Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
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Can the occurrence of certain digits be proven/disproven, for any arbitrary irrational number?
$\pi$ is perhaps the most famous irrational number. We know it contains all decimal digits from 0-9, just by virtue that all digits occur, at least once, within 32 decimal places:
$\pi = 3....
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Is there a better way to compute first digits of very large numbers?
The currently known method of finding the first digits of $a^b$ is multiplying $\log_{10} a$ by b, and extracting the fractional part. This allows us to compute the first digits of quite large numbers ...
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Prove that: $n$ has a prime divisor which is not smaller than $11$.
Let integer $n>10$ such that $n=\overline{a_{m}a_{m-1}a_{m-2}...a_{0}}$ where: $a_i \in {(1;3;7;9)}$, $i=\overline{0,m}$
Prove that: $n$ has a prime divisor which is not smaller than $11$.
Here or ...
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Does anyone know why this is happening with 1/7? [duplicate]
(I've never posted on here before, so apologies for any formatting problems)
I had always noticed this property of the decimal form of $1/7$ ( $0.14285714...$ ) where the decimals went $14$ , then $28$...
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Show that the number $A$ is irrational
question
For each $n$ natural number we denote by $a_n$ the first digit a
of the number $n^3$.
Show that the number $A = 0, a_1a_2 ... a_n ...$ is irrational.
my idea
A thing is clear....between $a_1$ ...
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Show decimal expansions
I can't wrap my head around this exercise:
Show that the rational number $\frac 94$ has two different decimal expansions, namely $2.2500000\dots$ and $2.2499999\dots$ by writing these decimal ...
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I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime
I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime
1/prime matches the last digit of the prime number itself for several primes. This pattern ...
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Is $\omega(n)=16$ the maximum?
What is the largest possible value of $\omega(n)$ (the number of distinct prime factors of $n$) , if $n$ is a $30$-digit number containing only zeros and ones in the decimal expansion.
I checked ...
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Decimal exponent
This is how I've attacked 3^1.0987, doing it with pen and paper. 3(3)^(987/10000) = 3*3^(((3)(7)(47))/10000). But from here I don't know what to do. I can't imagine trying to take the 10,000th root ...
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Sum of squared digits
Let $s(n)$ denote the sum of the squares of the digits of $n$. For example, $s(14) = 1 ^ 2 + 4 ^ 2 = 17$ Determine all integers adding n for which $s(n) = n$ holds.
I bound it to $243$ due to $9^2 *4 &...
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Representing number, where digits are cubes
Let $n$ is integer and we want to express it in terms of $10$ as $$n=R^3_k10^k+R^3_{k-1}10^{k-1} \cdots+ R^3_0$$ where $R_i\in \{\pm0,\pm 1,\pm 2,\ldots,\pm9\}$
Example : $37= 1^3\times10+3^3$
...
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Approximate summation formula of time to count numbers from 1 to N
When calculating how much time it takes to count from $1$ to $n$, it is normally used the approximation that it takes about $1s$ to say a number out loud, so it would take $n$ seconds, but there's a ...
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Arbitrary decimal value of $A(n)=\left(\frac{11}{10}\right)^n$
For $n\in\mathbb{Z}$ consider the number
$$A(n)=\left(1+x\right)^n\bigg{|}_{x=\frac{1}{10}}=\sum_{k=0}^\infty\binom{n}{k}10^{-k}$$
which we have expanded by the Taylor series. It is found that
$$a_1=\...
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Probability that the first $m$ digits in $2^n$ are $k_1k_2\dots k_m$
I want to find the probability that the first $m$ digits of powers of 2 are a given combination $k_1k_2\dots k_m$. So far, here's my reasoning:
A number $2^n$ will have the first $m$ digits of the ...
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Deleting Digits from Champernowne's Constant
As some may know, Champernowne's constant is one of the only known constants proven to be normal. The number is constructed by concatenating whole numbers as you count up and appending them behind a ...
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Two questions about emirp's
Define $r(n)$ to be the reverse of a positive integer , that is the number emerging if the decimal expansion is written down in reverse order. Emerging leading zeros are of course omitted , but this ...
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Is there a known explanation for the Feynman point?
The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, ...
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Similarities in the digits of the powers of 2 and 5
Many may have noticed that the negative powers of 5 contain the same digits as the positive powers of 2:
This pattern intrigued me. I started to wonder if it exists in different number bases. I soon ...
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Is this the smallest Proth - emirp of the desired form?
