For questions about decimal expansion, both practical and theoretical.

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-11
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0answers
33 views

93388•9546985339 [on hold]

This girl purchased 3 pounds of fruit. Her sister Evelyn purchased 1/10 the amount how much did each girl purchase?
4
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1answer
19 views

Properties of the Digit Product + Digit Sum of a number

The other day I started messing around with some properties and noticed a pattern emerging when the digit product and digit sum of a number were added together. For example, 15. (1+5)+(1*5) = 11. If ...
7
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1answer
2k views

Is 4 the second or third digit of pi

If someone says that they know 10 digits of pi, does that mean that they know ten digits starting with the 3 in 3.14 or with the 1 in 3.14?
5
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2answers
74 views

Proving the irrationality of the concatenation of the $n$th powers of primes

Note: The apostrophes are meant to separate different groups of digits. Like, $0.{1^2}'{2^2}'{3^2}'{4^2}'\cdots=0.14916\cdots$. I wasn't able to come up with something better. It is easy to show ...
1
vote
1answer
26 views

Sum of digits of polynomial smaller than of factorial

I'm trying to prove this : Let $f \in Z[X]$ then for sufficiently large $n$ we have $$s(f(n))<s(n!)$$ where $s$ is the sum of digits function. What I have so far : I thought this must be true ...
1
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2answers
20 views

Fixed points of iterates of a certain map $\Bbb N \to \Bbb N$

I have stumbled onto chains of numbers that are interesting in that, when they are split up into their digits, summed, squared, and repeating some number of times, yield the original number. This ...
4
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0answers
43 views

What's the order of growth of the 'double-and-rearrange' numbers?

This question asks about the reachability of some specific numbers via a procedure that starts from the number 1 and where a valid step is to either double the current number to yield a new number, or ...
19
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1answer
244 views

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
3
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2answers
51 views

Squares of a number yields a palindrome?

I was doing my statistics homework when I observed an interesting pattern: $ 11^2 = 121 $ $ 111^2 = 12321$ $ 1111^2 = 1234321 $ $ 11111^2 = 123454321 $ $ 111111^2 = 1.234565432 \times 10^{10} $ ...
5
votes
1answer
104 views

The smallest integer whose digit sum is larger than that of its cube?

79 is an example of a number whose digital sum is greater than that of its square (6241). Which is the least number, if any, whose digital sum is greater than that of its cube?
3
votes
2answers
101 views

Is the number $0.1234567891011121314\ldots$ a rational or irrational number? [duplicate]

Is the number $0.1234567891011121314\ldots$ a rational or irrational number? The number has a very clear pattern but however in order for the number to be a rational number it would have to be ...
5
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1answer
39 views

Question on recurring decimal digits

In my discrete maths class, I have come across an interesting phenomenon for which I can't find an explanation! If we divide $1$ by $13$ we obtain $0.07692307\ldots$ If we divide $3$ by $13$ we ...
2
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1answer
51 views

Factorization of the semi-palprime $N = pq$

I define semi-palprime be a prime number that remains the prime when its digits are reversed, like $p = 13$, and its mate is $q = 31$. I know that number $N$, $ N ...
1
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0answers
27 views

Rational analogue of expansion to base b

As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely. Less well known is that we can do this for ...
0
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1answer
31 views

Converting decimal fractions to binary

I know that if we multiply the fraction by 2 repetitively and take out the integer part every time, we will get the binary form. But why does this method work? Why should we multiply by 2 for the ...
-7
votes
4answers
190 views

Difference between ${2\over 9}$ and ${22\over 99}$? [closed]

The fractions ${2\over 9}$ and ${22\over 99}$ both have the same decimal value $0.22222\ldots$ But obviously they are not equal. What causes this situation? And also, what is the correct rational ...
0
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0answers
32 views

Finding sequences in $\pi$ [duplicate]

I recently came across some programs which were able to calculate exactly, when a particular sequence of digits appears the first time in the decimal expansion of $\pi$. This made me wondering, if it ...
3
votes
1answer
43 views

Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$ s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}} $$ Where $a_0(k)$ is the ...
2
votes
2answers
103 views

Digit sum equals 44 of a squared number

Is there a whole number $x$ such that the sum of the digits of $x^2$ equals 44? I would like someone to tell me if my thoughts are correct. The remainder of a number a divided with 9 is the same as ...
0
votes
4answers
120 views

How many $0$s does the number $30!$ have? [duplicate]

I want to find out the number of $0$s in the number $30!$, what should I do? Is there any trick that would work for a general question of this type, like number of $0$s in $50!$ ?
5
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4answers
146 views

Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
2
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1answer
20 views

Are there more convenient ways of getting the number of digits of a positive integer?

