# Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### $2^n$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 ...