Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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4
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0answers
88 views

Impossible to double an integer by moving a the initial digit to last

Link to the other post about this problem Prove that there does not exist an integer which is doubled when the initial digit is transferred to the end. So today I started working on this fairly nice-...
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24 views

there exist infinitely many pairs $i<j$ such that $S_2(a_j-a_i)=k$.

For a real number $\lambda > 100$, let $f(\lambda)$ denote the smallest positive integer $k$ satisfying the following property. For any integer sequence $0<a_1<a_2<...$, if $a_n\leq \...
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1answer
95 views

27 and 37 as repeating decimals [closed]

This is interesting... numbers 27 and 37, when you divide 10 by one of them, you have the other as a repeating decimal. Is there a name for this? $$\begin{eqnarray} 10 / 27 &=& 0....
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1answer
41 views

digit product is $20$, digit sum is $12$, find the least number

This is from a math olympiad: The product of the digits of positive integer $n$ is $20$, and the sum of the digits is $12$. What is the smallest possible value of $n$? I started with the prime ...
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3answers
151 views

Doesn't $0.\overline9=1$ lead to consequences like $a-0.\overline01=a$ and $2=1.\overline931415926$?

I'm just starting to learn calculus, but this was the first idea presented: $$0.\overline9=1$$ This would mean that this is true: $$a-0.\overline01=a$$ When I thought about it, then I realised that if ...
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3answers
41 views

Sum of digits of sum of digits of powers of 12345

Sum of digits of $12345$ is $1+2+3+4+5=15$. The sum of digits of sum of digits is $1+5=6$. I have plotted the sum of digits of powers of $12345$ with blue dots (x-axis is the power). As the average ...
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0answers
31 views

how can I display 245276777556527592946 in scientific notation rounded to 13 significant digits?

How to round 245276777556527592946 and can be represented in scientific notation? (13 significant digits, 12 after the decimal point)
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29 views

Question concerning a product of digits algorithm

Define a function $f\colon \mathbb{N} \to \mathbb{N} $ as follows: for a positive integer $n$, we express $n$ in terms of its decimal expansion, say $$n=a_0 + a_1 10 +\cdots +a_d 10^d $$ We define $...
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3answers
49 views

Why the first $x$ decimal places of $(5+\sqrt{26})^{x}$ are following a pattern?

$\sqrt{26}$ is irrational number, so the decimal places should show no pattern. But $(5+\sqrt{26})^{x}$ has these values: ...
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2answers
33 views

Prove that for an integer n larger than or equal to 2, the period for the decimal expression of the rational number $\frac{1}{n}$ is at most n -1.

I am currently working my way through "A Concise Introduction to Pure Mathematics" by Martin Liebeck, and have been stuck on an excersise for a couple of days now. The question reads: "Show that for ...
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11 views

Identifying a number is an integer, fraction or decimal.

Q- Is $\dfrac{24}{6}$ an integer,fraction,decimal? I think it is an integer and a fraction. Its an integer because $\dfrac{24}{6}$ can be reduced to get an integer. It is not decimal because it doesn'...
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2answers
69 views

Why $c$ closed to $-2\times10^n$ in $(1-c^2)^3+(c^3+10^nc^2-1)^3+(10^n c^2-1)^3=n$ for $n >1$?

I have tried many times to evaluate $(1-c^2)^3+(c^3+10^nc^2-1)^3+(10^n c^2-1)^3=n$ for $n >1$ as polynomial for some values of integer $n$ which are greater than $1$ for the solution of the titled ...
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17 views

What does it mean that two numbers are within two decimal places of each other?

Sorry that this seems like an easy question but I find myself getting confused. What does it mean that two numbers are within two decimal places of each other? Does it mean that when these two numbers ...
0
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1answer
16 views

Number equal to the product of its digits

I'm trying to find a positive integer $n>10$ such that $n=p(n)$ Here $p(n)$ is defined as the product of the digits of $n$. Example: $p(15)=1 \times 5=5$ I actually don't know where to start. ...
4
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4answers
169 views

real number and decimal expansions

For any real number $x$ we define its decimal expansion as $N\cdot x_1x_2x_2\cdots$ where $N=\lfloor x\rfloor$ and $$x_i=\left\lfloor 10^i \left(x- \left(N+\sum_{j=1}^{i-1}\frac{x_j}{10^j}\right) \...
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1answer
32 views

How would one notate the subset of the rationals with terminating decimal expansions?

Is there a convention for notating a subset of the rationals with restrictions on the denominators? I'd prefer there to be a relatively intuitive and concise notation for the set $\{\frac{n}{10^m}:n,m\...
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1answer
46 views

Is 5.0 an integer or decimal number?

Is 5.0 an integer or decimal number? I was asked by one of my friends, we got both confused. I said by definition integer contains no or zero decimal part so it should be an integer. But he said that ...
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2answers
28 views

Is there any numerical representation in which each rational has only one representation?