An emirp is a prime number that keeps prime if the digits in base $10$ are written down in reverse order. A Proth-prime is a prime number of the form $2^n\cdot k+1$ with positive integers $n,k$ , $k$ ...
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Estimating how many of the first $10,000$ Fibonacci numbers start with the digit $9$
Consider the problem of estimating how many of the first $10,000$ Fibonacci numbers begin with the digit $9$.
The only ideas I have so far:
Obviously, if we assume that the every first digit is ...
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What is the chance of getting same decimals of percentage 4 times in a row with RNG?
I was rolling some random percentage with RNG between 0 and 1; if you multiply the result to 100, you're getting the percentage between 0 and 100, decimal amount was 4. And then I got 0.7171% the ...
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Do permutations on the decimal expansions of irrational numbers retain the property of irrationality?
Suppose we have an irrational number with the following decimal expansion:
$$A = a_0 \ a_1 \ a_2 \ a_3 \ a_4 \ a_5 \ a_6 \dots $$
Now, construct a new real number through a permutation on the decimals ...
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Irrational Numbers and Surjection from $[0,1)$ to $[0,1)^2$
I am searching for some clues or solutions of the question below:
For $\sqrt{2}=1.41421356\cdots$,
Is $1.1236\cdots$ irrational?
To say more formally:
Let $f$ a function from $[0,1)$ to $[0,1)^2$ ...
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two forms of infinite decimal which are rewritten from finite decimal
How to transform a finite decimal into a infinite decimal? It sounds nothing to discuss. But I have seen two forms to deal with it which make me confused.
Let's consider a finite decimal $x_0.x_1x_2\...
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If $\{n_k\}$ is the set of natural numbers with no 0 in their decimal expansion, $\sum_{k=1}^\infty \frac{1}{n_k}$ converges to a number less than 90 [duplicate]
Let ${\{n_1,n_2,…\} }$
be the set of natural numbers that do not use the digit 0
in their decimal expansion. Then, the series
$\sum_{k=1}^\infty \frac{1}{n_k}$
converges to a number less than 90.
Is ...
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What is the measure of the set of numbers in $[0,1]$ whose digits have a mean of $0$?
I've been playing around with ways to systematically define a continuum of dense and uncountable subsets of real numbers in a (somewhat) intuitive manner, and tried the following characterization:
For ...
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How many subsets of $\{0,1,2,3,4,5,6,7,8,9\}$ could be realized as the set of distinct digits of a prime?
How many subsets of $\{0,1,2,3,4,5,6,7,8,9\}$ could be realized as the set of distinct digits of a prime?
Possible solution
Obviously, the empty subset is not realizable.
Five singleton subsets are ...
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Carmichael-numbers with only one odd digit
Here I ask for a third Carmichael number with only odd digits in their decimal expansion. Far more Carmichael numbers seem to exist with the property that in the decimal expansion there is only one ...
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Numbers such that $(\overline{a_1\dots a_n})^2=\overline{x_1\dots x_m}$ and $(\overline{a_n\dots a_1})^2=\overline{x_m\dots x_1}$.
Recently, I had the pleasure of finding out that
$$13^2=169\quad\text{and}\quad 31^2=961.$$
It had me wondering . . .
The Question:
What pairs of distinct natural numbers $r,s$ have decimal ...
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Prove that This Property Does Not Hold for Any Other Pair of Digits.
Given a positive integer $n,$ prove that there is a positive integer $m$
that to base ten contains only the digits $0$ and $1$ such that $n|m.$ Prove that the same holds for digits $0$ and $2,$ or $0$ ...
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Is there a third Carmichael number with only odd digits?
Upto $2^{64}$ , there are two Carmichael numbers with only odd digits :
$$53711113=157\cdot 313\cdot 1093$$ and $$3559313513953=29\cdot 113\cdot 337\cdot 673\cdot 4789$$
In the first case, the prime ...
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Is there any perfect power in the sequence $12,123,1234,12345,...$?
Inspired from the question Is there any perfect square in the sequence $12,123,1234,12345,...$?, there is no perfect square other than $1$ in the sequence of Smarandache numbers. But I wonder if are ...
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To prove that the Cantor function maps the end points of the intervals removed to a same point
For the function f in 1.6D i.e. $$x=0.b_1b_2....(3)$$ then
$$f(x) = y= 0.a_1a_2.....(2)$$ where $a_i=b_i/2$
Here (2),(3) represents the binary and ternary expansions of $x$ respectively.
Show that if $...
