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 ...
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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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-3 votes
1 answer
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Last two digits of $x^y$, when the units digit of $x$ is $1$

I want to know why the statement below is true. Let $x$ and $y$ be two positive integers. And consider that: the units digit of $x$ is $1$; the ten's place digit of $x$ is $t$; and the units digit of $...
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2 votes
2 answers
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A $17$-digit number and the number formed by reversing its digits are added together. Show that the sum has a even digit.

A $17$ digit number is chosen, and it's digits are reversed, forming a new number. These two numbers are added together. Show that there sum has at least one even digit. The solution given in the book ...
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How does one express $0.0\overline{1410}$ as a fraction?

How does one express $0.0\overline{1410}$ as a fraction? I know the answer is $141/9999$ but I am not sure how to derive it using some formula from below. Wikipedia says assume $x=0.a_1a_2...a_n.$ $...
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The first $0$ in a base $b$ expansion

Consider the following function $Z \colon \Bbb{Z}_{\ge2} \times [0,1) \to \Bbb{N} \cup \{\infty\}$. If $b \in \Bbb{Z}_{\ge2}$ is an integer greater than $1$ and $x \in [0,1)$, let $Z(b,x) \in \Bbb{N} \...
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2 votes
1 answer
119 views

Eventually-prime decimal expansions

Let $w$ be a right-infinite word over the alphabet $A = \{ 0, 1, \dots, 9\}$, with a distinguished decimal point after at most finitely many symbols from the left (i.e. $w$ is in $A^\ast . A^\omega$). ...
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Consider $\frac{abc-defc+fagh-iafg}{6}=337$ ($abc$ means $100a+10b+c.$ Find the maximum value of $iafg.$ [closed]

$\frac{abc-defc+fagh-iafg}{6}=337$ The string of letters are numbers so when the equation had $fagh$ it represents $1000f+100a+10g+h.$ Try to find the maximum of $iafg.$ Good luck!
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Finding an injection $(0,1)^2 \to (0,1)$

I'm trying to write down and prove that a map defines an injection from $f: (0,1)^2 \to (0,1)$. Here is my attempt. We define $f: (0,1)^2 \to (0,1)$ as follows. Given $x,y \in (0,1)$ with decimals ...
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Is $\pi$ in the infinite set $ \{ 3, 3.1, 3.14, 3.141, 3.1415, ...\} $?

Does $\pi$ exist in the following infinite set. Apologies for lack of set notation, and i'm hoping its not necessary to help me understand the nature of infinite decimal expansion. I know sets are ...
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1 answer
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Two different decimals representing the same number

Under what condition two different decimals represent the same number? Is it "Two different decimals represent the same number if and only if one of them has 9 as its non-terminating decimal ...
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2 votes
1 answer
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Is there an increasing arithmetic sequence the digit sum of its terms forms again an increasing arithmetic sequence?

Is there an increasing arithmetic sequence with $10000$ terms such that the digit sum of its terms forms again an increasing arithmetic sequence? This problem is from the (Problems from the book) ...
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if $S(a^n+n)=1+S(n)$ for any sufficiently large $n$ if and only if $a$ is a power of $10$

let $a$ be a positive integer such that $$S(a^n+n)=1+S(n)$$ for any sufficiently large $n$ if and only if $a$ is a power of $10$ where $S(n)$ is digit sum of a positive integer $n$ if $a$ is a power ...
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2 votes
3 answers
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Minimal number of games to narrow down the win rate

This just popped up in my head, so there might be a famous known solution/algorithm for the solution. The situation is this. Suppose know the (not exactly accurate) win rate(in percentage) of your ...
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Prove that there are infinitely many special numbers of the form $10^n+b$ if and only if $b-1$ is special.

While digging through some old notebooks today, I found a problem from a long time ago that I was never able to solve. It involves a sequence of positive integers called the “special numbers” defined: ...
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3 votes
2 answers
150 views

What is a decimal expansion?

I'm an adult trying to get back to school after a few years; I'm trying to prepare for college. In reading I have come across the terms 'decimal expansion' and 'decimal representation', which by the ...
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4 votes
3 answers
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What is the sum of natural numbers that have $5$ digits or less, and all of the digits are distinct?

$1+2+3+\dots+7+8+9+10+12+13+\dots+96+97+98+102+103+104+\dots+985+986+987+1023+1024+1025+\dots+9874+9875+9876+10234+10235+10236+\dots+98763+98764+98765=$ The only thing I can do is to evaluate a (bad) ...
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2 votes
0 answers
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Can a perfect square N be composed by only 0 and 1? Where N's prime factor are only 3 and 7.

I would like to know if someone knows how to prove that there are no perfect square composed only by zeros and ones in their decimal representation whose prime factors are only 3 and 7 (so of the form ...
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2 votes
2 answers
68 views

sum of the digits of an integer

Assume that a positive integer $n$ can be written in decimal notation as $${n_k}\cdots {n_1}{n_0} = {n_k}10^{k} + \cdots + {n_1}10 +{n_0},$$ and define $${\sigma}(n)={{\sum}^{k}_{j=0}}{n_j}.$$ If ${\...
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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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-4 votes
1 answer
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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 $11$...
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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 ...
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1 vote
1 answer
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Rounding $1731.9146$ to $2$ decimal places?

