# Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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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 ...
1 vote
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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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### irrationality of a decimal expansion

Consider the real number in $(0,1)$ having the decimal expansion $${\alpha} = 0.{a_1}{a_2}{a_3}\cdots$$ where $a_j$ is obtained by adding up the digits in the decimal expansion of the positive ...
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### Difference between binary division and its decimal division [closed]

Suppose I have one decimal number $23$ which decimal representations is $10111.$ Now $10111$ treated as dividend and divisor is $3$ which binary representations is $11.$ When $10111$ is divided by $11$...
1 vote
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### Cyclic repeating decimals

I was thinking today that if some fraction $1/n$ where $n$ is an integer has a digital period of $n-1$ then it must be a cyclic number. But Wikipedia says that this does hold but only states it true ...
1 vote
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### Is there a way to calculate a specific digit of PI

Is there any mathematical I could find a specific digit of 𝛑 If I had f(x) = ... what would the function to return the x digit ...
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### Proof of 'No natural number whose multiplication of digits is equal to 3570' [closed]

I have to prove that there is no natural number whose multiplication of digits is equal to $3570$ What would be the proper mathematical solution to this question?
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### Is the sum of infinite recurring decimals also a recurring decimal?

I am curious to know if $N=0.12233344444455555...$ is a rational or an irrational number. I see that, since it can be obtained by the sum of $0+0.1+0.022+0.000333+...$, it could be obtained by this ...
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### Where to stop when doing decimal division?

We know that $\frac{1}{8} = 0.125$ via calculator; however, if I didn't have access to a calculator and wanted to find this via long division, why would I stop at 3 decimal places? Why not 2 or 4? For ...
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### Finding an infinite set of irrational numbers between two given numbers.

I’m trying to do a question which asks: ‘find an infinite set whose elements are irrational numbers between $0$ and $0.0001$’ I understand that there are infinite irrational numbers between any two ...
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### Natural numbers with unique digit sum and product

This question is inspired by this poorly received question. For a given base $b$, every natural number has a unique representation in that base, and a corresponding digit sum and digit product. If a ...
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### Show, with proof, that the minimum value of $c$ such that $c^n + 2014$ has all digits less than 5, where $c, n \in \mathbb N$, for all values of $n$

Question: Determine, with proof, the minimum value of $c$ such that $c^n + 2014$ has all digits less than 5, where $c, n \in \mathbb N$ for all possible values of $n$. Note that $\mathbb N$ does not ...
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### Powers of full repetend primes in finding the longest period

For $n \in (7,20000)$, $x < n$ is such that $\forall y<n \text{, period} \frac{1}{x} > \text{period} \frac{1}{y}$. Then $x$ is either a full repetend prime, or a full repetend prime to the ...
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### Correct comparison of real number for n digits precision (absolute vs relative difference)

To compare if $2$ real numbers are equal, we define a desirable precision e.g. $n$ digits and then check if the following condition holds: $-\frac{1}{10^n} \lt x - y \lt \frac{1}{10^n}$ Now I was ...
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### Decimal expansions of $0.999\cdot\cdot$ and $1.000\cdot \cdot$ (infinite digits)

I am reading a passage from the book Foundation of Mathematics by Ian Stewart, and I need some help to make sure I understand it properly. A real number can be expresed by the following unique decimal ...
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### Period of the decimal expansion of $\frac{1}{9801}$

I have to show that $\frac{1}{9801}= 0.\overline{000102030405060708\dots9799}$. Here bar denotes Period. My Attempt: I have shown $\frac{1}{9801}= {0.000102030405060708........9799.........}$ using ...
I calculated $\left(\frac17\right)^2$ and the calculation returned a decimal where a series of numbers going up by an exponent of 2 were all concatenated together at the end of the decimal. the number ...