# Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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### If $u$, $v$ and $w$ are the digits of decimal system, then the rational number represented by $0.uw\overline{uv}$ is?

Context I was recently giving a math test and encountered the following question: If $u$, $v$ and $w$ are the digits of decimal system, then the rational number represented by $0.uw\overline{uv}$ is? ...
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### Forced factors of numbers like $100001000110002\cdots 10447$?

In an old factoring project I currently attack again I dealt with positive integers that emerge if one writes down the numbers $10\ 000$ to some number $k$ with $10\ 000\le k\le 99\ 999$ in increasing ...
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### Perfect cubes with digit-average at least $7.5$

I found so far the following perfect cubes with a digit-average (in base $10$) with at least $7.5$ : ...
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### Is 1/3 included in the sequence 0.3, 0.33, 0.333,...? [closed]

I assume that $\frac{1}{3}$ is equal to $0.3333...$. Let's define a sequence as follows: $0.3$, $0.33$, $0.333$, $0.3333$,... Question: is $\frac{1}{3}$ included in this sequence? Every item in the ...
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### What is the algorithm to represent any number in place-value subtrahends?

I'm going to ask a rather basic, but still curious question about arithmetic. It's a widely known fact that any number has a unique representation in place-value summands. For example, a number $137$ ...
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Let's suppose that $$10,a+15,ba +15,3 = 2\times (20, ab)$$ where numbers are in their decimal representation, and so $ab$ and $ba$ are two digit numbers. Is there a straightforward way to evaluate $a\... 2 votes 1 answer 144 views ### Looking for an algebraic number with a balanced sequence of digits It is generally believed that all irrational algebraic numbers$\alpha$are normal, in all bases. In base$2$that implies that there are arbitrary large$n$such that for the binary expansion $$\... • 54.9k 1 vote 0 answers 38 views ### Expected value of index of an n-digit number found in pi Let E(n) be the expected value of the index of finding an n-digit number in the digits of pi e.g. Number 0 is found at index 32 Number 1 is found at index 1 Number 2 is found at index 6 Number 3 is ... • 936 1 vote 1 answer 121 views ### Is it possible to calculate the first digits of this number? Is it possible to calculate the leading decimal digits of 3 \uparrow\uparrow 5\ = 3^{3^{3^{3^3}}} = 3^{3^{7,625,597,484,987}}? Using currently known methods, this would require knowing the complete ... • 229 2 votes 1 answer 55 views ### How is this summation expression transformed? I am solving one math problem. I could not understand the following transformation described below. I am guessing the denominator is re-written as a factorial of 2K and unnecessary even terms are ... 0 votes 1 answer 41 views ### Solution to x^2 + y^2 = x' + y' where x' and y' have the same digit-representation as x and y but with a trailing digit 2 respectively. So I was thinking of a particular problem the other day and started experimenting with it. It goes like this: I want to find natural numbers x and y such that: x is an n-digit number, and y ... 0 votes 1 answer 56 views ### Integers not containing a fixed substring in their decimal representation Prove that for any fixed string of digits S (each digit from 0 to 9), there are at most o(N) integers in \{1,2,\ldots,N\} such that their decimal representation does not contain S. This is ... • 2,733 1 vote 2 answers 148 views ### First digits of the iterated powers of 2 I wanted to show that the first digits of (2^{2^j})_{j=1}^\infty are not periodic. By the standard Dirichlet trick I can show that any array of digits forms the initial digits of some number of the ... • 281 1 vote 2 answers 103 views ### Prove that any positive integer has a multiple whose decimal expansion involves all ten digits. [closed] This is something I've been struggling with - I've been having trouble even wrapping my head around what is being asked. It feels like the answer might be on or around something extremely obvious, but ... 0 votes 0 answers 60 views ### Can the occurrence of certain digits be proven/disproven, for any arbitrary irrational number? \pi is perhaps the most famous irrational number. We know it contains all decimal digits from 0-9, just by virtue that all digits occur, at least once, within 32 decimal places: \pi = 3.... 3 votes 0 answers 106 views ### Is there a better way to compute first digits of very large numbers? The currently known method of finding the first digits of a^b is multiplying \log_{10} a by b, and extracting the fractional part. This allows us to compute the first digits of quite large numbers ... • 229 0 votes 0 answers 47 views ### Prove that: n has a prime divisor which is not smaller than 11. Let integer n>10 such that n=\overline{a_{m}a_{m-1}a_{m-2}...a_{0}} where: a_i \in {(1;3;7;9)}, i=\overline{0,m} Prove that: n has a prime divisor which is not smaller than 11. Here or ... 