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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Can there be a sensible unique subset of all definable real numbers? Why is it unable to serve as a replacement of the canonical set?
I've heard that reals is the least (well, I'll say it in intentionally vague way, so it will precisely reflect my current understanding) number-like set that is closed under an operation of taking ...
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Proving, using Dedekind cuts, that $C(0)$ is an additive identity for addition on cuts.
My source is Franck Ayres' Modern Algebra. The author states the fact under discussion (Chapter 7, "Real numbers") but does not provide a proof.
My question is about the second part of the ...
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Clarification about Cantor's Diagonal argument compared to Natural Numbers
I'm not a mathematician but I am a software engineering student. From what I've understood so far, the Cantor diagonal argument proves that the real numbers are infinite and uncountable. My biggest ...
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Dedekind cuts : establishing the equivalence of 2 definitions of addition on positive cuts
My source is Ayres, *Modern Algebra", $1965$ ed , ch. $7$, § "additive inverses" , p.$68$.
A cut in $\mathbb Q$ is defined as a non empty proper subset $C$ of the rationals such that (...
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Can a straight line be drawn through a single node on an infinite square grid without passing through any other nodes?
The problem is from an advanced 8th grade math curricula, and marked with a star:
*The topic is "Real numbers"
The plane is covered by an infinite square grid. Is it possible to draw a ...
- 570
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$-a=(-1)\cdot a,\forall a\in\mathbb{R}$ using axioms
I've been trying to prove
$$-a=(-1)\cdot a$$
for every $a\in\mathbb{R}$ using only axioms, however, every demonstration I found use one of these two properties:
$$0\cdot a=0,\quad\forall a\in\mathbb{R}...
- 2,084
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Are there any theorems that use the uncountability of the reals in their proof?
Can we use the uncountability of the reals as a tool to prove any theorem? Can we use this to calculate anything? Suppose I was trying to convince a pragmatist that uncountability is useful.
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Unable to come up with correct bounds for showing sequence convergence. Analysis I, Terence Tao, Theorem 6.1.19, part (g)
I was trying to prove the following (part (g) of the Theorem):
Theorem 6.1.19 (Limit Laws). Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numbers, and
let $x,y$...
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Measure theory, prove the countable additivity of measure
I am reading Cohn's Measure Theory, here is an exercise from Chapter 1, Section 2.
Let ($X,\mathcal A,\mu$) be a measure space, and define $\mu^{\star}:\mathcal A\rightarrow[0,\infty)$ by
\begin{...
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Prove that for any rational number $t$ , there is a solution of the equation $ax^2+by^2=t$.
Let $a$ and $b$ be two non-zero rational numbers such that the equation $ax^2+by^2=0$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the ...
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Complex numbers : finding real & imaginary numbers and the magnitude
enter image description here
My Thoughts are:
Using geometric series $S_n= a(r^{n+1} -1) /(r-1) )$ for $a=1 , r=2i , n=26$
Now as I know that $i^2=-1$ , so $i^{27} = i^3 =-1$
$\implies (2i)^{27} = (2^...
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Convergence of rapidly decaying series
Let $0<\lambda_1<\lambda_2<\dots$ be an increasing sequence of positive real numbers with $\lim_{k\to\infty}\lambda_k=\infty$.
Let $a_1,a_2,\dots$ be a sequence such that for all $m>0$,
$$...
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Can a non-trivial continuous function "undo" the discontinuities of another function?
Apologies for the unclear title, I have no idea if the property I'm looking for has a better name.
I'm wondering if there exists a pair of functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ such ...
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How do I write this Theorem with quantifiers?
Here is the theorem from Steven Abbot's Understanding Analysis.
Theorem. Two real numbers $a$ and $b$ are equal if and only if for every real number $\epsilon > 0$ it follows that $|a - b| < \...
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How to construct a nonzero real number between two given nonzero real numbers?
Statement:
Let $$X=$$
$$\{(a,b) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}\setminus \{0\}:a<b\}$$
There exists a function $f:X \rightarrow \mathbb{R} \setminus \{0\}$ such that for all $(a,b) ...
