Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
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fastest way to find number of digits in number [closed]
What's the fastest way to determine the number of digits in a number?
Would it be like so:
...
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If you write down all the numbers from 1 to n, how many digits would you have written down?
I've seen the question for numbers like 50, 100 or 1000, but not for $n$. Although I found a formula that might be the answer, but I don't know the name of it or the proof for it. I couldn't find it ...
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Convert Base16 to Base10
I am currently trying to use Y-Cruncher to calculate Pi.
Upon calculating 100 digits of Pi, I get this result:
...
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What is the decimal form of 1/3? [duplicate]
I don't understand. 1 divided by 3 is thought to be 0.333333... but, 0.33... times 3 is 0.999999999999999999999...
I don't see a way that you can represent this as a decimal, and you probably can't, ...
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How to Solve Number System Problem
The Problem
Below is a problem dealing with number systems:
My Attempt
Since traowo,ptae,tarumpao appear repeatedly and in consistent positions, I assumed they were conjunctions similar to "and&...
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How to find how frequently a specific digit appears in a number?
Supposed I have the number $N$ that is composed from the digits $\underline{d_1d_2d_3...d_i}$, then I define $N^*$ as $d_1+d_2+d_3+...+d_i$, where $d_i$ can take integer values only from $0$ to $9$. ...
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How to expand $\pi$ in non-decimal bases?
I was reading about $\pi$ converted into other base systems. We all know $\pi$ in base $10$:
$$\pi_{10} = 3.1415926535897932384626433(...)$$
but how has this been converted into, say, base $11$, where ...
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A function from [0,1] to [0,1] which seems to be continuous but still cannot be.
I can't figure out where the following reasoning fails:
Define a function $h:[0,1] \to [0,1]$ by letting $h(0)=0$ and $h(1)=1$,
and $h(0.d_1d_2d_3d_4\ldots)=0.d_1d_1d_2d_2d_3d_3d_4d_4d_5d_5\ldots$
The ...
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Are there any perfect squares of the form 88...81 (in decimal, at least two 8's)?
I saw this problem recently and it is deceptively hard. The usual mod 4 trick won't work, and indeed there will be perfect squares whose last n digits will be 88...81, for any n.
I can show that if ...
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Average period of the decimal expansion of reciprocals of prime numbers
Let $P(n)$ be the period of the decimal expansion of $1/n$, for prime $n$ (e.g. $1/7=0.\overline{142857}\rightarrow P(7)=6$). The value of $P(n)$ fluctuates heavily, but it seems to have an average ...
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Prove $S(1981^n) \geq 19$
Let $S(n)$ denote the sum of digits of a non-negative integer n. Prove that $S(1981^n) \geq 19$
I tried to use induction. I know that $S(pq) \leq S(p)S(q)$ so $S(1981^{n+1}) \leq 19S(1981^n)$ but it ...
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Number theory approach to Project Euler's "Large Sum" problem?
I am refreshing some of my skills by solving problems on the Project Euler site. It is a repository of problems that usually require some mathematics knowledge and programming knowledge to solve ...
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Looking for an efficient way to multiply "powered digits"
For a program, I have numbers expressed as a vector of "powered digits". The first element is the power of $0$ (number of $0$ in the number, usually $0$), the second is power of $1$, number ...
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Problem with decimals about Stirling approximation/expansion .
Problem :
Let :
$$f(x)=n!$$
And :
$$g(x)=\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}(1+1/(12n)+\cdots)$$
Call $k$ the number of decimal before the comma for $f(x)$ and $K$ for $g(x)$
What is the exact ...
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Calculating expected number of attempts to find a large palindromic prime in digits of $\pi$
I want to find the greatest palindromic prime in the known digits of $\pi$. Currently there are $100$ trillion digits calculated. There are $9$ trillion palindromic primes of $25$ digits, and the ...
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Are there two unequal decimal numbers where the integer part is the same and the decimal expansion is reversed such that there sum is an integer?
Given a decimal number $d$ we define the function $D$ to be the number where the integer part is the same and the decimal string is reversed.
For example, $D(5.879)=5.978$, $D(-800.5924)=-800.4295$ ...
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Reference request: A. Schinzel on digital sums of powers
I am searching for the earliest published proof of the following result:
$$\lim_{k\to\infty} s(2^k) = \infty$$
where $s(n)$ denotes the sum of the decimal digits of $n$.
This problem has been ...
