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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.

The field of real numbers, usually denoted by $\mathbb{R}$ or $\mathbf{R}$ is a field equipped with an order, which is complete with respect to that order. Moreover, it is the only ordered field which is complete (up to isomorphism). The real numbers are used as basis for measuring "length".

The real numbers can be classified in various ways: rational and irrational numbers; algebraic and transcendental numbers; computable and non-computable numbers; etc.

The real numbers carry a natural topology, which is generated by the order. The topology can be induced by a naturally arising complete metric. See more on Wikipedia.

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An irrational number $x$ such that $x^x$ is rational

I can show that there exists an irational number $x$ such that $x^x$ is rational. But I have no example. Can you give a pricise example of such number $x$?
user390026
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Find the smallest $n$ such that an integer lies between $nx$ and $ny$ for real numbers $x$ & $y$.

Say I'm given two real numbers as inputs: $x$ and $y$, with $x < y$. I want to find the smallest natural number n such that there's at least one integer between $nx$ and $ny$ (inclusive of $nx$ and $ny$). The largest $n$ can possibly be is…
Dave
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prove a number is irrational

If $x$ and $y$ are irrational numbers then $x$ to the power of $y$ is irrational I am asked to prove or disprove this statement. To do so I got an idea to use the contra-positive, for that I need to prove if $x$ to the power of $y$ is rational then…
Ampatishan Arun
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Proof of If $A \subseteq \Bbb{R}$ with $A$ inductive set then $\Bbb{N}\subseteq A$

In calculus class, I saw a proof of this, but I am not convinced. Note: A$\subseteq \Bbb{R}$ is inductive if and only if 1$\in A$ and $\forall x \in A \Rightarrow x+1 \in A$. I tried to do a proof here: Suppose that $A$ is a inductive set. By…
Jose Vega
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What makes a number representable?

The set of real numbers contains element which can be represented (there exists a way to write them down on paper). These numbers include: Integer numbers, such as $-8$, $20$, $32412651$ Rational numbers, such as $\frac{7}{41}$,…
user80458
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Showing $\sqrt{2}, e, \pi$ are real numbers in the axiomatic approach to defining $\mathbb{R}$

I would appreciate if someone could demonstrate how to show $\sqrt{2}, e, \pi$ are real numbers in the axiomatic approach to defining $\mathbb{R}$ (without reference to a model). The Wikipedia page for Real Numbers gives a summary of the different…
muaddib
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Math Homework Question

I think I might be overthinking this question and I need some help Assume only the knowledge of addition and multiplication of real numbers. What do we mean by $u^{-1}=v$ ? Can this equality be verified? Assume only the knowledge of addition and…
user245858
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Proving that a rational cut is a Dedekind cut

I'm new here, so I don't know how to do the fancy symbols. Sorry... This is for my intro. to adv. math class, and I've been struggling this entire semester. I kinda understand the concept being asked, but I have no idea how to go about proving…
David
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Do there exist "random" real numbers?

According to a common perception of the real numbers, it contains any totally random sequence of digits that a monkey could type on a keyboard in infinite time, like $$0.298403840284023840238402480234802348023480240480328402348230\ldots $$ But does…
Gaussler
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What does $\mathbb{R}^J$ mean?

Let J be some index set. I was watching a lecture by Mikhail Gromov and he made a passing comment about why $\mathbb{R}^J$ makes sense because J is a set, but $\mathbb{R}^n$ does not because $n$ is a number. I thought we just used $n$ because it…
o0BlueBeast0o
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given x>0 and n contained in N show show that there exists a unique positive real number $r$ such that $x = r^{n}$

I have proved half of this proof but I'm stuck on the other half because it is a little harder than the first due to negatives. I have considered that $(r+h)$ so that $(r+h)^n$ is less than $x$ and $h$ is between $0$ and $1$. I proved that half.…
Ayoshna
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A sort of partition of real numbers

I'm looking for two injective functions $f, g:\mathbb R\to\mathbb R$ with $f(x)+g(x)=x$ for all $x\in\mathbb R$ and $\operatorname{Im} f\cap\operatorname{Im} g=\emptyset$. I've tried nothing and I'm all out of ideas.
T.J. Gaffney
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Find all finite set S such that any $a, b, c \in S$, $ab + bc + ca \in S$

Find all finite set of real numbers S such that: $ab + bc + ca \in S$ with any distinct $a, b, c \in S$ I just can solve the problem when $\exists$ at least $3$ elements $\in S\ge 1$ In another case, I got stuck.
Xeing
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Are there "number systems" that have real numbers as digits?

I was thinking about how one learns about the different types of numbers. First there are just whole numbers like $1, 2, 3, ...$ Eventually you write numbers in decimal notation like this: $$6.283185307...$$ And I even found some systems that have…
NiveaNutella
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Alternative Constructions of the Reals

I have been reading about the construction of the reals (Tao's Analysis I, Classic Set Theory by Golderi, and bits of extra info from Naive Set Theory by Halmos and Rudin's Principles of Mathematical Analysis) starting from the axiom of infinity,…
ghost
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