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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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Relation between lubs of a set and its subset

If $A\subset B$ and $B$ is bounded above, show that $\operatorname{lub}A \leq \operatorname{lub}B$. This seems very obvious but I am not able to write a proper solution for this. Does it really require a proof or we can logically conclude this?…
V2002
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rational exponents inconsistency

Why is there an inconsistency for (-1)^0.6 (or (-1)^(6/10)). I know in school I was taught to simplify the exponent to 3/5, which results in -1. However, when I leave the exponent as 6/10, I get a either the 10th root of (-1)^6 = 1 or…
E.Eckstein
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Decimal expansion proof

Prove that if $x
user781232
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How to prove that a set is not a finite union of intervals?

Consider the set $\mathbb{R} - \mathbb{Z}$. It is certainly a countable union of intervals. How does one prove that it is not in fact a finite union of intervals?
user107952
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Is 0 a real number?

I am just curious if 0 is a real number. The definition of a real number is all rational and irrational numbers. And the def definition for rational number is that "$\mathbb{Q}={a\div b|a,b\in\mathbb{Z}}$". But in done say that $0$ is a whole number…
user803596
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$ a_1, a_2, ...... ,a_n $ are real numbers. Prove that there is a real number $ k$ such that $ a_1 + k, a_2+k,\ldots, a_n +k $ all are irrational.

$ a_1, a_2, ...... ,a_n $ are real numbers. Prove that there is a real number $ k$ such that $ a_1 + k, a_2+k,......, a_n +k $ all are irrational. My thinking: If $ a_1, a_2, ...... ,a_n $ these are real numbers and $ k$ is irrational then the…
Chris
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Compare $10.1$ and $5+\sqrt{26}$

How would you compare $10.1$ and $5+\sqrt{26}$? If we square, $102.01$ $?$ $51+10\sqrt{26}$...
Knowledge Greedy
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Is there a non-trivial distinction between $0.999...$ and $1$?

Although I fully accept that $0.999...=1$, I have always wondered if this means that they always have identical properties. For example: Does $3-2=0.999...$? Can $0.999...$ be used to describe the cardinality of a set (even if this is contrived and…
Joe
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What is the intersection of $\Bbb{R}^2$ and $\Bbb{R}$?

$\Bbb{R}^2=\Bbb{R}\times\Bbb{R}$, if I then try and imagine the intersection of $\Bbb{R}^2$ and $\Bbb{R}$, I see a plane built by fixing two real number lines perpendicular to each other and try to imagine how $\Bbb{R}$ intersects with this. Does…
Charlie
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Order proof using axiomatic

I started today studying real numbers axiomatic and I'm a bit stuck showing a proof of a property of order in real numbers. This is the problem: if $r<0,$ then $-r>0$. I have to do it just using axiomatic. Thank you for any ideas.
Richard Kingston
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Is it true that any collection of non-negative extended real numbers have a sum?

As I understand the set of extended real numbers ($\bar{\mathbb{R}}_+\left[0, \infty \right]$) has the property that any subset of it has a unique least upper bound. In the following $[n]$ with $n \in \mathbb{N}$ denotes $\{ k \in \mathbb{N} \mid k…
Necrosovereign
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Decimal expansion of 1

Let $E$ be the set of all x $\in [0,1]$ with decimal expansion containing 0s and 9s. Is E closed? Or more specifically do I consider the limit point 1 being part of set E or not?
Jia Cheng Sun
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Prove that there is no positive rational number a such that a^2 = 3.

I was doing some exercises from the book called Basica Mathematics - Serge Lang and here's the two problems: 1) Prove that there is no positive rational number a such that a^2 = 3. You may assume that a positive integer can be written in one of the…
bl4ckch4ins
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Definition of Real Numbers in Infinity?

Real numbers is a set of all subsets A $⊂$ $Q$ with this features: A $ ≠ ∅$ , A $≠ Q$ A is closed from underneath, ( $∀$ x,y $∈ Q$) ( x < y $∧$ y $∈$ A) $⇒$ x $∈$ A A doesn’t have the biggest element. My question is how can A be closed from…
Mia09
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Convexity in one dimension

For which values of $x ∈ \mathbb R$ is the set $A=[0,2] ∪ [x,x+2]$ convex? (a) $−2 < x ≤ 1 $ (b) $−1 ≤ x ≤ 2 $ (c) $−2 ≤ x < 2$ (d) $−2 ≤ x ≤ 2 $ My attempt: both $b$ and $d$ by using $(1 − t)x + ty$? But there should only be 1 correct answer.
convex1
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