Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
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Is it true that $0.999999999\ldots=1$?
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
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Is an automorphism of the field of real numbers the identity map?
Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map?
If yes, how can we prove it?
Remark An automorphism of $\mathbb{R}$ may not be continuous.
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Induction on Real Numbers
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.
Of course you have to change ...
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Completion of rational numbers via Cauchy sequences
Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
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Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?
Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
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Why are integers subset of reals?
In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). ...
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Proving that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$
Let $a$ and $b$ be positive reals. Show that
$$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
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Is "$a + 0i$" in every way equal to just "$a$"?
I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$).
I say this is practically the case, so in every calculation you ...
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Is the real number structure unique?
For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.
In my analysis class, our book ...
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Do we really need reals?
It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.
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Definable real numbers
Reading this Wikipedia page I found this definition:
A real number $a$ is first-order definable in the language of set
theory, without parameters, if there is a formula $\phi$ in the
language ...
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Why does an argument similiar to 0.999...=1 show 999...=-1?
I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc.
Can anyone point me to resources that would explain what the ...
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Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?
I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.
But does it ever make sense to compare a real number and a complex/imaginary one?
For example, ...
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Why does the Dedekind Cut work well enough to define the Reals?
I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals.
I just can't get the ...
user11470
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What is a natural number? [duplicate]
According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the ...
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