I want to define $n$th power of $10$ for a positive integer. Say for $43$ it would be $2$, for $5$ it would be $1$, for $9999$ it would be $4$. As for $1$, $10$, $100$, ... I am still shifting between ...
1
vote
1answer
58 views

Doubling sequences of the cyclic decimal parts of the fraction numbers

Is there any theory, why and when doubling sequences of the decimal part of the fraction numbers occur? Take for example these small numbers: ...
2
votes
1answer
48 views

For any $n$ is there $n$ consecutive $0$ in the decimal expansion of $2^m$ for suitable $m \in \mathbb N$?

We have define a function $f:\mathbb N \rightarrow \mathbb N$ such that $f(n)=$ { smallest $m \in \mathbb N$ such that decimal expansion of $2^m$ have $n$ consecutive $0$ }. I computed some values of ...
1
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3answers
127 views

What is $\left\lfloor0.\overline{9}\right\rfloor$? [duplicate]

We know that $0.\overline{9} = 1$ but then what is $\left\lfloor0.\overline{9}\right\rfloor$? My thought process went: $0.\overline{9} = 1$ so therefore $\left\lfloor0.\overline{9}\right\rfloor = ...
5
votes
3answers
122 views

Generalize multiples of $999…9$ using digits $(0,1,2)$

The smallest $n$ such that $9n$ uses only the three digits $(0,1,2)$ is $1358$, giving a product $12222$. For $99n$ this is $11335578$, giving $1122222222$. Similarly, ...
0
votes
1answer
36 views

Upper bound of digit sum of powers

Take $x \in \Bbb N$, $x \le9$ and $m \in \Bbb N$. Now we define a function $d_s(n): \Bbb N \to \Bbb N$ as the digit sum of $n$ in base $10$. Now let's say we have a lower bound $b_l$ and an upper ...
1
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4answers
121 views

Find numbers $\overline{abcd}$ so that $\overline{abcd}+\overline{bcd}+\overline{cd}+d+1=\overline{dcba}$

Find the numbers $\overline{abcd}$, with digits not null that satisfy the equality \begin{equation}\overline{abcd}+\overline{bcd}+\overline{cd}+d+1=\overline{dcba}\end{equation} where ...
0
votes
2answers
23 views

Denote a number as a sum of two in terms of the base

Technically, in decimal base, $$445 = 44 \cdot 10 + 5.$$ Lets say there's a number of base $B$ where I want to segregate the number into last digit and rest of the first digit as a representation ...
4
votes
2answers
55 views

A positive integer is equal to the sum of digits of a multiple of itself.

Let $n$ be a positive integer, prove there is a positive integer $k$ so that $n$ is equal to the sum of digits of $nk$. I'm not really sure how I should approach this problem, I tried to do a ...
2
votes
3answers
64 views

Why does the division algorithm work for converting between number bases?

I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this ...
2
votes
1answer
29 views

Is the set of real numbers in $[0,1]$ with digits $1$ and $3$ only in their development in base $5$, dense in $[0,1]$?

Let $E$ denote the set of real numbers in $[0,1]$ with digits $1$ and $3$ only in their development in base $5$. How to prove that $E$ is dense in $[0,1]$? Is this the right way to see that E is ...
3
votes
3answers
51 views

Is this formula true for $n\geq 1$:$4^n+2 \equiv 0 \mod 6 $?

Is this formula true for $n\geq 1$:$$4^n+2 \equiv 0 \mod 6 $$. Note :I have tried for some values of $n\geq 1$ i think it's true such that :I used the sum digits of this number:$N=114$,$$1+1+4\equiv ...
1
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0answers
35 views

Procedure converting decimals to rationals.

