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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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Real Analysis. Bounds, Infimum and Supremum

Given that $S=\left\{x \mid 4x^2 > x^3 + x\right\}$. (1) Determine whether $S$ is bounded. (2) Determine their supremum and infimum. I divided the equation by $x$ to have a quadratic. Then my roots are in decimals doesn't look correct. Thank you!
Soledolu Oluwatayo
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Dedekind cuts and multiplication

A common way to define multiplication for Dedekind cuts is to first define it for pairs of positive reals, and to then extend it to general pairs of reals case by case. Is there an alternative definition that is less ham-fisted? edit: I suppose one…
user235033
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What's this number called and what are its properties?

What is the following number called and what are its known properties? $$.12233344445555566666677777778888888899999999910101010101010101010...$$ (I think you get the pattern.) P.S.Also, see what I did with the integer part there? :)
user132181
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When 0 is multiplied with infinity, what is the result?

Any number multiplied by $0$ is $0$. Any number multiply by infinity is infinity or indeterminate. $0$ multiplied by infinity is the question. Answer with proof required.
Murtuza Vadharia
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Mathematical definition of exponent

What does it mean by a^b in real number system? How is it defined mathematically? It is clear in case of exponent being an integer. i.e., a real number a is multiplied b times where b belongs to Z If b is a rational..say b=p/q, then a^b can be…
vara
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Solve in the set of real numbers the equation :$[x]\cdot\{x\}=2007x$

Solve in the set of real numbers the equation :$[x]\cdot\{x\}=2007x$, where $[x]$ is the whole part of x and $\{x\}$ is the fractional part of x First thing to mention is that $\{x\}\in(0,1)$, we can simply check if the equation when verifies when…
IONELA BUCIU
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Solve the equation $x=1-5(1-5x^2)^2$

Solve the equation $$x=1-5(1-5x^2)^2$$ ###My work Let $f(x)=1-5x^2$. Then we have tha equation $f(f(x))=x$. But in this case we don't use the equation $f(x)=x$ because $f(x)$ is not monotonic function
Roman83
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Find the smallest possible number of different numbers between $a + b, b + c, c + a, ab + 1, bc + 1, ca + 1, abc$

Find the smallest possible number of different real positive number between $a + b, b + c, c + a, ab + 1, bc + 1, ca + 1, abc$ $a$, $b$, $c$ are real positive numbers and they're all different. From that, I can tell that $a+b \ne b+c \ne c+a$,…
vlnkaowo
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What does ${\mathbb R^n}$ mean?

I know that $\mathbb R$ refers to all real numbers. But what does $\mathbb R^2$ mean? For that matter, what does $\mathbb R^n$ mean when $n$ equals any natural number?
Xavier Stanton
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What are superreal numbers?

I am trying to understand the extensions of the real numbers. Nothing too serious like a course or research. Especially since the wikipedia article on superreal numbers is pretty empty. (https://en.wikipedia.org/wiki/Superreal_number). Are the…
The Bosco
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Power tower of $10$

Is $10^{10^{10^{10}}}$ the same thing as $10000000000^{10000000000}$? Thought this from writing it as $(10^{10})^{(10^{10})}$. Can it be done like this or should it be written another way?
Henry Lee
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How can we be certain that ℝ fills the entire number line?

It can be a stupid question, but how do we know that ℝ fills the numberline Initially we thought that fractions were enough, that is until we found the irrational numbers. Can't it happen again with a new type of number? Like infinitesimal…
Luis Dias
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Well-Define of Summation of Real Number : Cauchy/Quotient Set Approach

Definition of $\Bbb R$ For two sequences $\alpha,\beta : \Bbb N \rightarrow \Bbb Q$ Define $\alpha \sim \beta$ when $\forall e\;\;$one can pick $N$ s.t. $\forall i \ge N$, $\lvert \alpha(i)-\beta(i)\rvert \lt e$ Since $\sim$ is equivalence…
Beverlie
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How does $y=\frac{2x+2}{x+2}$ contribute to demonstrating the Dedekind cut for $\sqrt{2}$?

In a previous question a respondent corrected my description a Dedekind cut of $\sqrt{2}$. This Wikipedia article states: "Showing that it is a cut requires showing that for any positive rational $x$ with $x^2 < 2$, there is a rational $y$ with $x <…
MathAdam
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Does a Dedekind cut define a real number into existence?

Suppose I cut the rationals greater than $-q$ into two sets, $L=(-q,q)$ and $R=[q,\infty )$. Take $L$ to be the set of all rationals whose squares are less than $2$, and $R$ to be the set of all rationals whose squares are greater than $2$. There…
MathAdam
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