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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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How to represent an arbitrary real number in $[0,1)$

Let $\{a_k\}_{k\geq 1}$ be a sequence of positive integers such that each $a_k > 1$. Show that every real number $x\in[0,1)$ can be represented as $$x=\sum_{k=1}^\infty\frac{x_k}{a_1a_2 \cdots a_k} ,$$ where $x_k\in\{0,1,...,a_k-1\}$. Edit by…
Alexander Lau
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Proving that between any two real numbers there exist a real number

Formally, I want to prove that if $x$ and $y$ are real numbers such that $x \lt y$ then there exists a real number $z$ such that $x \lt z \lt y$. I want to know whether, in constructing the proof, I should think in terms of continuity of the real…
MHall
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How to name extremely large numbers?

Currently I'm dealing with extremely large numbers and so I've been wondering how to name them... I know of the usual Definitions, like a Decillion is 10^33 or 10^66 on the short and long scale respectively (Here, I'm just using the short…
BloodEchelon
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If $a-b\le c<\infty$ then $a<\infty$

Given arbitrary $a,b$ in the extended reals do we really know that; If $a-b\le c<\infty$ then $a<\infty$
Number4
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Definition of multiplication by an irrational

I suppose I'm looking for more of an intuitive answer here. Recall if we raise a number $x$ to an $a$ power where $a$ is an irrational, then $x^a$ can defined as the limit of $x^{q_n}$ with $a$ as the limit of the sequence {$q_n$}. Since we know…
Ecotistician
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Prove that if $x =\frac{p}{q} \in (0, 1], q > 1$, then the period $P$ of repeating digits of $x$ is in fact less than or equal to $q − 1$.

Prove that if $x =\frac{p}{q} \in (0, 1]$ is a rational number, $q > 1$, then the period $P$ of repeating digits in the decimal representation of $x$ is in fact less than or equal to $q − 1$.
Nykis
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Need to show that the Euler Constant lies in the interval (1/2,3/5)

Consider the sequence 1+1/2+...+1/n - log n. I have shown that it is strictly decreasing and bounded below by zero. I need to show that the limit of this sequence ( the Euler Constant) lies in (1/2,3/5).
nandi
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How to define natural numbers in terms of real numbers?

Some problems of theory of real numbers are about proving that some real numbers are irrational. But I find it a bit confusing to prove that some real number cannot be expressed (equal) to the quotient of two intgers, when you are working in the…
Юрій Ярош
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How to prove that for $0 < a \leq 1$ and $a \not = 1/n$ there exists the maximal $k$ s.t $k\cdot a < 1$

In this article of IIME Journal, at page 16, it is claimed that for $0 < a \leq 1$ and $a \not = 1/n$ for any integer $n$, there exists an integer $k$ such that $k\cdot a < 1$ and $k$ is the greatest integer satisfying this condition. However, how…
Our
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If $a,b,c$ are rational numbers and $a\sqrt2+b\sqrt3+c\sqrt5=0$, show that $a=b=c=0$

If $a,b,c$ are rational numbers and $a\sqrt2+b\sqrt3+c\sqrt5=0$, show that $a=b=c=0$ I can solve for two terms only but for additional $c\sqrt5$ term, I got stuck. I also want to know how to solve this for general $n$ terms i.e.…
DREAM ISI-CMI-IIT
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To show that $x^2+y^2\propto xy$.

Let $x$ and $y$ are variables satisfying $3x-4y\propto\sqrt{xy}$. Then show that $x^2+y^2\propto xy$, where $y∝x$ implies $y$ directly proportional to $x$. My attempt... $3x-4y=k\sqrt{xy}$ for some real $k$. This implies $(3x-4y)^2=k^2xy$ implies…
abcdmath
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Is the construction of real numbers using Dedekind cut a consequence of completeness of real numbers?

I am a bit confused on whether completeness of real numbers allows for their Dedekind construction or whether Dedekind construction dictates the completeness of real numbers. In other words, is completeness an inherent property of real numbers, or…
Mathnewbie
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$a,b,c,d\in\Bbb R$; $a+b+c+d=0,M=ab+bc+cd,N=ac+ad+bd$; prove that at least one of $20M+17N$ and $20N+17M$ is non-positive

$a,b,c,d$ are real numbers summing to zero. Let $M=ab+bc+cd$ and $N=ac+ad+bd$. Prove that at least one of the sums $20M+17N$ and $20N+17M$ is non-positive.
Aytajdadashova
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Calculate the numbers of characters to display 512Bit of information

Let's say I want to use an alphabet of 6.000 symbols to display 512 Bit of information. With my rudimentary math skills, I figured out this equation should give the answer: 6000^x =…
hdev
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