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What numbers have a multiple that in base 10, can be written with only the digits $2$ and $5$.
I was working on the classic number theory problem "for any integer $n$, it has a multiple whose base $10$ representation consists of only $1$s and $0$s" (for anyone who stumbles on this ...
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Proportion of a digit in an algebraic number's binary expansion
We know that
A real number is rational if and only if it's binary (or base $n$ expansion, for all $n$) is eventually periodic. Therefore, the proportion of each digit (0 or 1 in the binary case) is a ...
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Perfect square in the sequence formed by consecutive numbers
I recently saw a question Is there any perfect square in the sequence $12,123,1234,12345,...$?
This led to thinking about a new question.
Consider sequence https://oeis.org/A057137 that is the ...
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Find the smallest positive multiple of $1999$ that ends in $2006$ (last four digits)
Find the smallest positive multiple of $1999$ that ends in $2006$ (last four digits)
Approach:
$1999N\equiv 2006\pmod{10000}$
(1) $9995N\equiv-5N\equiv30\pmod{10000}$
(2) $-N\equiv 6 \pmod{2000}$, so ...
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Let $t$ be a positive real number. Prove that there is a positive integer $n$ such that the decimal expansion of $nt$ contains a $7$.
From this question, I'm honestly not too sure on how to prove it.
I've thought about an approach, which is to take 3 numbers from t. Then prove that the number in the middle of the product of those 3 ...
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why does the sequence of n/7 always have the same six digits in the same order in the decimal part?
I realized the other day that in the decimal representation of $\frac{n}{7}$, where $n=\{1,2,3,4,5,6\}$, the decimal part always has the same sequence of six digits but shifted to start at a different ...
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Analogue to emirp for Carmichael numbers
An emirp is a prime number with the property that if we write down the digits in base $10$ in reverse , we again get a prime number. Trivial emirp's are the palindrome primes (if we write down the ...
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Proof that a repeating decimal has non-repeating digits after decimal iff denuminator has factors of 2 or 5 besides other prime factors
I'm studying a lesson about fractions. It classifies rational numbers into three categories based on their decimal representation.
Terminating Decimal: If a reduced fraction'denuminator has only ...
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How many numbers less than N have a prime sum of digits?
I'm working on solving Project Euler's Problem 845. It's asking us to find the $10^{16}$-th positive integer number that has a prime sum of digits. Adopting a 'naive' solution, I compute the sum of ...
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Formula for calculating the difference between sums of digits in the same base for $(n-1)$ and $n$
I am looking for a formula to calculate the difference between the sums of the digits of $(n-1)$ and $n$ in a given base, denoted as $sb(n-1) - sb(n)$. Specifically, I want to find a formula that ...
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Calculating the length of a decimal expansion in constant time
Is there a way to calculate the length of a decimal expansion as a result of a division operation in constant time?
$\frac{1}{256} = 0.00390625$ therefore the expansion length is $8$.
$\frac{1}{357} = ...
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Proving that there are infinite primes with digit sum 8 in base 10
I recently wrote about a problem I cam up with while thinking about number theory, which you can find on this post. Long story short, I'm trying to prove there are infinite natural numbers such that ...
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Call $n\in\Bbb N$ "balanced" if the sum of its digits equals the count of its divisors. How many "balanced" numbers are there up to $m$?
I recently stumbled across a problem about numbers' divisor count (more specifically, how many positive integers are equal to the square of their divisor count - answer was 2: they are 1 and 9).
But I ...
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Proof of $\inf\left \{ \frac{\mathrm{d} (n^2)}{\mathrm{d} (n)} \; \bigg| \; n \in \mathbb{N} \right \}=0$, where $d(n)$ is the sum of digits of $n$
So I wanted to find the infimum of the set described in the title, and I'm pretty confident on what the subsequence should be to ensure a 0 infimum.
But a proof of this eludes me. I tried some funky ...
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Question about the collection of the prime factors of a fibonacci number
A positive integer $n$ is called pandigital , if every digit from $0$ to $9$ occurs in the decimal expansion of $n$.
Conjecture : The largest non-pandigital fibonacci-number (a fibonacci-number with ...
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Useful length of Pi?
Not sure where this really fits, so am trying Mathematics first. Feel free to migrate to another StackExchange forum if more appropriate elsewhere. So I was listening to a podcast yesterday that was ...
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Is there a name for such rounding algorithm?
When I compute a number where an approximation is needed but no accuracy is specified, I usually scan the first few digits and if I find a zero, then I truncate the decimals before the first zero. For ...
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