously if $x$ were rational, the set would be finite and therefore would have no limit points, ...
1
vote
1answer
27 views
What operations do I do in an inequality to raise a number to the variable?
Suppose we have $f(x)=x^2+2x$ and an interval of $[0,1]$. What I normally do to find a range in that interval is:
Ok. that's the easy part. I need do the same, but with $f(x)=3^-x$
Here is my ...
-1
votes
0answers
31 views
How many times digit 9 appears between 1 to 10000 [closed]
While writing numbers from 1 to 10,000 how many times the digit 9 will be written ?
4
votes
0answers
129 views
Roth's theorem: contradiction?
Roth's theorem says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions to this:
$$\displaystyle \left|\alpha-\frac pq\right|<\frac 1{q^{2+\epsilon}}...
-1
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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} \ \...
1
vote
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, +\...
1
vote
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 &...
1
vote
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 āproveā $x<y$ using only $+$ and $=$? And if not, is there a more or ...
0
votes
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
votes
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 \...
4
votes
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\...
1
vote
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
votes
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. ...
0
votes
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 ā 0. Then, there exist unique
integers q and r ...
0
votes
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 ...
1
vote
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.
1
vote
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.
0
votes
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 ...
0
votes
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.
4
votes
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 mathematiciansāeg ultrafinitistsādisagree ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 $y \in A$} \}$...
0
votes
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
votes
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 $\...
0
votes
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 ...
0
votes
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
votes
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, ...
-1
votes
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
votes
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
votes
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
