For questions about decimal expansion, both practical and theoretical.

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0
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3answers
131 views

Find the number of zeros after decimal point in $0.2^{25}$ [closed]

Find the number of zeroes immediately after decimal point in $(0.2)^{25}$,given that $\log 2=0.30101$ My attempt: I found the answer as $17.\dots$ Should we add $1$ as $17$ is the characteristic ...
0
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3answers
64 views

Show that number of units of $2^{2^n}+1$ is always $7$

Given that $n$ is a natural number such that $n\geq2$, show that number of units of $x=2^{2^n}+1$ is always $7$ For $n=2$ we have $x=17$. For $n=3$ we have $x=257$ We can show that the number of ...
2
votes
2answers
103 views

Length of digits before the period in decimal expansion for rational numbers

I'm a newbie with number theory and I've been reading this page and trying to figure out how to calculate the length of the digits before the period and digits of the period of a rational number of ...
0
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0answers
23 views

How to know if addition of factorials changes the amount of trailing zeroes?

In this specific puzzle, it is $603!+604!+605!$. If it was $603!*604!*605!$ instead, I would simply count all the occurrences of multiples of 5, 25 and 125, which would be $148$ for $603!$ and ...
4
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3answers
52 views

Understanding expansion on base $k$

There are some times when one might need to use expansion of real numbers on some base $k$. One example is when dealing with Cantor's set, one uses expansion of the numbers inside $[0,1]$ on base $3$. ...
14
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1answer
132 views

Multiplication of doubly-infinite decimal numbers

Summary of the question: Doubly-infinite decimal numbers are "numbers" whose decimal expansions are allowed to extend infinitely both to the left and right of the decimal point, for example ...
1
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3answers
57 views

Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits

I have just proved the divisibility rule for 3 and 9. Let $n\in\mathbb{N}$. Let $m$ be a number that appears when you shuffle the digits in $n$. Show that $n-m$ is a multiple of 9. Can anyone offer ...
1
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0answers
43 views

Numerical Mathematics and Computing - 6 digit rounding arithmetic

This is a homework problem on a mathwebsite (webwork) which states: For the function f(x)=sqrt(x+4)−2, find an alternative formula that can accurately evaluate its value for small x. To which I ...
0
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1answer
25 views

Isolating Decimals

I'm in need of isolating the decimal part of a number using maths only, no excel functions or anything like that, but it's proving to be much harder than I thought it would be. For example, I have ...
0
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0answers
28 views

Operations on binary numbers, that have equal numbers of two available digits, that preserve this property?

I'm looking for operations on binary numbers that have the same number of the available digits, $0$ and $1$, such as $100011$, $100101$ and $10$. These could be integers as the examples given are, or ...
0
votes
1answer
21 views

Sequences of consecutive digits of arbitrary length in irrational ternary numbers (take two).

Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Furthermore, suppose the only digit ...
5
votes
5answers
756 views

Does every irrational number contain arbitrarily long sequences of some digit?

Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Does this imply there are arbitrarily ...
0
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1answer
75 views

Decimal expansion of $x\in [0,1]$

This is an exercise from Royden Real Analysis: Let $p$ be a natural number greater than 1, and $x$ a real number, $0 \leq x \leq 1$. Show that there is a sequence $\{a_n\}$ of integers with $0 \leq ...
4
votes
1answer
55 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
votes
2answers
80 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
37 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
vote
2answers
24 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
votes
0answers
73 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 ...
20
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1answer
265 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
votes
2answers
63 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
111 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
154 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
votes
1answer
61 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
61 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
32 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
votes
1answer
173 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 ...
-6
votes
4answers
224 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
votes
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
53 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
128 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
137 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
votes
4answers
186 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
votes
1answer
24 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
87 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
51 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
vote
3answers
153 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
148 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
41 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
vote
4answers
126 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
27 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
100 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
213 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
30 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
57 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
36 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
votes
3answers
187 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
vote
2answers
58 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
votes
2answers
61 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
49 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 ...