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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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show that there is no a positive integer $n$ for which $\sqrt{n+1} + \sqrt{n-1}$ is rational

I do not understand how to do this? I have tried to prove it by contradiction and I proved, assuming $\sqrt{n+1}+\sqrt{n-1} = \frac{a}{b}$, that $2n + \sqrt{n^2 - 1}$ divides $a$ and $b$ and so it is not rational but that does not prove it (both $a$…
Potato
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Is it true that a subset $U \subset \mathbb{R}$ is uncountable if and only if it contains an interval?

Is it true that a subset $U \subset \mathbb{R}$ is uncountable if and only if it contains an interval? I feel like this should be so but there could be some wonky counterexample.. Thanks! edit: The motivation for my question comes form "An…
user637978
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Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable?

Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable? In particular, I am thinking of a subset of $\ \mathbb{R}\ $ where some condition was imposed on the…
Adam Rubinson
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How does one write $0$ as a fraction in lowest terms?

In Spivak's Calculus, two established properties of real numbers are: (1) If $a$ is any number, then $a+0=0+a=a$ (2) For every number $a$, there is a number $-a$ such that $a+(-a)=(-a)+a=0$. Looking at (2) and letting $a:=0$, we have: $0…
S.C.
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Negation of the Completeness Axiom

There are a lot of theoretical results in Real Analysis, Calculus, Topology, etc. that depend on or use this result below. This has been a question that has bothered me for a long time and seems to me very similar to Euclid's Fifth Postulate in…
W. G.
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Prove that addition under "real numbers" is well defined.

Problem: If we denote real numbers as Cauchy sequences and: $$[\{a_i\}]+[\{b_i\}] = [\{a_i+b_i\}] ; i∈ N$$ Show that "$+$" is well defined under real numbers. My try: Assume that: $$ \begin{cases} [\{a_i\}]=[\{a_i'\}]\\ \\ …
Arian Ghasemi
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Pointing out $\sqrt 2$ on number line

We know that using a compass and a ruler we can point out $\sqrt 2$ on the number line. But we don't know the last digit of $\sqrt 2$. So how can we be sure that the pointed number is $\sqrt 2$?
Shreya Chauhan
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Proving for all $n\in \mathbb{N}$, $(2\pi k, 2\pi k+1/n) \cap\mathbb{N} \ne\emptyset $ for some $k \in \mathbb{N}$

I have been trying to prove the following for quite a while but have been unable to do so: For all $n\in \mathbb{N}$, $(2\pi k, 2\pi k+1/n) \cap\mathbb{N} \ne\emptyset $ for some $k \in \mathbb{N}$. I believe $2\pi$ is not something special in…
ashK
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Can the sum of two irrational roots of two coprime natural numbers greater than 1 be an irrational root of a rational number?

After thinking more about my last question and reading the answers I reformulated it. I'm positive I wanted to ask the following: Let $x,y\in \mathbb{N};\: x,y>1;\: gcd(x,y)=1; \: \sqrt{x},\sqrt{y}\notin\mathbb{N}$ and let $z\in\mathbb{Q}; \:…
DaifM
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How to solve the "four" variables problem

Given x, y, z, and w are real numbers which satisfy these three equation: $$x^{2} + 5z^{2} = 10$$ $$yz - xw = 5$$ $$xy + 5zw = \sqrt{105}$$ Find the value of $y^{2} + 5w^{2}$ Can anyone give a hint? I don't even know where to start
Anas Ghazi
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Nested interval principle

I am asked to prove the nested interval principle by using the axiom of completeness. That is, for a decreasing sequence of nested closed intervals $I_1, I_2, I_3,...$, there exists exactly one $x ∈ R$ such that $∀n ∈ N : x ∈ I_n $. The way I…
user600210
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Archimedean field with non total order

I am currently looking for the simplest way to characterize the real numbers. Usually they are described as the complete archimedean field, showing that all such fields are isomorphic. Archimedean means that for all $x$ in the field, there exists a…
V. Semeria
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Does every real number have a unique binary representation?

I'm actually intereseted in real numbers that belong to the interval $(0,1)$, but a more general answer will be great
McLovin
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A bound for a real rational function

Let $r\geq 1, $ and $\theta$ be any real number. Then \begin{equation}\frac{1-r\cos\theta}{1+r^2-2r\cos\theta}\leq \frac{1}{2}.\end{equation} For, wehave $2\leq 1+r^2\implies2-2r\cos\theta\leq1+r^2-2r\cos\theta,$ which establishes the above…
user159888
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The number of real solutions of complicated surd

I have tried to solve this question, and I found the one real root, x=0, but I do not understand how to show that there is only one root: Find the number of real solutions of $x+\sqrt{x^2+\sqrt{x^3+1}}=1$ The solution to the question, is 1 real…
frog0101
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