Mathematics Stack Exchange
Mathematics Stack Exchange

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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Does same number of numbers exist between 0 to 1 and 1 to 2 and 2 to 3?

Consider the numbers between 1 and 2. The reciprocals of the numbers will be between 0.5 and 1. Now consider the numbers between 2 and 3. The reciprocals of those numbers will lie between 0.333... and 0.5. And the reciprocals of numbers between 3…
KeSHAW
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Sum of $n$ real numbers

It is quite well-known that $(x_1 + \ldots + x_n)^2 = x_1^2 + \ldots + x_n^2 + 2 \sum_{i=1}^{n}\sum_{j=1}^{i}x_ix_j$. Is there a similarly elegant way to express $(x_1 + \ldots +x_n)^4$?
max_121
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Is the set of all non-zero real number bounded?

Define the set of all non-zero real number as $A = \{a \in \mathbb{R} : a \neq 0\}$. So the obvious gap of the set is the point zero. I am a bit confused by the definition of upper/lower bound, because a tutor of my class said this set $A$ is…
juzmathnerd
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Locating exact position of Real Numbers on a Real Number Line

According to Cantor–Dedekind axiom, corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number. I learnt in high school, the method for finding…
user961447
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Axioms of Real Numbers using Groups

Some texts define the field of real numbers as a set $\mathbb{R}$, on which there are two operations defined (called Addition $+$ and Multiplication $\cdot$, respectively), such that: $(\mathbb{R},+)$ is an abelian group with neutral element…
PAT
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Let a,b and n be natural numbers. If a|n, b|n and g.c.d(a,b)=1 then prove that ab|n.

I am have difficulty carrying out the following proof: Let a,b and n be natural numbers. If a|n, b|n and g.c.d(a,b)=1 then prove that ab|n. I understand that if a|n this implies n=ca for some natural number c and that b|n implies n=mb for some…
Zach Jamison
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On a subset of real numbers

Let $\alpha$ be a real number such that $1$ and $\alpha$ are rationally independent. I need a subset $U\subseteq\mathbb{R}$ such that for $x,x' \in \mathbb{R}$ and $x-x'\not \in \mathbb{Z}$, there exists $m \in \mathbb{Z}$ such that $x-m\alpha -u…
budi
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Is this a good example of an uncomputable number?

Let $BB(n)$ be the $n$-th Busy Beaver number, let $S(2)=BB(2)$ and for $n>2$ let $S(n)=1+S(n-1)+BB(n)$. Then is this an uncomputable…
user3635700
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How to find the values of $k$ if $2k \in \mathbb{Z}$ and $\frac{2}{k}\in \mathbb{Z}$

Let $k \in \mathbb{R}$. What are the possible values of $k$ Under the condition that $2k \in \mathbb{Z}$ and $\frac{2}{k} \in \mathbb{Z}$. I know that the answer is $k \in \lbrace \frac{1}{2}, 1, 2 , -\frac{1}{2}, -1, -2\rbrace$, But I don't know…
Mira
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How to get the two-digit number which is 3/8 of the number I get by swapping the digits of the original?

I know that 27 satisfies that requirement in the title: $$72\cdot\frac{3}{8}=27$$ However, I've obtained this solution by a short python script checking all 2-digit numbers. How can I do better and find it fast by hand? For those interested in the…
zabop
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What is wrong with my bijection from $\mathbb{N} \to \mathbb{R}$ and consequent proof/argument $\mathbb{R}$ is countable?

The most common way to construct R is as a set of equivalence classes of Cauchy sequences, where two sequences are considered equivalent if their term-wise difference converges to zero. These Cauchy sequences can be described in mathematical…
hmmmmmmm
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Understanding Theorem 182 in Landau's "Foundations of Analysis".

I do not get what Landau wants to do in the final remarks of his Theorem 182. He says something about a third case and divides the problem in subcases, but this tell me nothing. Because I can not post pictures, the statement and the proof from the…
Alejandro
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A dense, uncountable subset of reals whose complement is also dense and uncountable in every interval

Does there exist a dense set of reals which is uncountable in every interval, whose complement is also dense and uncountable in every interval? So, for example, the union of positive rationals and negative irrationals would not work for my question.
user107952
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Prove that there are positive real solutions to $x+y=xyz \land xy=x+z \land y+z=yz$

For $x,y,z \in \mathbb{R}_{> 0}$, how can I prove that there are solutions to this set of equations, $$x+y=xyz \land xy=x+z \land y+z=yz$$ in a legitimate way, not by feeding it to WolframAlpha? I have found out that $x = \frac{y}{yz-1} =…
vlnkaowo
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Is this statement true: $x^2<0 \ \land \ x^2 \in \mathbb{R} \ \Rightarrow \ x \in \mathbb{C} \setminus \mathbb{R} $?

So basically, is there a set containing elements not in the complex numbers that squared is a negative real number? $ \{ x \ \vert \ x^2 \in \mathbb{R} \land x^2<0 \} \stackrel{?}{=} \emptyset $
user852404
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