Mathematics Stack Exchange
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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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Are there real numbers that are neither rational nor irrational?

I wouldn't have asked this question if I hadn't seen this image: From this image it seems like there are reals that are neither rational nor irrational (dark blue), but is it so or is that illustration incorrect?
Alex Azazel
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What is so wrong with thinking of real numbers as infinite decimals?

Timothy Gowers asks What is so wrong with thinking of real numbers as infinite decimals? One of the early objectives of almost any university mathematics course is to teach people to stop thinking of the real numbers as infinite decimals and to…
TripleA
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Prove that any real number can be expressed as the sum of two irrational numbers

Prove that any real number $r$ can be expressed as the sum of two irrational numbers $x$ and $y$. Progress: I have a specific example for any rational number $r$: $x = r-\pi$ and $y = \pi$ (or replace $\pi$ with any irrational number.) However, I…
Jadelin Martin
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Is there a website like OEIS for real constants?

I'm hoping an online service exists where I can type in say 3.14159 and it then shows a bunch of 'interesting' (however one would define that) numbers. Naturally in that instance it would bring up $\pi$ first as the most likely candidate, then…
Ben Crossley
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Is there a sense in which $\mathbb{R}$ does have 'holes'?

The somewhat intuitive idea of 'holes' all depends on the definition of the word of course. In a much more naive notion of 'holes', $\mathbb{Q}$ doesn't have any, as for any two $p, q \in \mathbb{Q}$ ($p < q$) one can always find another $r \in…
SvanN
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For what $x$ is there some $k$ such that $x\uparrow\uparrow k$ is an integer?

Apparently it is unknown whether $\pi \uparrow\uparrow 4$ is an integer (I learned this from this tweet). I'm curious about which real numbers have some power tower which is an integer. That is, facts about the set: $S = \left\{x \in \mathbb{R}…
Davis Yoshida
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How can I mathematically split up a 3 digit number?

For example, if I have 456, How can I split this and then let each column value be a separate number? The only way I can think of doing this would be to subtract '100' n times until that column is '0' and storing the number of subtractions, leaving…
b20
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Solve the equation $2^{1-x} + 2^{\sqrt{2x-x^2}}=3 $

Solve the equation $$2^{1-x} + 2^{\sqrt{2x-x^2}}=3 \tag 1$$ on reals, using elementary knowledge (using trigonometry or logarithms is allowed, but without limits, differential calculus etc.) We have to find solutions on $[0,2]$ interval. Two…
user261263
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Why should I believe that the real numbers model distances along a line?

Taking the real numbers to be a complete ordered field, why do we believe that they model distances along a line? How do we know (or why do we believe) that any length that can be drawn is a real number multiple of some unit length?
user832339
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Distinguishing properties of $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ that lead to differing cardinalities?

I have what many on here would consider an elementary question, but I would very much appreciate responses that use only elementary ideas, if possible, so that I can understand them. I would also appreciate detailed rather than brief responses. By…
layman
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Infinite number of decimal places?

here's my question: Let's say you have two numbers. For this example, it can be 24 and 25. Now, as I understand it, there can be decimal intervals between them, such as 24.2, 24.34 and even 24.788843. Now, what I'm wondering is: Is there the…
CinnabarToffee
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Prove (or disprove): $a\times b=c\times d$, the solutions for $x$ in the equation $\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$ is only $\pm 1$.

Prove (or disprove): If $a,b,c,d$ are positive real numbers with $a\times b=c\times d$, then the only solutions for $x$ in the equation $$\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$$ are $x = \pm 1$. Other than the obvious $a=b=c=d$ solution.
GohP.iHan
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How to check whether the following inequality holds or not?

I am given with real numbers $r,k > 1$. Is the following true? $$\frac{rk}{r+k-1} > 1$$ I have taken a few reals and the result holds for them. Does this inequality holds in general or not?
PAMG
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Discrete Math Proofs Involving Real Numbers

I am stuck on these two problems. $1$. Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a}+\frac{1}{b} + \frac{1}{c}) \ge 9$. $2$. Prove that for every three positive real numbers a, b, and c that $a^2 + b^2 +…
mrQWERTY
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Converse to a proposition on finite sets of real numbers

Every finite set of reals has the property that every non-empty subset of it has a maximum and a minimum. Does this property characterize finite sets of reals? That is, is it the case that if $S$ is a subset of the real numbers where every non-empty…
user107952
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