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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Proving the arithmetic-geometric mean inequality using just only the field and order axioms

Using just the axioms, prove the arithmetic-geometric mean inequality: $$\sqrt{ab}\leq\frac{a+b}{2}$$ for any $a, b \in\mathbb R$ with $a > 0$ and $b > 0$. (Assume, for the moment, the existence of square roots.) The only legal move I seem to be…
Sadio
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Bound on the absolute difference

Let $A$, $\overline{A}$, $\underline{A}$ and $B$ be real numbers, and let $\underline{A}\le A\le \overline{A}$. Is the following correct? $$ |A-B| = \max\{A-B, B-A\} \le \max\{\overline{A}-B, B-\underline{A}\} $$ Apologies for the elementary…
user52227
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Is it possible to list two real numbers that lie immediately next to one another on the number line?

This question stems from my general understanding of Cantor's diagonal proof, which from my understanding suggests this is impossible. At the same time it seems like such an elementary idea, to list two numbers that lie immediately next to one…
IgnorantCuriosity
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What's the supremum of the set ${\{x:x<2}\}$?

By the definition of $\mathbb{R}$ as the Dedekind complete well ordered field, there should be such a supremum. What is it?
user285146
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Proof of $a \le 0$ $\Leftarrow \Rightarrow$ $\forall \epsilon > 0$ $a < \epsilon$

In analysis class I saw a proof but I would like see another Proof: $\Rightarrow$ Suppose that $a \le 0$ $\land$ $\epsilon > 0$ if $a=0$ by hypothesis $a < \epsilon$ if $a<0$ $\Rightarrow$ $a < \epsilon$ $\Leftarrow$ Suppose that $\epsilon > 0$…
Jose Vega
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proof of 1) $|a| = 0 \iff a = 0$ and 2) $|a| \ge 0$

On the internet there are many proofs but very summary Definition $|a| = $ $a$ if $a \ge 0$ $-a$ if $a < 0$ Proposition 1 $|a| = 0 \Leftarrow \Rightarrow a = 0$ $\Rightarrow$) Suppose that $|a| = 0$ This implies that $a\ge 0$ xor $a<0$ Case…
Jose Vega
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Proof of $\Bbb{N}$ is inductive

Here there is a proof but I think it is incomplete(missing $1\in\Bbb{N}$) Note: A⊆R is inductive if and only if 1∈A and ∀x∈A⇒x+1∈A. By definition, $\Bbb{N}$ is the intersection of all inductive sets I tried to do a proof here: $\Bbb{N} = \bigcap…
Jose Vega
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Non-zero Number Multiplication

How would you solve this problem: If $a,b,c$ are non-zero real numbers such that $\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}$, and $x=\frac{(a+b)(b+c)(c+a)}{abc}$, and $x<0$, then $x$ equals $\textbf{(A) }-1\qquad \textbf{(B) }-2\qquad…
Ayush
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Uniqueness of floor and ceiling

I was reading Calculus I by Tom Apostol and it has a question as following: Let $ x $ be any real number, prove that there exists exactly $ 1 $ integer $ n $ such that $ n \leq x < n + 1. $ In this exercise I am allowed to use the fact that the sets…
user298251
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Existence and uniqueness of $y$ such that $xy=(xy)^2$ for every real $x$

I've been asked to prove this theorem: There exists a unique $y \in \mathbb{R}$ such that for every $x \in \mathbb{R}$, we have $$xy = (xy)^2$$ Now, this may seem rather silly, but I'm wondering if this theorem is even true. Clearly, if $y = 1/x$,…
ShinyPebble
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Prove that $\min(S)$ does not exist for $S=(0,1)$

Prove that $\min(S)$ does not exist for $S=(0,1)$. I'm taking the proof by contradiction route i.e. assuming m = min(S) then trying to find some sort of contradiction. I've tried take m=2m-1 and take m = (m-1)/2 but neither seem to work?
Mals T
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Finite Interval under Nonnegative Real Numbers

I want to construct a generic finite interval under $\mathbb{R_+}$ - it should be bounded, closed and it should include $0$ as the lower bound. This would allow me to choose an element from this compact set (i.e. $ x \in [0, b] \subset R_+$ where b…
mathos
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Calculate $\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}$

I need help with calculating the following,$$\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}$$ i have tried to solve it as $$\sqrt{\left(\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}\right)^2}$$ but i've come nowhere.
zermelovac
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Density of computable numbers

Proposition: for any real number, there exists a computable number that is arbitrarily close to it. Is this proposition true/false/undecided?
Dave
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Set of positive bounded numbers in 2 dimensions, excluding zero

I want to state that a robot needs to be inside a bounded area. I would like to have a nice short expression like the one below: $$x \in \mathbb{R}_+^2$$ But what I actually mean is the following: $$ x_1 \in (0,x_{lim}] \\ x_2 \in (0,y_{lim}]…
Wouter Kuijsters
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