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Define function that tells me if an integer is a zero_special
I'd like to define $zeroes(n)$ as the number of zeros in the decimal expansion of the integer $n$. A number $n$ is zero_special if $$zeroes(n) > zeroes(n-1)$$ Can I write a function that determines ...
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Finding large palindromic primes in the decimal expansion of pi
I'm trying to complete a coding challenge that involves finding large palindromic primes in the decimal expansion of $\pi$. I'm at the second stage of this challenge, which asks me to find the first ...
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Digit sum formula $n - 9 \sum_{i\ge1} \left \lfloor \frac{n}{10^i} \right \rfloor$
Let $S(n)$ be the sum of digits of n
Prove that $S(n) = n - 9 \sum_{i\ge 1} \left \lfloor \frac{n}{10^i} \right \rfloor$ for all natural numbers n
I started with induction which works easily if $10$ ...
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In how many ways can 3 distinct numbers fill 6 blanks (each repeated exactly twice)?
I have a basic knowledge of permutations and combinations but using different approaches to this problem lead me to different answers.
Basically, there are 6 blanks and 3 distinct digits a,b, and c. ...
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Surprising Patterns in Numbers whose Digit Sum is equal to their Square Root in an arbitrary Base.
In base 10, $\sqrt{81} = 8 + 1 = 9$. It turns out that 81 is the only number in base 10 that has this property. I wanted to find out if there are other numbers with this property in other bases. ...
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Can we define addition of numbers which are **NOT** eventually all zero as we go to the left?
I am struggling to define addition of objects which are similar to decimal-expansions.
In this post, we refer to the decimal-expansion-like things as "wumbers".
Our goal is to write ...
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Can you prove that proof-by-induction is invalid for the real interval [0, 1]?
We have a special function $S$ from the real interval $[0, 1)$ to the real-interval $(0, 1]$ which I will define near the end of this post.
Someone claims that the following proof-schema is valid:
We ...
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Find, in terms of $\epsilon$, the smallest positive integer $n$ such that $|\sqrt{n}-\lfloor{\sqrt{n}}\rfloor-\frac12|<\epsilon$.
Find, in terms of $\epsilon$, the smallest positive integer $n$ such that $|\sqrt{n}-\lfloor{\sqrt{n}}\rfloor-\frac12|<\epsilon$ where $0<\epsilon<\frac12$.
Numerical experimentation suggests ...
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Decimal Expansions - Pugh Exercise 1.18
I have been working through an exercise from Pugh's 'Real Mathematical Analysis' and although there is an answer on this site discussing the exercise (Pugh exercise 1.18) there is a part of the ...
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Test of rational number as a terminating decimal number using $2^m \cdot 5^n$
I do not understand why $q$ needs to be equal to $2^m \times 5^n$ for $\frac{p}{q}$to be a terminating rational number.
Why cannot $q$ be equal to $2^m\times 3^n$?
Is there any documentation that I ...
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What is the most efficient way to generate first K digits of $n!$ without calculating the whole factorial? [closed]
K around 15 would suffice,
And n upto 10^9 should suffice.
Everyone says its easy in python but I am still not able to .
My attempt using Stirlings approximation is ...
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Given any two normal numbers, can we always find a find a truncation point at which the digits can be rearranged from one to the other?
I just watched a video showing that you can re-arrange (or permute as it is called in the video) the first nine decimal digits of the reciprocal of pi, to give the first nine decimal digits of the ...
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Is there a formula that allows you calculate the n-th decimal digit of pi without calculating the previous digits?
I know that there is a formula that allows one to calculate the n-th hexadecimal digit of pi without calculating the previous digits. But is there an analogous formula for decimal digits? I conjecture ...
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Find all integers $x$ that are at most $100000$ so that $S(11x) = x$.
For a positive integer $x,$ let $S(x)$ denote the sum of the squares of the digits of $x$.
Find all integers $x$ that are at most $100000$ so that $S(11x) = x$.
Prove or disprove whether there are ...
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some decimal numbers with fractional part that can be written down easily in base 10 cannot be in base 2 without using infinite series of number(0.2)
i am not an expert in maths so pardon me for my insufficiency in math vocabulary but noticed a behavior and was curious to know more about it.
some numbers like 0.2 can be written easily in base 10 ...
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Proving that a real number has at most two decimal expansions
I'm trying to prove that any real number has at most two decimal expansions. I have an idea of how to prove it, but I'm not fully sure if every step of my proof works. Here is my attempt.
Let $x \in \...
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How to show that there are no decimal representations of $1$ other than $1.000\dots$ and $0.999\dots$? [duplicate]
I know that $1.000\dots$ and $0.999\dots$ are two different decimal representations of $1$, but how can I prove that they are unique?
