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Questions tagged [real-numbers]

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

4
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1answer
79 views

Proof that x^3 = 2 has a real solution using completeness theorem?

I am trying to prove that x^3 = 2 has a solution in the real numbers using completeness theorem. I was trying to base myself on a proof seen before for x^2 =2 but it seems to have many steps that are ...
0
votes
0answers
17 views

When do a real $x$ have many decimal representation? How to prove there is a maximum of two decimal representation? [duplicate]

I was asking myself when a real number has two decimal representation. Looking around, it seemed this was true if and only if one of its decimal representation end with only zeros $x_D=...0000000$ (I ...
0
votes
2answers
30 views

Every real number has a decimal representation.

I was reading this answer explaining why all real number have a decimal representation. I think it is really a nice explanation but I don't really see were (I think it is a little hidden) we use the ...
0
votes
1answer
31 views

For all $0<a\in \Bbb R$, there exists a unique $0<x\in \Bbb R$ such that $x^2=a$

For all $0<a\in \Bbb R$, there exists a unique $0<x\in \Bbb R$ such that $x^2=a$. Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for ...
0
votes
0answers
28 views

Multiples of triangular numbers

I know that $\;2T_2=T_3\;$ is a triangular number. Can anyone suggest me other triangular numbers whose two multiple is again a triangular number?
-2
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2answers
55 views

Why is $\mathbb{C}$ algebraically closed, but $\mathbb{R}$ isn't [duplicate]

Is there an intuitive explanation for why the complex numbers $(\mathbb{C}$) are algebraically closed, but $\mathbb{R}$ isn't?
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0answers
39 views

Euler's phi function.. [on hold]

Does Euler's phi function continuous?
0
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0answers
51 views

what do infinitesimals look like?

i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions (1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a ...
3
votes
1answer
31 views

Diagonalizability over $\mathbb{C}$ and $\mathbb{R}$ respectively

I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing: $$A = \begin{Bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&...
-1
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1answer
44 views

How to prove given sequence converges to 0? [duplicate]

I am given a sequence {$a_{n}$} of nonzero numbers converges to infinity. How can I use this to prove that the sequence {$\frac{1}{a_{n}}$} converges to 0? I can intuitively see why $\frac{1}{\infty}$...
1
vote
1answer
66 views

The isomorphism between two complete ordered fields is unique

The isomorphism between two complete ordered fields is unique. Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! My attempt: ...
1
vote
1answer
20 views

How to find the split in the given set of numbers.

Beforehand I would like to express apologies if this sort of question is not suited here, but I desperately need it answered. I have a set of numbers: [2, 5, 6, 2, 2, 81, 72, 77] From the above ...
0
votes
1answer
45 views

Applying the chain rule on vectors and matrices

I need to find $\frac{dy}{dx}$ for the following y = $||A^Tx - b||_2^2$ where $A \in R^{3x3}, b \in R^{3x1}, x \in R^{3x1}, y \in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = ...
6
votes
4answers
63 views

Need help with the proof {needed for my algorithm}

I am not from pure math background. I am working on an algorithm which works good for all the practical reasons based on the following assumption. that, if ab = cd and a+b = c+d then, either a = c ...
0
votes
2answers
27 views

May someone help me with a functions questions?

I was wondering if someone could help me with this question. It is genuinely not homework, it is extra work set by my angry asian mum, and I am desperately in need of help for just this one question. ...
2
votes
1answer
48 views

How many digits are there in a number($x$) that contains only $3$,$4$,$5$ & $6$ when the sum of digits of $x$ and $2x$ is 900?

$x$ is a positive integer such that its digits can only be $3,4,5,6$. $x$ contains at least one copy of each of these four digits. The sum of the digits of $x$ is $900$ and the sum of the digits of $...
-2
votes
1answer
48 views

Let $v_1$, $v_2$, $v_3$ be a basis of the $\mathbb{R}$-vector space $\mathbb{R}^3$

I'm not that good at math and would be very happy if you could give me some hints and so on So my task is: Let $\{v_1, v_2, v_3\}$ be a basis of the $\mathbb{R}$-vector space $\mathbb{R}^3$ show ...
1
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2answers
50 views

Explain this “contradiction” of the proof of $x > 0$ iff $x \in \mathbb{R}^{+}$

I am working through Apostol's Calculus I and just read about the order axioms. He presents a new undefined concept called positiveness, gives the axioms and then defines symbols $<, >, \leq, \...
1
vote
1answer
57 views

Proof clarification - If $ab = 0$ then $a = 0$ or $b =0$

I came across a proof for the following theorem in Apostol Calculus 1. My question is regarding (1) in the proof, why is this part necessary? I don't see why you can't begin with (2) Theorem 1.11 If ...
0
votes
2answers
60 views

Real Analysis Inequality Proof Involving Reals and Rationals $0 < |r - q| < \varepsilon$

I'm having difficulties making progress in proving: $$\forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$ To clarify, $r$ is a real number and $q$ is a ...
0
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2answers
16 views

Validity of Proof for 'Possibility of Subtraction' from Apostol 1

I attempted a proof before reading the solution Apostol provides. I don't think it is valid but I am trying to determine why Theorem 2, Possibility of Subtraction: Given $a \text{ and } b$, there ...
1
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1answer
40 views

Definition of rational numbers from real numbers

Usually the set of numbers are introduced starting from integers, from wich the rational numbers are defined using equivalence classes of couples of integer numbers. Than, from these rational numbers, ...
3
votes
2answers
51 views

Regarding $x < y \Rightarrow x^n < y^n$ proof rigor.

