# Tagged Questions

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag. Do not use this tag for questions on solving equations.

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### Can a change of basis modify irrationality/transcendance?

Fix a real number $x$. We can consider its binary expansion, for instance $x = (0.01101001100101101001011\ldots)_2$. Now we consider the real number $y = (0.01101001100101101001011\ldots)_{10}$ : we ...
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### P(P(N)) is equinumerous to the set of real functions

I need to show that (the power set of power set of N) P(P(N)) is equinumerous to the set of real functions. I know that P(N) is equinumerous to R, thus it is equivalent to show that $\{0,1\}^R$ is ...
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### Proving that $\sin(1/x)$ is not continuous at 0

Let $$f(x) = \begin{cases} 0 &\text{ if x=0}\\ \sin(1/x) &\text{ otherwise} \end{cases}$$ Prove that $f$ is discontinuous at $0$ My proof goes like this: for the function to be continuous ...
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### How Can You Define Successors Over The reals?

I've been going through Introduction To Modern Set Theory by Judith Roitman, and am confused by her exposition of well orderings. She gives the following proof that every element $x$ of a well-ordered ...
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### Brief overview of the foundations of math?

I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don'...
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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 ...
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### Is this matrix going to be real or complex?

I hope that this is the right forum where to post this question (and not here). I have a Chi-Square Kernel Matrix (using the second version, which is positive-definite) ...
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### Geometric basis for the real numbers

I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients. But I never actually think of real numbers in this way. I ...
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### How to prove $a,b,c \in \mathbb{R} \mid a+b+c \geq abc \implies 3abc(a+b+c) \geq 3(abc)^2$?

I'm working through some proofs from Cvetkovski's "Inequalities", when I came across a more difficult one (for newbies like me). Given $a, b, c \in \mathbb{R} \mid a+b+c \geq abc$, how can we prove ...
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### What benefits do real numbers bring to the theory of rational numbers?

Complex numbers make it easier to find real solutions of real polynomial equations. Algebraic topology makes it easier to prove theorems of (very) elementary topology (e.g. the invariance of domain ...
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### Help determining if a finite subset of $\mathbb R$ is closed and bounded.

If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
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### A topology on the set of lines?

Of course any set $X$ can have a topology, but are there more natural topologies, metrics or similar on the set of straight lines in $\mathbb R^2$?
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### Find the smallest $n$ such that an integer lies between $nx$ and $ny$ for real numbers $x$ & $y$.

Say I'm given two real numbers as inputs: $x$ and $y$, with $x < y$. I want to find the smallest natural number n such that there's at least one integer between $nx$ and $ny$ (inclusive of $nx$ and ...
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### Is the Fibonacci constant $0.11235813213455…$ a normal number?

Recall that a normal decimal number is an irrational number $\alpha \in \mathbb{R}$ such that each digit 0-9 appears with average frequency tending toward $\frac{1}{10}$, each pair of digits 00-99 ...
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### Nearest neighbor of an irrational number

I am confused in my thoughts about the irrational numbers in real line. My confusion is: If $x\in$$\mathbb R$$-\mathbb Q$ then for $\epsilon>0$ as small as you please, the element ($x+\epsilon$) ...
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### For an arbitrary uncountable set of irrational numbers, can I always construct a sequence from them that converge in the rationals?

Suppose you have a set $S$ of uncountably many irrational numbers. Can you construct a sequence of $S$ that converges to a rational number? What I have tried: Since $S$ is uncountable, the inf of ...
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### Listing real numbers as countable like listing rational numbers [closed]

like proving the set of positive rational numbers are countable, where we list the rationals as the following list, why can't we represent real numbers like the same? If positive Rational numbers (p/...
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### Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$\operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases}$$ In physics, the sinc function has ...
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### approximate irrational numbers by rational numbers

I want to prove this below: (1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$...
If a bounded closed interval $[a,b]$ is covered by finite open intervals $\bigcup\limits_{j = 1}^n {({c_j},{d_j})}$, I want to prove $b - a < \sum\limits_{j = 1}^n {({d_j} - {c_j})}$. It seems ...