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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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First number in an open interval?

What is the first number in an open interval? For example, if I have the open interval (0, 1), what is the lowest number in that interval? Does this question even make sense with real numbers?
Publius
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Prove the completeness of the real numbers

It confused me for a while. Here is the question. Prove that given any Cauchy sequence of reals, there exists a Cauchy sequence of rationals that converges to the same value.
Sk8er_ZZZ
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Simplify this expression involving Gamma function

Simplify $\dfrac{2a\cdot\Gamma(2a)}{\Gamma(2a+1)}$. Where $a$ is any positive real number.
SA-255525
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Prove that equality occurs if and only if x = y

Prove the arithmetic geometric mean inequality. That is, for two positive real numbers x,y we have sqrt(xy) is less than or equal to (x+y)/2. Furthermore, equality occurs if and only if x = y. I have proved the first part but I was wondering if…
Ayoshna
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"$\leq_\Bbb{R}$" is restriction of "$\leq_{\overline{\Bbb{R}}}$" to $\Bbb{R}$ ?!

In definition of Affinely Extended Real Numbers, I think that "$\leq_\Bbb{R}$" is restriction of "$\leq_{\overline{\Bbb{R}}}$" to $\Bbb{R}$ Is it correct?... Thanks in advance!
mle
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Real number multiplicative inverses expressed in another form

I've been asked to express the multiplicative inverse of $3 + \sqrt{5}$ in the form $c + d\sqrt{5}$, where $c,d$ are rational numbers. I understand that for some rational numbers $c,d$ we must have: $$1 = (3 + \sqrt{5})(c + d\sqrt{5}).$$ I was able…
user121947
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Open balls centered at infinity in the extended reals?

Is an open ball of some radius $\delta >0$ defined at $+\infty$, if $+\infty$ is in the extended reals? (I have the same question for $-\infty$.)
jack jacob
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constructing $\mathbf R$ via binary search

Using the function $$f(x) = \begin{cases}\frac{x}{4} &\text{if }|x|\le 2\\[0.3em] \frac{|x| - 1}{x} & \text{otherwise}\end{cases}$$ one can map $\mathbf Q$ into the interval $[-1, 1]$ in a way amenable to binary search. Suppose $x$ is the real…
node196884
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Dedekind cut implies Archimedean Principcle

Similar questions have been asked before. This is problem of Foster's Analysis 1. Prove the Dedekind cut: Let A, B non empty subsets of $\mathbb {R} $ and $A\cup B=\mathbb{R}$ such that $x \lt y$ for all $x \in A$ and $y \in B$, then there is…
Peter Szilas
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Does ...999.999... = 0?

Does the number $...999.999... = 0$? My reasoning for asking this is outlined below. First, for base 10 it can be shown that $...999 = -1$. In addition it is common knowledge that $0.999... = 1$. This seems to imply that $...999.999... = (-1 + 1) =…
Inquisitive
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Isomorphism between equivalence classes of Cauchy sequences and Dedekind cuts

Real numbers can be defined in terms of equivalence classes of Cauchy sequences and in terms of Dedekind cuts. It can be shown that both of these constructions give a complete ordered field. It is also known that up to isomorphism, there is only one…
wiktoria
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Order of operation: division and multiplication

I always thought division and multiplication have the same order of operation, so it does not matter which of these you do first. But in helping a young student with her math I realized that it does matter. Example 1: 8 x 5 / 10. Multiplication…
Mauritz Hansen
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Meaning of ℝ (\$\Bbb{R}\$)

What is the meaning of "blackboard bold" letters, such as $\Bbb{R}$ (written in MathJax as $\Bbb{R}$)? I saw this letter here: MathJax basic tutorial and quick reference ...and in a machine learning text like this: Any other insight beyond my…
Gabriel Staples
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Shifting Finite Sets to Cover Points without Intersection

I am interested in finite subsets $S\subseteq\mathbb{R}$ that may be shifted to cover any point not contained within themselves on the real line without overlap. Formally, these are sets $S\subseteq\mathbb{R}$ such that for each…
Thomas Anton
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Proof that all reals can be written as the sum of an integer and a real between 0 and 1

I found this statement in my textbook and I wasn't able to prove it: $$\forall x \in \mathbb{R} ; \exists (k, r) \in \mathbb{Z}\times [0, 1) / x = k + r $$ I tried proving it by contradiction and succeeded in doing it for numbers in $\mathbb{Z}$,…
HP Impact
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