I came across the implication $$x < y \Rightarrow x^n < y^n$$ $$x,y>0, n\in Z^+$$ in a textbook and came up with the following proof. Proof Since $x<y$ the following chain of inequalities ...
0
votes
1answer
27 views

Is it possible for $\langle \Bbb Q,<,+,\cdot,0,1 \rangle$ to be isomorphic to a proper subfield of itself?

It is well-known that every ordered field contains a subfield that is isomorphic to $\langle \Bbb Q,<,+,\cdot,0,1 \rangle$. Let $\langle A,<,+,\cdot,0',1' \rangle$ be an ordered field. I would ...
0
votes
1answer
26 views

The smallest subfield of an ordered field is isomorphic to $\langle \Bbb Q,<,+,\cdot,0,1 \rangle$

This is an exercise from Chapter 9. The Sets of Real Numbers from textbook Introduction to Set Theory by Hrbacek and Jech. The textbook does not provide solution and I would like to verify my attempt. ...
0
votes
1answer
33 views

Prove that if we take away supremum axiom of real numbers, then we would have such a function

Prove that if we take away supremum axiom of real numbers, that is to say, if there exists a subset of real numbers that is bounded from above but does not have a supremum, then there exists a ...
2
votes
2answers
42 views

Do the axioms of ordered field imply that $a\cdot 0=0$ and $0<1$?

We introduce some definitions: A structure $\mathfrak{A}=\langle A,<,+,\cdot,0,1 \rangle$ where $<$ is a linear ordering, $+$ and $\cdot$ are binary operations, and $0,1$ are constants such ...
2
votes
3answers
58 views

Real numbers definition problem

In the definition of the limit, real positive numbers are used as distance, so we can not use the limit in the definition of real numbers. the definition of the limit. But dumb numbers are defined as ...
0
votes
0answers
46 views

Is this a typo in the hint for theorem “The complete ordered field is unique up to isomorphism”

In Chapter 10. Sets of Real Numbers from textbook Introduction to Set Theory by Hrbacek and Jech, the authors mention: Let $\mathfrak{R}=\langle \Bbb R,<,+,\cdot,0,1 \rangle$ be the structure ...
0
votes
2answers
89 views

Is $0.\sqrt9$ a valid number?

Is $0.\sqrt9$ valid number? Are such numbers allowed? First, I thought the value of the above number can be 0.3 but then it occurred how I would interpret this number: $0.65\sqrt2$ or $0.65\sqrt229$ ...
1
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0answers
24 views

For every $x\in\Bbb R$ and $n\in\Bbb N\setminus \{0\}$, there exists $r,s\in\Bbb Q$ such that $r< x<s$ and $s-r< 1/n$

For every $x\in\Bbb R$ and $n\in\Bbb N\setminus \{0\}$, there exists $r,s\in\Bbb Q$ such that $r< x<s$ and $s-r< 1/n$. This is a lemma in Chapter 10 from my textbook Introduction to Set ...
2
votes
1answer
41 views

Let $x\in\Bbb R$. Then there exists a unique $w\in\Bbb R$ such that $x+w=0$

Let $x,y\in\Bbb R$. We define addition operation $(+)$ on $\Bbb R$ by $$x+ y:=\inf\{r+ s\mid r,s\in\Bbb Q \text{ and } x<r \text{ and } y<s\}$$ Then there exists a unique $w\in\Bbb R$ such ...
2
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0answers
31 views

Let $x\in\Bbb R$, $a=\inf \{r \mid r\in\Bbb Q,x<r\}$, and $b=\inf\{-s\mid s\in\Bbb Q,s<x\}$. Prove that $a+b=\inf\{r-s \mid r,s\in\Bbb Q,s<x<r\}$

Let $x,y\in\Bbb R$. We define addition operation $(+)$ by $$x+ y:=\inf\{r+ s\mid r,s\in\Bbb Q \text{ and } x<r \text{ and } y<s\}$$ Theorem: Let $x\in\Bbb R$, $a=\inf \{r \mid r\in\Bbb Q,...
3
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0answers
48 views

Prove that $\forall x \in \Bbb R^+:x+0=x\cdot 1=x$

Let $\Bbb R^+=\{x\in\Bbb R \mid x>0\}$ and $x,y\in\Bbb R^+$. We define multiplication operation $(\cdot)$ on $\Bbb R^+$ and addition operation $(+)$ on $\Bbb R$ by $$x\cdot y:=\inf\{r\cdot s\mid r,...
3
votes
5answers
136 views

