Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
4,559
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equality involving supremum and infimum of two disjoint sets that partition the real Line
Let $A$ and $B$ two subsets of $\mathbb{R}$ such that : $A \cup B = \mathbb{R}$ and $A \cap B = \emptyset$.
We suppose that $A$ has a supremum that we denote $\alpha$ , and $B$ has an infimum that we ...
- 393
1
vote
1
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Shifting Index of Recursive Sequence
If I have a recursive sequence defined by:
$a_0 = 7$
$a_n = a_{n-1} + 3 + 2(n-1),$ for $n \geq 1$
How is this recursive sequence the same as the one above. Isn’t $n+1 \geq 2$?
$a_0 = 7$
$a_{n+1} = a_{...
- 139
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0
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Is subtraction on the reals isomorphic to division on the positive reals?
I know that the magma $(\mathbb{R};+)$ of addition on the real numbers is isomorphic to the magma $(\mathbb{R}^+;\times)$ of multiplication on the strictly positive real numbers. I wonder, is it the ...
- 21.3k
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Logical consistency in proof for Real Cauchy sequence implies convergence
I have doubts about this proof I reproduced from a text I have been following.
Any help in full would be appreciated as it would really help me to get more familiar with the trickiness of analysis ...
- 139
0
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1
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Why do we use real numbers for (for example) masses in physics and how do we verify product axioms? [closed]
I have studied the definition of real numbers as V.A.Zorich explains it in his first book Mathematical Analysis I. Basically, he says that any set of objects that respects a certain list of properties ...
- 67
2
votes
1
answer
72
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Showing the supremum squares to 2 [duplicate]
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 ...
- 127
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4
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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 ...
- 341
2
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1
answer
57
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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
1
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0
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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
1
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1
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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
1
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0
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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
2
votes
1
answer
32
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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} ...
- 1,391
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
140
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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