4 votes 1 answer 158 views ### Does anyone know why this is happening with 1/7? [duplicate] (I've never posted on here before, so apologies for any formatting problems) I had always noticed this property of the decimal form of 1/7 ( 0.14285714... ) where the decimals went 14 , then 28... • 41 3 votes 2 answers 314 views ### Show that the number A is irrational question For each n natural number we denote by a_n the first digit a of the number n^3. Show that the number A = 0, a_1a_2 ... a_n ... is irrational. my idea A thing is clear....between a_1 ... • 826 2 votes 1 answer 86 views ### Show decimal expansions I can't wrap my head around this exercise: Show that the rational number \frac 94 has two different decimal expansions, namely 2.2500000\dots and 2.2499999\dots by writing these decimal ... 0 votes 1 answer 240 views ### I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime 1/prime matches the last digit of the prime number itself for several primes. This pattern ... 2 votes 1 answer 98 views ### Is \omega(n)=16 the maximum? What is the largest possible value of \omega(n) (the number of distinct prime factors of n) , if n is a 30-digit number containing only zeros and ones in the decimal expansion. I checked ... • 85.1k 0 votes 0 answers 29 views ### Decimal exponent This is how I've attacked 3^1.0987, doing it with pen and paper. 3(3)^(987/10000) = 3*3^(((3)(7)(47))/10000). But from here I don't know what to do. I can't imagine trying to take the 10,000th root ... 0 votes 0 answers 25 views ### Sum of squared digits Let s(n) denote the sum of the squares of the digits of n. For example, s(14) = 1 ^ 2 + 4 ^ 2 = 17 Determine all integers adding n for which s(n) = n holds. I bound it to 243 due to 9^2 *4 &... 2 votes 1 answer 129 views ### Representing number, where digits are cubes Let n is integer and we want to express it in terms of 10 as$$n=R^3_k10^k+R^3_{k-1}10^{k-1} \cdots+ R^3_0$$where R_i\in \{\pm0,\pm 1,\pm 2,\ldots,\pm9\} Example : 37= 1^3\times10+3^3 ... • 2,697 1 vote 1 answer 39 views ### Approximate summation formula of time to count numbers from 1 to N When calculating how much time it takes to count from 1 to n, it is normally used the approximation that it takes about 1s to say a number out loud, so it would take n seconds, but there's a ... 5 votes 3 answers 352 views ### Arbitrary decimal value of A(n)=\left(\frac{11}{10}\right)^n For n\in\mathbb{Z} consider the number$$A(n)=\left(1+x\right)^n\bigg{|}_{x=\frac{1}{10}}=\sum_{k=0}^\infty\binom{n}{k}10^{-k}$$which we have expanded by the Taylor series. It is found that$$a_1=\... • 51 2 votes 1 answer 74 views ### Probability that the first$m$digits in$2^n$are$k_1k_2\dots k_m$I want to find the probability that the first$m$digits of powers of 2 are a given combination$k_1k_2\dots k_m$. So far, here's my reasoning: A number$2^n$will have the first$m$digits of the ... • 1,090 1 vote 0 answers 40 views ### Deleting Digits from Champernowne's Constant As some may know, Champernowne's constant is one of the only known constants proven to be normal. The number is constructed by concatenating whole numbers as you count up and appending them behind a ... • 183 1 vote 0 answers 60 views ### Two questions about emirp's Define$r(n)$to be the reverse of a positive integer , that is the number emerging if the decimal expansion is written down in reverse order. Emerging leading zeros are of course omitted , but this ... • 85.1k 8 votes 4 answers 1k views ### Is there a known explanation for the Feynman point? The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, ... • 717 5 votes 2 answers 446 views ### Similarities in the digits of the powers of 2 and 5 Many may have noticed that the negative powers of 5 contain the same digits as the positive powers of 2: This pattern intrigued me. I started to wonder if it exists in different number bases. I soon ... • 87 0 votes 0 answers 22 views ### Is this the smallest Proth - emirp of the desired form? An emirp is a prime number that keeps prime if the digits in base$10$are written down in reverse order. A Proth-prime is a prime number of the form$2^n\cdot k+1$with positive integers$n,k$,$k$... • 85.1k 3 votes 2 answers 162 views ### Estimating how many of the first$10,000$Fibonacci numbers start with the digit$9$Consider the problem of estimating how many of the first$10,000$Fibonacci numbers begin with the digit$9\$. The only ideas I have so far: Obviously, if we assume that the every first digit is ...
Suppose we have an irrational number with the following decimal expansion: $$A = a_0 \ a_1 \ a_2 \ a_3 \ a_4 \ a_5 \ a_6 \dots$$ Now, construct a new real number through a permutation on the decimals ...