Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
4,750 questions
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Does "the preimage of a closed interval is a finite union of closed intervals" imply $f:\mathbb{R}\to\mathbb{R}$ is continuous?
Suppose I have a function $f:\mathbb{R}\to\mathbb{R}$ with the property that for any closed interval, its preimage is a finite union of closed intervals. Can I conclude that $f$ is continuous, or do ...
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Solve $[x]+[x^2]=[x^3]$
the problem
Solve $[x]+[x^2]=[x^3]$
my idea
using the fact that $ x=[x]+ \{ x \} $ we can write the equation as $x^3-x^2-x=\{x^3\}-\{x^2\}-\{x\} \in (-2,1)$ because ${x} \in [0,1)$
Now we can solve $x^...
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Domain and range of $f(x)=\frac{\sqrt{x-5}}{\sqrt{3-x}}$ — is the domain empty or not?
I’m a high-school student working on finding the domain and range of the following function
$$f(x)=\frac{\sqrt{x-5}}{\sqrt{3-x}}$$
My reasoning (straightforward conditions):
For the numerator to be ...
- 221
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How $\min(x,y)$ works in reasoning
I came across this proof from a reliable book, showing that the set $X=\lbrace x \in \mathbb{R} \mid x \gt 0 \text{ and }\:x^2\leq2\rbrace$ has a supremum $a$ which verifies $a^2=2$. We have proven ...
- 177
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Axiomatization similarities between Naturals and Magnitudes
Given certain Peano-like axiomatizations of the naturals (but not all of them), the axioms of Well-ordering and Induction are equivalent in that you could use either one and get the same system.
Well-...
- 557
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Quotienting the real line by conjugate elements of real affine group $\mathrm{Aff}(\Bbb R)$
Consider the real affine group $\mathrm{Aff}(\mathbb{R})$ and elements $p,q,z\in \mathrm{Aff}(\mathbb{R})$ such that $$p=z^{-1}q z.$$ Consequently, $p,q$ are by definition conjugate. Assume that $\...
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finding the maximum integer value in a quadratic with one real number and one integer
I have the following equation:
$$
n^2 = \frac{49000000000000\epsilon^2-2814154000000\epsilon+40405422121}{1000000000000\epsilon^2+5656854000000\epsilon-705671}
$$
where $n$ is a positive integer, and $...
- 8,251
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What real numbers correspond to second order logic definable Dekekind cuts?
This question spurred from a thought I had: does every (lower) Dedekind cut have a (finite) second order logic formula that defines it?
Fix the usual setting: the domain is $\mathbb{Q}$ with the order ...
- 2,584
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Monotonicity of odd powers of reals in strictly constructive/choice-free setting
Working with the Dedekind real numbers, in a fully constructive, choice-free context: can we show that if $f(x) = x^{2n+1}$ is monotonic? That is, if $x \le y$, then $x^{2n+1} \le y^{2n+1}$?
I have ...
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Gnuplot gave me some trouble plotting the Fourier series of $f(x) = e^{-|x|}$. Anyone else have this experience when plotting a Fourier series? [closed]
I wanted a plot of:
\begin{equation}
f(x) = e^{-|x|}
\end{equation}
and I wanted to compare $f(x)$ to its Fourier series ($n = 1,3,20$):
\begin{equation}
F(x) = \frac{e^{\pi}-1}{\pi e^{\pi}} + \frac{2}...
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Are the constant functions in $\mathcal C (X, \mathbb R)$ first-order definable?
This question is partially inspired by this one, which also deals with model theory on rings of continuous functions $\to \mathbb R$.
Consider the ring $\mathcal C(X)$ of continuous functions $X\to \...
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On the cardinality of Cartesian product of infinite sets
I was arguing with a friend over whether the cardinality of $\Bbb C$ equals the cardinality of $\Bbb R$. A proof I found stated that $ |\Bbb C| = |{\Bbb R} \times {\Bbb R}| = |{\Bbb R}|^2 $. Hence, ...
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Is the number system for $x$ assumed beforehand when proving the quadratic formula?
When proving the quadratic formula (or any other mathematical equation, definition, formula, etc., from like all the way from basic math to advanced calculus), do we have to assume/declare the number ...
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On the Necessity of Constructing Number Systems from First Principles in Real Analysis
When studying real analysis, is it necessary to go as deeply as Terence Tao does in Analysis I, for example by constructing the natural numbers, integers, and rationals from first principles? Or is it ...
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Is ℝ uniquely determined by the “point removal splits it into two copies of itself” property?
Let $X$ be a non-empty topological space satisfying the following conditions. Is $X$ homeomorphic to the topological space $\mathbb{R}$?
$X$ is $T_1$.
$X$ is connected.
For any point $x$ in $X$, the ...
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