# 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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### Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
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### Showing that $\sqrt{9+9\sqrt{9+9\sqrt{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$?

$$\sqrt{9+9\sqrt{9+9\sqrt{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$$ In the second nested radical, the repeating pattern is $(-,-,+)$. I approached this ...
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### Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
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### What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
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### Both variables algebraic/ both transcendental?

Suppose $X\subset \Bbb Z_{<0}$, and $a=\sum_{n\in X} 2^n$ is algebraic, then is $b=\sum_{n\in X}3^n$ also algebraic? (To clarify, what I mean is: does it hold for all such $X$?) We know that $a$ ...
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### How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the (relatively)...
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### Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
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### Existence and uniqueness up to isomorphism of the real numbers from axioms

Pretty much what the title says: how does one prove the existence and uniqueness of the real number system from the ordered field axioms together with the least-upper-bound property (or maybe some ...
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### The axiomatic method to real number system VS the constructive method(genetic method)

According to book Georg Cantor: His Mathematics and Philosophy of the Infinite - Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the ...
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### Expectation of a matrix over random vectors

Suppose I have a matrix A which is for argument's sake 3 by 3 and has different eigenvalues, let's say 1, 2 and 3. If I was to generate vectors on a unitary sphere $\eta$ and then calculate the dot ...
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### Existence of two subsets of $\mathbb R$ with a certain property, undecidable in $ZF$?

Inspired by Question # 2420627 "Prove that there is a bijection $f:\mathbb R\times \mathbb R\to \mathbb R$ in the form of $f(x,y)=a(x)+b(y)$" and the answer and comments by Thomas Andrews. ...
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### How to prove that the intersection of countably many open sets of reals is either at most countable or of cardinality continuum?

I've learned that any Borel set in $\mathbb R^n$ is either countable or of cardinality continuum from Springer's Problems and Theorems in Classical Set Theory. I wonder if there exists some more ...
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### Sequence in 10-base decimal representation of $\pi$

For some time ago Wolfram launched MyPiDay, which lets you find your 6-digit birthday string in the 10-base decimal representation of $\pi$. For example, my birthday string "970524" starts at the ...
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### infinitely small vs arbitrarily small

In a question I recently asked, Definitions of supremum it was pointed out that my usage of infinitly small was misguided, and the proper terminology was arbitrarily small. Could someone explain what ...
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### Coloring the real plane with a countable number of colors.

We color the real plane with a countable number of colors (each vertex gets a color). Can we always find a rectangle such that all of its vertices have the same color? (the edges of the rectangle need ...
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### Notation for the interval between two arbitrary numbers

Does anyone know of a non-awkward way to notate the interval between two numbers when you don't know which number is larger? For example when describing the remainder term in Taylor's theorem, one ...
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### $f(f(…f(x)…))$ $a$ times, where $a\in\mathbb{R}$

Take $f(x)$ and do a "double-call": $f^2(x)=f(f(x))$ I use this notation here to explain my problem. This can be easy calculated for any function. Also $f^{100}(x)$ is not really a problem. This ...
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### Proof of If $A \subseteq \Bbb{R}$ with $A$ inductive set then $\Bbb{N}\subseteq A$

In calculus class, I saw a proof of this, but I am not convinced. Note: A$\subseteq \Bbb{R}$ is inductive if and only if 1$\in A$ and $\forall x \in A \Rightarrow x+1 \in A$. I tried to do a proof ...
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### What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
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### Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc

I wonder what is the name of a mathematical system extending the real numbers that includes signed zero along with unsigned zero as well as other "limit targets", such as $1^+=1+0^+$, $5^-$ etc, so ...
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### Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
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### Archimedean field with non total order

I am currently looking for the simplest way to characterize the real numbers. Usually they are described as the complete archimedean field, showing that all such fields are isomorphic. Archimedean ...
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### An introduction book for analysis with the axioms of Tarski

Briefly: Is there an introduction book for analysis with The axioms of Tarski as basis? (I found them very elegant. And the most important thing for me is that the axioms are "obvious", not like the ...
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### System of six equations in real numbers

Let $a,b,c,d,e,f$ be real numbers. Solve the following system of equations: $\begin{cases}a+b=-e\\ ab=f\\ c+d=-a\\ cd=b\\ e+f=-c\\ ef=d\end{cases}$ I got stuck trying to solve this problem by ...
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### Vector field in $\Bbb Q^2$ and $\Bbb R^2$

I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a ...
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### Prove there is a square of a rational number between any two positive real numbers

I'm doing this problem and the solution is below. I don't understand why this solution proved $m^2 \le x$ firstly? Any help, thanks.
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### Is there a Dedekind-theory of real numbers?

Someone said that Dedekind had a "theory of real numbers", and I wonder about the truth level of that statement. He published the well-known construction of real numbers in 1872, partly inspired by ...
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### Existence of a Precise axiomatization of Eudoxus theory of magnitude

Is there a precise axiomatization of the Eudoxus theory of proportions? For example, a) (D, +, <) is a structure such that < is a strict linear order, b) + is an order-preserving ...
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### Is it true that every open set in Euclid space is a countable union of open (closed) cubes?

There is a theorem in the book but it only applies to $\mathbb{R}$ and open intervals. The proof seems to be unnecessarily complicated. Can I just say I can cut the space countably many times (...
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### Dedekind Cuts and Rationals

I have a question regarding how you might construct the reals from the rationals by taking Dedekind cuts. My basic understanding is that a Dedekind cut is a bipartition of the rationals such that ...