In positional representations, there are always some rational numbers which have multiple representations. For example, in base 10, 1 can be written as 1 or as $0.\overline{9}$. Do there exist any ...
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1answer
39 views

Algorithm to approximate decimal expansion for fraction

Let's say I have some fraction $\frac{n}{m}$, which is fully reduced. how can I approximate its decimal expansion to a given accuracy? Like $\frac{1}{7}$ is 0.143 if you want 3 decimal places of ...
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2answers
51 views

How I can prove that the last digit of $1+6^n+2\times 3^n+7^n+4^n+3\times9^{n}+4\times8^n$ is $3$ or $9$?

I have checked the first $14$ digits of Golden ratio, and I have found some attractive properties. I have defined the sequence as $6^n+1^n+8^n+0^n+3^n+3^n+9^n+8^n+8^n+7^n+4^n+9^n+8^n+9^n$. Some ...
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1answer
17 views

Why $(n^x +n^{x+2})$ is divisible by $5$ for some $n$ and not for others.

Why for numbers with last digits $0, 2, 3, 5, 7$ and $8, (n^x +n^{x+2})/5$ is a whole number and for numbers with last digits $1, 4, 6$ and $9, (n^x +n^{x+2})/5$ is not a whole number?
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78 views

What’s the 100-th digit of $2^{10000}$?

I found this question on a Chinese programmer forum. They solved it by brute-force method like 2 ** 10000[99] in python. The solution is 9. I’m wondering if we can solve it in a better way? Do we have ...
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2answers
47 views

Is the last digit of $2^{2^{n-1}(2^n-1)}-1$ always $5$ for $n >3$?

Is the last digit of $2^{2^{n-1}(2^n-1)}-1$ always $5$ for $n >3$? I did modification to the Mersenne numbers (Even perfect numbers) foruma I put that formual to be the power of 2 have got : $2^{...
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1answer
292 views

What's the next base-ten non-pandigital factorial number after 41!?

By pandigital number I mean a number for which each digit in a given base occurs at least once (some definitions that state each digit must occur exactly once), and since I looking for numbers that ...
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1answer
23 views

how to get n digits of a given number?

I have very large number. I have to operate first n digits from left. Is there a command in Mathematica that will give n digits of a given number something like xxxxx[123456789, 5]=12345? ...
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2answers
58 views

Find the last digits of $a_{2009}$, and of $b_{2009}$.

Define the sequences $a_1, a_2,...$ and $*b_1, b_2,...*$ by $a_1 = b_1 = 7$ and $$a_{n+1} = {a_n}^7, \\ b_{n+1} = 7^{b_n}$$ for $n\ge 1$. Find the last digits of $a_{2009}$, and of $b_{2009}$. What ...
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3answers
42 views

Find all 6-digit squares which are the concatenation of three 2-digit squares

I am looking to find perfect squares $\overline{abcdef}$ with the property that $\overline{ab}$, $\overline{cd}$ and $\overline{ef}$ are perfect squares. I ran a quick program to find that the only ...
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0answers
60 views

When writing in base 7, show that any infinite sequence of digits of the form 0.d1d2d3 , where di ∈ {0, 1, . . . , 6}, represents a number [0,1]

So I believe I understand how to write a number in [0,1] using this (0.345 will become 0.(3/7^1)(4/7^2)(5/7^3)), but I don't understand how to show this a sequence.
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1answer
23 views

Show that a decimal with a repeating pattern represents a rational number?

Exercise 1.3.1 (Guide to Analysis, Hart, p.4) Consider a decimal of the form $x=0.a_1a_2...a_n$ with a repeating pattern of $n$ digits. Write $x = 0.a_1a_2 ... a_n$. Express $10^nx$ as a decimal. Then ...
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63 views

Formulated this question while on a break from a break (repeated digit sum and composite numbers related question)

Suppose $a_1...a_m$ is some composite natural number with at least two different prime factors written in decimal notation. Is there an infinite number of composites with at least two different ...
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1answer
100 views

Can the sequence of successive digits of $\pi^{18}$ ever give a prime?

In this question First $k$ digits of $\pi^n$ and compositeness it is asked for some $\ n\ $ giving late or possible never a prime number. A good condidate is $\ n=18\ $. According to my ...
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1answer
73 views

First $k$ digits of $\pi^n$ and compositeness

Let $\text{fd}(k,x)$ be first $k$ digits of some real number $x$. For $\pi=x$ we have the sequence $,3,31,314,3141,3141,31415,...$ (in base $10$) For $\pi^2=x$ we have $9,98,986,9869,...$ (in base $...
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1answer
37 views

Are there any more numbers that are the sum of ascending powers of their digits?

