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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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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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 some things in the inductive step, when you want…
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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 below argument that shows $999\ldots = -1$ is…
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All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$

I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$. I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with infinite periods denoted by brackets. $$2=\sqrt{2 +…
Yuriy S
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Is there a domain "larger" than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and multiplication always generate natural numbers,…
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Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never deal with irrational numbers, but only rational…
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Cover of "Gödel, Escher, Bach"

Consider the cover image of the book "Gödel, Escher, Bach", depicted below. The interesting feature is that it shows the existence of a subset of $\mathbb{R}^3$ which projects onto $\mathbb{R}^2$ in three different ways to form the letters of the…
Kind Bubble
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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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Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?

I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition. A real number is a cut, which parts the rational numbers into two classes. Let $\mathbb{R}$ be the set of cuts. A cut is a…
God bless
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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, could one say that $5+2i> 3$ because the real part of…
Juanma Eloy
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Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots together. The square root, multiplied by the cube root,…
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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.
Makoto Kato
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Why are real numbers useful?

A question (by a fellow CS student taking a first course in calculus, presumably after the lecture in which continuity was introduced: was as follows. In the real, physical world, we deal with numbers that are sort of “finite” or “discrete” by…
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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. Since almost all real numbers are not computable,…
Emilio Novati
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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 just assume that "$a + 0i$" is equal to the real…
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