Proof explanation of $``\exists x\in\mathbb{R}$ with $x^2=2"$

Can someone please help me break down the proof below from $(*)$ onwards. I'm lost at what is going on and where the proceeding steps are coming from. Is this a proof by contradiction? Why are we ...
1
vote
1answer
54 views

Prove that $\forall x,y,z \in \Bbb R^+:(x\cdot y)\cdot z = x\cdot (y\cdot z)$

Let $\Bbb R^+=\{x\in\Bbb R \mid x>0\}$ and $x,y\in\Bbb R^+$. We define the multiplication operation $(\cdot)$ on $\Bbb R^+$ by $$x\cdot y:=\inf\{r\cdot s\mid r,s\in\Bbb Q \text{ and } x<r \text{...
-1
votes
1answer
46 views

$\omega^\omega$ correspondence with $\mathbb R$-irrationality

Here in the second comment I do not understand why $\omega^\omega$ corresponds to irrational numbers? : In my experience one typically identifies $ω^ω$ with the irrational elements of R; and then we ...
0
votes
1answer
50 views

limsup of a series

The series is from Rudin's *Principles of Mathematical Analysis$ ("Baby Rudin"), p.67. $$\frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+\frac{1}{3^2}+... $$ and Rudin claims that $$\limsup_{n\to \infty} (...
4
votes
2answers
44 views

Maximum of the leading coefficient when $\text{deg}P=6$, $0\leq P(x)\leq1$ for $-1\leq x\leq 1$.

Let $P(x)$ be a real polynomial of degree 6 with the following property: For all $-1\leq x\leq 1$, we have $0\leq P(x)\leq 1$. what is the maximum possible value of the leading coefficient of $P(x)...
1
vote
1answer
710 views

Proving that between any two real numbers there exist a real number [closed]

Formally, I want to prove that if $x$ and $y$ are real numbers such that $x \lt y$ then there exists a real number $z$ such that $x \lt z \lt y$. I want to know whether, in constructing the proof, I ...
0
votes
0answers
94 views

Is the set $(\{n\cos(n)\},\{n\sin(n)\})$ dense in $[0,1]^2$

Let $\{\cdot\}$ be the fractional part. $\Bbb N$ is the set of positive integers. Is the set $\{\ (\{n \cos(n)\},\ \{n \sin(n)\}): n \in \mathbb{N}\}$ dense in $[0,1]^2$? It is known that the set $\...
0
votes
0answers
61 views

The Distribution of Weird Numbers in the Natural Numbers

I have been studying the set of weird numbers, the first 10,000 elements of which are given by https://oeis.org/A006037/b006037.txt, and I noticed something about their distribution that I cannot ...
1
vote
1answer
53 views

Motivating topological proofs

Hi I'm working on Rudin's real analysis textbook and I motivate the proofs about topology using $R$ $R^2$ and $R^3$ in my mind but I know he proves it for $R^n$. Even though I know if it works for the ...
1
vote
1answer
55 views

$\omega^\omega$ correspondence with $\mathbb R$

How does the natural continuous bijection between $\omega^\omega$ and $\mathbb R$ look like? I.e. why elements of $\omega^\omega$ are called reals?
0
votes
0answers
33 views

Finding messages in decimals

A puzzle asked if there is message in the decimal expansion of $\pi$, starting after the decimal point. The answer, with $\pi$= 3.1415, using A1-Z26 conversion, split 14|15 is no. Trying this for $...
2
votes
1answer
58 views

Partition of positive reals with each part closed under addition without choice

It is an easy exercise using transfinite recursion to prove the following (in ZFC): There exists sets $S,T$ that partition $\mathbb{R}_{>0}$ such that each of $S$ and $T$ is closed under ...
1
vote
0answers
41 views

Showing $\mathbb{Q} \cap [a,b]$ is an open set in $\mathbb{Q}$ for irrational $a$, $b$.

I came up with this lemma (although not confident enough about it) while solving Baby Rudin. In the chapter "Basic Topology", I attempted to solve question 16, in which $\mathbb{Q}$ is regarded as the ...
0
votes
2answers
23 views

Absolute Value and Exponents

In my homework I've been accustomed to assuming that $|x|^a = |x^a|$ Recently however, I've begun to doubt that. Take the following example: $$ \begin{equation*} \begin{split} |\sqrt{-|x|} | &= ...
0
votes
0answers
23 views

Trying to extend distributive property of modulo operation to real numbers

Here Wikipedia states that modulo operation is distributive: $$a \cdot b\ mod\ n = (a\ mod\ n)\cdot (b\ mod\ n)\ mod\ n$$ Which is true for every natural number. Unfortunately it is not for rational ...
-3
votes
1answer
62 views

How to prove $|x − y| ≤ |x| + |y|$, proof and reasoning [duplicate]

Prove that, for all $x, y ∈ \mathbb{R}$, we have $|x − y| ≤ |x| + |y|$. Can I say . $|x − y|^2 ≤ (x − y)^2$, and work from there? Thank you