Suppose I have been given a rational number in decimal format (since decimals of rationals repeat, finite precision presentation suffices), what is the most effective way to write it in form of ratio ...
0
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3answers
129 views

Find the smallest natural number $n$

Find the smallest natural number $n$ such that rightmost digit is $6$ and when we deleted that digit $6$ and add it to the left of the number we get $4n$. Example of the operation: $123456$ becomes ...
1
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2answers
43 views

Finding number of digits of decimal number

I'm looking for a way to calculate the number of digits in a decimal number, such as $600.045$. I'm aware of the $1+\mathrm{int}(\log(x))$ formula for finding number of digits of an integer, but this ...
0
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2answers
59 views

What determines what base the right side of this base coversion will be?

Referring to this example of positional notation on Wikipedia: There are several examples $$465\;\;\text{(base 10)} = 465\;\;\text{(base 10)}$$ But then $$465\;\;\text{(base 7)} = ...
1
vote
1answer
41 views

How to find if 1/n will be recurring decimal expansion?

How to determine if the expansion of $1/n$ would be a recurring decimal expansion or not? for example, $1/7 = 0.\overline{142857}$ but $1/8=0.125$. So, how to find if $1/n$ would be a recurring ...
0
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0answers
19 views

Iterated digit product

A very interesting calculator at http://www.micmaths.com/defis/defi_01.html repeatedly calculates the product of the digits of a number and stops when it reaches a single digit. It asks what is the ...
6
votes
6answers
127 views

What are the last two digits of $77^{17}$?

I'm trying to solve current task referenced the following but I stuck at $\displaystyle77^{17}\equiv x\pmod{100}$. As it is described on above link it uses Binomial Theorem. But I read a lot about the ...
2
votes
2answers
157 views

Does this weird series converge?

$\sum_{n\in S}$$\frac{1}{n}$, where S consists of those positive integers whose decimal expansion does not contain the digit 1. This was a part(b) question. Part (a) was an evaluation of the ...
1
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1answer
54 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
0
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4answers
39 views

Recurring decimals

Does anybody have any idea why when you divide a any number, say, by 11 (excluding multiples of 11 of course) you obtain a recurring decimal? I know that it must either terminate or recur, but why ...
6
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2answers
91 views

What are the necessary conditions for UPC primes?

0 68000 00027 7 is a UPC that the Hershey Company could use for some candy bar or other product. It happens that $6800000027$ is a prime number. But $68000000277$ ...
2
votes
3answers
197 views

What is the last digit? [closed]

Consider all 100 digit numbers, i.e., those between $0$ and $10^{100} - 1$ (inclusive). For each number, take the product of non-zero digits (treat the product of digits of $0$ as $1$) , and sum ...
4
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2answers
62 views

Product of digits

Find all natural numbers $x$ ($x$ in base $10$) so that the product of its digits is $x^2 - 10x - 22$. Here is what I did so far: I took two cases. The first case was considering one or more ...
0
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1answer
67 views

converting decimal to hexadecimal using division method

Okay so I know the basic procedure of converting a decimal number to any base-r is to divide by r and keep up with reminders until you reach zero. The reminders form the new base that is equivalent to ...
4
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0answers
96 views

Does $\pi$ contain infinitely many “zeros” in its decimal expansion?

Some number doesn't contain $"7"$ in its decimal expansion. For example Liouville's constant $$L=\sum_{n=1}^\infty\frac{1}{10^{n!}}=0.11000100....$$ contains only $0$ and $1$. It is well-known ...
24
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4answers
4k views

What does .9 with a line above the 9 mean?

What does this mean? $$\Large.\overline9 $$ I've never seen this notation before.
0
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2answers
23 views

Looking for reading material on: Numbers, in whose internal decimal places appear all natural numbers as a sequence of digits

For example one can have the number 0.12 and can look at the sequence of the digits of it's internal decimal places and see, that 0.12 contains the numbers 1,2 and 12. It is also easy to construct a ...