So, I got this number $1731.9146$ and I need to round it to $2$ decimal places. The answer should be $1731.91$, but I had always thought, for some reason, that the answer (if precise) should be $1731....
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-3 votes
1 answer
223 views

Proving that 31123319 is the largest number with a self-accounting property (A047841)

22 is special because it contains the number of its numbers. The next smallest number with this self-accounting property (see A047841) is 10213223. What is the largest such number? I've figured out ...
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7 votes
3 answers
133 views

When is $X/R(X)$ an integer where $R(X)$ is the reverse of an integer $X$?

My question concerns reverse numbers (e.g. $1234 → 4321$). Is it possible to find integer solutions greater than $1$ for such numbers when you take their ratio? I am not interested in trivial ...
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Ternary and decimal expansion of an integer solely consists of $2$ and $0$.

Each digits of the decimal expansion of the integer $2022$ (this year) consists of $0$ or $2$ and also, each digits of the ternary expansion of the same integer $2022$ (which is $2202220_3$) consists ...
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4 votes
1 answer
85 views

What is the highest number of digits so that this number of digits in a specific power of 2 are exactly 10%?

I accidentally found out that in $2^{{10}^{6}}$ (==$2^{1000000}$), there are exactly 10% digits of 6 (in the decimal form). And I would like to know - are there powers of 2 in which all the digits ...
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1 answer
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How to demonstrate the method used to convert decimal to any other base?

Let's say we have an arbitrary number in base $b$, $(x_3x_2x_1x_0)_b$. We can write the equivalent of this number in base $10$ as follows: $(x_3x_2x_1x_0)_b = x_3*b^3+x_2*b^2+x_1*b^1+x_0*b^0$ So, let $...
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2 votes
1 answer
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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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0 votes
1 answer
92 views

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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3 votes
1 answer
89 views

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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0 votes
0 answers
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How are results guaranteed in algorithms that include intermediate approximation?

Let me elucidate this vague question with an example. Consider for example the following Gauss sum of roots of unity $N=(e^{2\pi i/5} + e^{2\pi i/4} e^{4\pi i/5} + e^{4\pi i/4} e^{8\pi i/5} + e^{6\pi ...
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1 answer
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Decimal representation of a multiple entirely consists of odd digits [duplicate]

Problem: Prove that for every odd integer $n$, exist a multiple $m$ of $n$ whose decimal representation entirely consists of odd digits. My work: +) For all $n:(n,5)=1$, $10^{\varphi (n)}-1$ works +) ...
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Nth root as value inside the root symbol is less than 1

$$L:=\lim_{x\to0^{+}}x^{a}=0~~~~~~~~~~~\left(a>0\right)$$ As$~a\geq1~$, it is quite obvious for me that the above limit converges to zero. The current problem for me is that$~a<1~$ For instance ...
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3 votes
1 answer
94 views

Prove that every real number $x \in [0,1]$ has a "certain" binary representation

I want to prove that every real number $x \in [0,1]$ has binary representation in the following way: Let $B$ denote the set of all sequences $b:\mathbb{N} \rightarrow\{0,1\}$. Consider then $f:B\...
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0 votes
3 answers
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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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0 answers
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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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1 answer
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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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4 votes
2 answers
67 views

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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0 votes
0 answers
25 views

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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2 votes
1 answer
216 views

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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1 vote
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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 ...
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3 votes
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0.0204081632... as a repeating fraction

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 ...
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1 vote
0 answers
28 views

Solve in $\mathbb{N}$ x $\mathbb{N}$ for $m$ and $n$ such that $m^n$ and $n^m$ have the same number of digits.

Let $\delta : \mathbb{N} \rightarrow \mathbb{N}$ be a function such that $\delta (x)$ denotes the number of digits in $x$. Find all pairs of natural numbers $m$ and $n$ such that $\delta(m^n)=\delta(n^...
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4 votes
1 answer
80 views

Convergence of $(b_n)$ if $b=b_0.b_1b_2...b_n...$

Suppose one has a decimal expansion of a real number $b:$ $$b=b_0.b_1b_2b_3...b_n...$$ where $b_n\in\mathbb{Z}$ and $0\leq b_n\leq9$ for $n\geq1$. For which $b$ does $(b_n)$ converge? Is the answer ...
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2 votes
1 answer
117 views

How is it truly determined the digits after the decimal don’t EVER repeat? (Is irrational.) Theoretically, the first million digits could repeat.

I was observing a high school algebra class and they were discussing irrational versus rational numbers. Irrational go on forever (the digits after the decimal) and don’t ever repeat. Rational repeats....
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11 votes
1 answer
728 views

Is there any proof that there are only 3 so-called "neon numbers"?

The phrase "neon number" is sometimes used for a number where: square the number, add the digits of that in base 10, and you get the original number. So, 9 is a neon number (-> 81, 8+1, 9)...
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-1 votes
1 answer
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The number of decimals numbers between different size number ranges equal? [duplicate]

For instance, is it correct to say that the number of decimals Numbers between $220$ and $289.999...$ and the number of decimal numbers between $297$ and $297.999...$ are equal? Even though the range ...
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