I mean, there is no other decimal representation $a_n\dots a_0. a_{...
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Is there a valid proof of one-to-one correspondence b/n the real interval $(0, 1]$ and the nonterminating decimal expansions $0.d_1 d_2 d_3 …$?
As per Michael J. Schramm, 1996, Introduction to real analysis, Theorem 6.6:
In an Archimedean ordered field in which the Nested Intervals property holds, there is a one-to-one correspondence between ...
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Bdmo 2014 question?
$N$ is a number that consists of $2012$ digits. If you take any consecutive $m$ digits $(m \leq 2012)$ from $N$ starting from any position in that number, there'll be another position in $N$ so that ...
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Dividing by $b^n-1$ in base $b$ results in repeating decimals. Can this be proven with modular arithmetic?
I'm trying to do a video about mathematical discovery and proof. Idea goes something like this, with appropriate demonstrations along the way:
notice that $\frac19 = 0.\overline1$, $\frac29 = 0.\...
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Decimal Cauchy sequence
Given a real number $x\in\mathbb{R}$ in decimal form
$$
x = C_0. C_1 \dotsc, \quad C_0 \in \mathbb{Z}, \quad C_i = 0, \dotsc, 9, \quad i > 0
$$
we may define a sequence of decimals
$$
q_n = C_0. ...
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Agreement in decimal expansions
As part of alarger proof, I'm trying to show that if $x$ and $y$ agree up to the $(N+1)$st digit in their decimal expansions (e.g., both are $1.41412 \ldots$ or elements in the sequence of successive ...
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Does a set of all decimal expansions of $\pi$ contains $\pi?$ [duplicate]
Let's say there is a set containing all finite decimal expansions of $\pi$:
$$A = \{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... \}$$
Does this set contains $\pi$?
I see that it is probably not true ...
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Position of specific value
Let's assume a have an arbitrarily long number, take π for example. Since we know π is infinite, there will at some point be a group of numbers like "2015201620172018...", correct? My ...
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Prove that when the denominator of a rational number is of the form $2^n * 5^m$ it is a terminating decimal
What is the proof for when the denominator of a rational number is of the form $2^n * 5^m$ it is a terminating decimal?
For example:
$7/8$, where $8$ is of the form $2^3 * 5^0$
Therefore, $7/8$ is a ...
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Last two digits of $x^y$, when the units digit of $x$ is $1$
I want to know why the statement below is true.
Let $x$ and $y$ be two positive integers. And consider that: the units digit of $x$ is $1$; the ten's place digit of $x$ is $t$; and the units digit of $...
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A $17$-digit number and the number formed by reversing its digits are added together. Show that the sum has a even digit.
A $17$ digit number is chosen, and it's digits are reversed, forming a new number. These two numbers are added together. Show that there sum has at least one even digit.
The solution given in the book ...
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How does one express $0.0\overline{1410}$ as a fraction?
How does one express $0.0\overline{1410}$ as a fraction? I know the answer is $141/9999$ but I am not sure how to derive it using some formula from below.
Wikipedia says assume $x=0.a_1a_2...a_n.$
$...
user817548
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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} \...
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Eventually-prime decimal expansions
Let $w$ be a right-infinite word over the alphabet $A = \{ 0, 1, \dots, 9\}$, with a distinguished decimal point after at most finitely many symbols from the left (i.e. $w$ is in $A^\ast . A^\omega$). ...
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Consider $\frac{abc-defc+fagh-iafg}{6}=337$ ($abc$ means $100a+10b+c.$ Find the maximum value of $iafg.$ [closed]
$\frac{abc-defc+fagh-iafg}{6}=337$
The string of letters are numbers so when the equation had $fagh$ it represents $1000f+100a+10g+h.$ Try to find the maximum of $iafg.$ Good luck!
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Finding an injection $(0,1)^2 \to (0,1)$
I'm trying to write down and prove that a map defines an injection from $f: (0,1)^2 \to (0,1)$. Here is my attempt.
We define $f: (0,1)^2 \to (0,1)$ as follows. Given $x,y \in (0,1)$ with decimals ...
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Is $\pi$ in the infinite set $ \{ 3, 3.1, 3.14, 3.141, 3.1415, ...\} $?
Does $\pi$ exist in the following infinite set. Apologies for lack of set notation, and i'm hoping its not necessary to help me understand the nature of infinite decimal expansion.
I know sets are ...
