# Tagged Questions

For questions about decimal expansion, both practical and theoretical.

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### How to find if $n$th root of $m$ is rational?

I know that if $m$ is an integer, then the $n$th root of $m$ must be an integer, else it is irrational. But what if $n$ and $m$ both are decimals - is there a way of easily telling if the $n$th root ...
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### What is the growth relationship of the number of digits a number has as numbers increase?

To clarify the question, since I'm sure the wording is awkward: In the decimal number system, to get from 1 digit to 2, it takes n=10 numbers. To get from 2 to 3, it takes 90 more numbers added to n. ...
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### Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
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### Equivalence of Repeating Decimal

GRE exam question asks what is greater $.\overline{717}$ or $.\overline{71}$ I believe both are equal, but GRE says that $.\overline{717}$ is greater. But why? If they repeat for infinity, isn't ...
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### why $10$ in any base number system written as $10?$

I am a student trying to write an article in number system can same one give me an idea why $10$ in any base number system written as $10$ $?$
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### Does $[0.9999…]=1$? [duplicate]

We all know that $0.99999...=1$ So does that imply $[0.99999...]=1?$ Or do we consider it as $0?$ My doubt is: any gif of the form $[0.xyz...]=0$. If $[0.99999...]=1$ won't that be contradicting? ...
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### Definition of real by infinite series instead of their Cauchy limits

Looking at Wikipedia´s definition of real numbers I choose a variant one of the alternative definitions, using Cauchy limits. However, Instead of taking a limit I choose the number to be represented ...
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### Last $500$ digits of $2015!-1$

As the title says, I'm looking for the last $500$ digits of $2015!-1$. I assume it's a repetition of zeroes from the factorial, so the final result is a lot of $9$-s, but I can't formulate a solution ...
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### How many trailing zeroes does 4617! contain? [duplicate]

I am getting $1151$ as answer on continuous division by $5$. Is it right? On each division by 5, some remainder is generated...doesn't that count? Example: 4617/5 + 923/5 + 184/5 + 36/5 + 7/5 ...
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### Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. My question is, Does it make any sense to prove this equality? Can one give any "meaning" of the symbol $0.999\ldots$ ...
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### Proof that for any $x$ there is a $y$ such that $xy$ is a palindrome

I'm wondering how I would prove For any $x$ there exists at least one $y$ such that $xy$ is a palindrome. For example: 91*99=9009
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### Compute Lebesgue measure of set of all real numbers in $[0,1]$ whose decimal representations don't contain the number 7

Consider measure space $(S, \Sigma, \mu) = (\mathbb R, \mathscr B(\mathbb R), \lambda)$. Let $V^C \subseteq S$ denote the set of all numbers in $[0,1]$ whose decimal representations don't contain the ...
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### Math induction problem with large numbers

I am trying to figure out how to prove $17^{200} - 1$ is a multiple of $10$. I am talking simple algebra stuff once everything is set in place. I have to use mathematical induction. I figure I need ...
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### Normal numbers and equidistribution

Does anyone know how to prove the following criterion? (Due, 1949) $x \in (0,1)$ is normal for base $10$ $\iff \{10^nx\}_n$ is equidistributed. "$\Leftarrow$" is quite easy using definition of ...
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### Does every finite digit-sequence appear in some factorial?

Suppose, some finite digit-sequence is given. Can we prove or disprove, that there is always some number $n$, such that the digit-sequence appears in the decimal-expansion of the number $n!$ ? If ...
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### Irrational numbers are non-terminating/non-repeating decimals [closed]

Why is it true that all irrational numbers are non-terminating/non-repeating decimals? By definition, an irrational number is one that can't be expressed as a ratio of integers.
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### Can I guess an irrational number formula from its digits?

Let us say I have 10,000 digits started from some point (lets say the 16th digit) of the decimal expansion square root of some arbitrary number, like 13. Is there any way I can get back the original ...
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### Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string

Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string? For example, given the string 123456, can I find a perfect square with leading digits ...
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### Conversion base $r$ to base $10$ (decimal) Algorithm

There's another algorithm for converting from base r to base 10? The only one I know is the following one: For example 20 (base 5) to base 10 is: $2X5^1 + 0x5^0 = 10.$
It seems obvious that infinite many primes can be formed only using two distinct digits. $67776767776667777777$ is an example for such a prime. Even if we allow only the digits $0$ and $1$, there ...
I define the function $d_{\mathrm{avg}} : [0, 1]\to [0, 1]$ such that for $0.x_1x_2x_3\cdots$ the decimal expansion of $x$ (defined such that $\nexists N : x_k = 9$ for all $k \geq N$), d_{\mathrm{...