Are there infinitely many numbers $abc...z$ with $d$ digits such that $a^k + b^{k+1} + c^{k+2} + \dots + z^{k+d-1} = abc...z$ for a positive integer k? For k=1 the largest is $12157692622039623539$, ...
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2answers
37 views

Find the leading digit(s) of a factorial

What are the better methods (algorithms) to computing the first number (or few leading numbers) of a large factorial. Wolfram alpha seems pretty fast and handles large numbers. Is it accurate? Does ...
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3answers
43 views

Is there exist a formula to calculate sum of digits of an integer

I'm the novice, sorry if I can't ask more specifically. If the given number is 2-digits integer. We have sum = number*20%199%19. Can you prove the above formula? ...
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1answer
230 views

Is it true that for any $b$ that is not a power of $3$, there exists at least one integer $n>0$ whose product of digits in base $b$ is equal to $n/3$?

Define a function $P_b(x)$ as $$P_b(x)=\text{"the product of digits of x in base b"}$$ Is it true that for any $b\in\mathbb{N}$, if $b$ is not a power of $3$, then there exists at least one ...
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1answer
93 views

Repeating pattern of power number in base (!).

Let $N\in\mathbb{Z}_+$ represent as $N = m!\cdot r_m+(m-1)!\cdot r_{m-1}+\cdots+2!\cdot r_2+1!\cdot r_1$ Where $0\le r_i\le i$ for $1\le i\le m$ Algorithm $$\begin{split} \frac{N-r_1}{2} &= q_1 ...
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1answer
30 views

Significant Digits: Rounding & Exponential form

Round the following numbers to $4$ significant digits and writeteh rsult in exponential form. $102.53070$ $656.980$ $0.008543210$ $0.000257870$ $-0.0357202$ $$$$ I have done the following: ...
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1answer
27 views

How to prove that two variables have the same first digit by modular arithmetic?

How could I find that x and y have the same first digit(Like, 52, 2 is the first digit here. Just an example) by using modulo arithmetic?
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2answers
128 views

What is the behaviour of the first $n$ digits of ${\underbrace{99\dots99}_{n\text{ nines}}}^{\overbrace{99\dots99}^{n\text{ nines}}}$ as $n\to\infty$

For a natural number $n$, let $f(n)$ denote the first $n$ digits of the decimal expansion of $${\underbrace{99\dots99}_{n\text{ nines}}}^{\overbrace{99\dots99}^{n\text{ nines}}}=(10^n-1)^{10^n-1}.$$ ...
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47 views

Why can't I concatenate a number and always get a prime?

Fix any $N$ a prime with decimal expansion $j_1j_2\dots j_k$. My question: why can't every concatenation of $N$ onto itself be prime? That is, why can't the set containing $$N$$ $$j_1j_2\dots ...
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1answer
21 views

How to prove a rule when a ration is periodic either not?

My son asked me, why division of integers sometimes produces periodic and sometimes decimal real numbers. What has come so far to my mind, is that while we use a decimal system, then every non-...
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1answer
51 views

Why is 1.4 - 1.3 == 0.9999+ but 0.4 - 0.3 == 1.000000003

I'm not sure if this is a maths question or a programming question or a how-does-your-computer-work question. Sorry about that. I remember from university that 0.999999 ... == 1 since 1 - 0.999999 ......
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1answer
242 views

Limit: ratio of the digit product and the number itself

Compute:$$\lim_{n\to\infty}\frac{a_n}{n},a,n\in\mathbb N$$ Where $a_n$ equals the product of the digits of $n$ in base $10$. source Math Analysis 1 exam, 2012 My attempt: The first idea that ...
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1answer
92 views

Is there a proof that there is no such number that makes a pythagorean triple with sum and product of its digits? [closed]

My programming professor recently gave us a task to make a program that prints "every integer smaller than given integer n for which its sum of digits, product of digits and itself make up a ...
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0answers
541 views

$2^n$th decimal place of $\sqrt{2}.$

Someone on Art of Problem Solving claims to know how to calculate the $2^{2020}$th decimal place of $\sqrt{2},$ and will tell us if everyone gives up. Brute force will not work, nor will a BBP style ...
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1answer
39 views

Digits & Squares

If $\overline{abcd} = (\overline{ab} + \overline{cd})^2$ and only $c$ can be $0$, find the sum of all possible values of $\overline{abcd}$. $(A) 13850$ $(B) 14051$ $(C) 14742$ $(D) 14851$ $(E) ...
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2answers
52 views

Does 1/3 have a unique decimal representation?

I think it does, but I’m not sure. And also there are rationals which have unique decimal representation besides irrational numbers. Am i right?
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1answer
318 views

Conjecture about the distribution of $0/1$ in the binary expansion of rational numbers

I suggest that you read the conclusion at the bottom, before reading this entire and very long post. Let $x=\frac{p}{q} \in [\frac{1}{2},\frac{3}{4}]$ be a rational number, with $p, q$ integers. Also,...

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