Questions tagged [normal-number]

This tag is used for questions about normal, simply normal, and absolutely normal numbers. Discussions regarding normality, simply normality, or absolutely normality of numbers are included (e.g., "Prove that a particular number is not absolutely normal").

A real number $x$ is said to be simply normal in a given base $b\in\mathbb{Z}_{>1}$ if, for each integer $t$ such that $0\leq t<b$, $t$ occurs in the sequence of the digits of $x$ in the base $b$ with the natural density $\dfrac{1}{b}$. For example, in the base $b=10$, $$x=0.\dot{0}12345678\dot{9}$$ is simply normal in the base $10$. On the other hand, $$x=0.\dot{1}$$ is clearly not simply normal in the base $10$.

We say that $x$ is normal in the base $b$, all possible string of a given length $n\in\mathbb{Z}_{>0}$ occurs in the base-$b$ representation of $x$ with the natural density $\dfrac{1}{b^n}$. The example $$x=0.\dot{0}12345678\dot{9}$$ is simply normal, but not normal, in the base $10$. On the other hand, Champernowne's constant $$C_{10}=0.1234567891011121314151617181920212223242526272829\ldots$$ (obtained by concatenating the representations of the natural numbers in order) is normal in the base $10$. It is known that almost all real numbers are normal in a given base $b$ (i.e., the set of all non-normal real numbers in the base $b$ has Lebesgue measure zero).

We say that $x$ is absolutely normal if it is normal in all bases $b\in\mathbb{Z}_{>1}$. Sometimes, an absolutely normal number is also called a normal number. Becher and Figueira proved that there exists a computable absolutely normal number.

A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base. If $f(2):=4$ and $f(n):=n^{\frac{f(n-1)}{n-1}}$ for every integer $n\geq 3$, then the number $$\alpha:=\prod_{m=2}^\infty\,\left(1-\frac{1}{f(m)}\right)$$ is known to be absolutely abnormal.

See also https://en.wikipedia.org/wiki/Normal_number.

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Calculate $\pi$ from digits of $\pi$

With a random normal distribution $\pi$ can be calculated with help of the PDF (probability density function). The method below apparently shows $\pi$ can be determined with random digits $[0,1,2,3,4,5,6,7,8,9]$. If this is correct and $\pi$ is a…
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Conjecture about rational numbers

Inspired by normal numbers I created the simple following problem: First take a rational number, for example $\frac{3}{4}$ which is equal to $0.75$; now add the first digit after the decimal separator to the digit $4$ we get $\frac{3}{4.7}$ which is…
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Probability that $n$ random walks (1D) intersect a single point

I am struggling finding a relation to determine the probability $n$ random walks (1D) intersect in a single point at step $s$. In the method below my attempts. My method is somewhat intuitive based. I am looking for more rigorous proof. note: This…
OOOVincentOOO
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The first $0$ in a base $b$ expansion

Consider the following function $Z \colon \Bbb{Z}_{\ge2} \times [0,1) \to \Bbb{N} \cup \{\infty\}$. If $b \in \Bbb{Z}_{\ge2}$ is an integer greater than $1$ and $x \in [0,1)$, let $Z(b,x) \in \Bbb{N} \cup \{\infty\}$ be the first place after the…
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Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

Successively: $\pi \to e \to \log(2) \to G$ were calculated/estimated by sampling uniform distributions. Method: With a normal distribution $\pi$ can be calculated with help of the PDF (probability density function). See: Calculate $\pi$ from digits…
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Can every natural number be written as $\lfloor 2^{2^s+1}/3^k \rfloor$?

Problem: Can every natural number be written as $\displaystyle \left\lfloor \frac{2^{2^s+1}}{3^k} \right\rfloor$ for some $s,k\in \mathbb{N}$? Context and thoughts: This is a generalization of these problems: Density of $2^a 3^b$ and Creating a…
AnilCh
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so what is the average of the sequence made from $\cos(2)$

so what is the average of the sequence made from cos(2)? $\cos(2)=0.4161468365471423869975682295007621897660...$ the first number in the sequence is the first number in the decimal expansion until it hits a $0$ so…
user808403
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What goes wrong when I try to create the following probability distribution on the first uncountable ordinal $\omega_1$?

This question was inspired by the well-known fact that there's no uniform probability distribution on a countably infinite set, because this would contradict $\sigma$-additivity of the probability measure. I can't have an countably infinite number…
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Is $\pi$ a "somewhat normal number"?

I am defining a "somewhat normal number" as such: A somewhat normal number is a number where every possible numerical string in base $10$ can be found in said number at least once, but the density of each $n$ digit string may not be equal to…
Kyky
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