Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
4,556
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Showing the supremum squares to 2
In class, my real analysis teacher defined $\sqrt 2 = \sup \{x \in \mathbb{R} | x^2 < 2\}$ and left proving that this definition implies $\sqrt 2 ^2 = 2$ as an exercise. But I haven't been able to ...
- 129
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3
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Is there a generalization of factoring that can be extended to the Real numbers?
I simply mean that factoring integers is well understood, but factoring an irrational or any real number does not seem to make sense, especially taking into account that a large integer many ...
- 313
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Existence of standard part
My favorite proof of the existence of standard part of a limited $x$ in the context of an extension $\mathbb R \subset {}^\ast\hskip-.5pt\mathbb R$ is to say that $x$ defines a Dedekind cut on $\...
- 44.1k
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How to combine the $4$-dimensions of spacetime into 1 dimension?
I have been thinking about the possibility of representing all points in a $4$-dimensional spacetime coordinate system $\mathbb{R}^{1,4}$, as points on one line $P$ (or axis of a $1$-dimensional ...
- 126
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+100
How many steps are needed to turn one "a" into at least 100,000 "a"s using only the three functions of "select all", "copy" and "paste"?
Suppose that at the beginning there is a blank document, and a letter "a" is written in it. In the following steps, only the three functions of "select all", "copy" and &...
- 531
1
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1
answer
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Proof that Dedekind Cuts are isomorphic to decimal expansions?
The real numbers $\mathbb{R}$ are defined as the superset of the $\mathbb{Q}$ with additional elements to yield Dedekind Completeness. Specifically, every subset with an upper bound in $\mathbb{R}$ ...
- 171
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vote
0
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31
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Field isomorphism between copies of $\mathbb R$
$
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\tld}{\tilde}
\newcommand{\tldt}{\mathbin{\...
- 4,252
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1
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Is the following method for proving density of irrational numbers in real numbers without using rational numbers density in real numbers rigorous?
The motivation for this question is:
I told my friend to use:
$\forall x_{1}, x_{2} \in \mathbb{R}, x_{1} < x_{2}, \exists r \in \mathbb{Q}: x_{1} < r <x_{2}.$
To prove:
$\forall x_{1}, x_{2} ...
- 1,415
3
votes
2
answers
100
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Imagining Rational Cauchy Sequences as Dancing Around a Real Number Instead of Converging to One
I'm trying to build my intuition regarding the Cauchy-sequence construction of the reals.
Essentially, do you think that it is more accurate to visualize a real number as being defined by sequences of ...
- 171
1
vote
1
answer
44
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The product of a Dedekind cut and its inverse equals one
Exercise 1.11 of the textbook Real Mathematical Analysis by Pugh asks what the inverse of a cut $x=A|B$ is ($x$ being positive).
The first step is to show that $x^*=C|D$ where $C=\{1/b: b\in B \space\&...
user1327236
7
votes
1
answer
139
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Is $\sum_{\substack{n_1+\ldots+n_k=2m\\ n_1,\ldots,n_k\in\mathbb Z_{\geq 0}}}x_1^{n_1}\cdots x_k^{n_k} \geq 0$ for all $x\in\mathbb R^k$?
Is the sum of all monomials of the same even total degree positive ?.
In other words:
$$
\mbox{Is}\ \sum_{\substack{n_1\ +\ \cdots\ +\ n_k\ =\ 2m\\[1mm] n_1,\ldots,n_k\ \in\ \mathbb Z_{\geq 0}}}x_1^{...
- 359
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Transcendental nature of natural log for proof validity?
I am following Understanding Analysis by Stephen Abott. I read a well bit into the book but I decided to go through and do the exercises through the book because I felt as if I wasn’t being rigorous.
...
- 1
2
votes
2
answers
60
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Why the property of exponents holds true even for fractional powers
How can we prove that the property of exponents a^m × a^n =a^(m+n) (where "^" this sign denotes the power a is raised to)holds true even if m, n and a are fractions? Like I can clearly see ...
- 49
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17
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A multiplicative calculation for real sets $(0,M)$ and $\emptyset\neq B\subseteq (0,1]$
Let $M$ be a positive real number, $\emptyset\neq B\subseteq (0,1]$, and put $A:=(0,M)$,
$\beta_0:=\inf B$, and $\beta_1:=\sup B$. Then it is easy to see that $BA=\beta_1 A$
(where $BA:=\{ba: b\in B, ...
- 2,491
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votes
1
answer
40
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Majorization equivalence
If $ x\in\mathbb{R}^n $, we note $x_{[1]}\geq\cdots\geq x_{[m]}$ the permutation that arranges $x$ in a nonincreasing order. For vectors $ x, y \in \mathbb{R}^n $, we note $ x \prec y $ if $\sum_{i=1}^...
- 147