For questions about decimal expansion, both practical and theoretical.

learn more… | top users | synonyms (1)

0
votes
4answers
98 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 $50!$ ?
4
votes
4answers
122 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
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 ...
0
votes
2answers
20 views

Working out the last digits of indices [closed]

Is there a general rule to work out the last digits of indices (without using a computer)? Ex: $3^{15}$
1
vote
1answer
41 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
47 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
121 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
117 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
31 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
3answers
82 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
20 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
47 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
51 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
vote
0answers
34 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
114 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
39 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
56 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
37 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
votes
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
122 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
154 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
vote
1answer
41 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
votes
4answers
36 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
votes
2answers
89 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
153 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
votes
2answers
60 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
votes
1answer
43 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
votes
0answers
92 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
votes
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
votes
2answers
22 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 ...
1
vote
1answer
79 views

Number theory / decimal representation

Prove that for any $n\in\mathbb{N}$ there exists a number $m\in\mathbb{N}$ such that the decimal representation of $m^2$ has $n$ ones at the beginning and some combination of $n$ ones and twos at ...
2
votes
3answers
62 views

Do asymptotes disprove 0.9 repeating equal 1?

I am in 9th grade and taking geometry. Several of my friends taking pre-calc say that 0.9999... does not equal 1, but is just an asymptote. I have not taken that subject yet and they don't give any ...
8
votes
0answers
74 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
1
vote
0answers
46 views

Multiple of power of 5 with only the digits 2,5,6

after helping a friend solve a homework, I asked myself the following question: $H\subseteq\{1,2,\ldots,9\}$, $T(H)=\{n\in\mathbb{N}:$ all the digits in the decimal representation in $n$ belong to ...
1
vote
1answer
22 views

Given $2^n$, what is the largest power of $2$ that will divide any random concatenation of base $10$ digits of powers of $2$ ending with $2^n$?

My first thought was that it would be $2^n$ itself, for example, if you concatenate $4$ and $2$ to get $42$, that's divisible by $2$ but not by $4$. But whit $2^9 = 512$, you can concatenate $16$ and ...
0
votes
1answer
22 views

Simple question: Writting numers into decimal

I want to know if there are numbers in the interval $[\frac{28}{100},\frac{29}{100}]$ Which got a $7$ in their decimal expansion. I would say "yes", because we can write $0.28=0.2799999999...$ But ...
3
votes
1answer
36 views

Real numbers and rationals - Decimal Expansion

How would one endeavor to show that A real number is rational if and only if its decimal expression ends in recurring digits?
14
votes
1answer
369 views

Numbers having in decimal representation no common digits with all their proper divisors

Let us call a positive integer having in decimal representation no common digits with all its proper divisors "a good number". $54$ is a good number : $1,2,3,6,18,27$ $48$ is not a good number : ...
3
votes
2answers
580 views

How can the decimal expansion of this rational number not be periodic?

I just noticed that dividing $1 \div 998$ gives me the apparently non-periodic $$0.001002004008016032064\ldots ,$$ which is $$10^{-3} + 2\times 10^{-6} + 4\times10^{-9} + 8\times 10^{-12} + \cdots = ...
3
votes
1answer
52 views

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
-3
votes
2answers
200 views

square of digits - why does it always contain 1 or 89 [closed]

I attempted project euler problem 92, while I passed it, my solution works, but had just...awful performance. So I would like to try again tomorrow. In the meantime understanding why the iteration ...
0
votes
0answers
12 views

Determine decimal places in the expansion in the expansion

I need to know how to determine how many places in the decimal of 1/800. No need to answer my exact question but how I can go about it, please. However, the exact question is below. "Note that the ...
5
votes
1answer
24 views

Are there longer runs of squares that look like different squares across consecutive bases?

I was trying to solve a different problem of representation across multiple bases, looking at this page http://primefan.tripod.com/BaseReps.html when I noticed this: $$11000100, 21021, 3010, 1241, ...
0
votes
2answers
274 views

Find whether a given rational number has a terminating decimal expansion

Without actual division find whether the rational number $\dfrac{1323}{264600}$ is terminating or non terminating. I know that to solve this, we have to convert the denominator into the formula ...
7
votes
3answers
146 views

Irrational Numbers : Show that $0.1248163264…$ is irrational

I was working through some basic Number Theory Problems in Rosen and came across the following problem : Show that the real number $0.1248163264...$ represented in ...
6
votes
1answer
66 views

Finding all the zeroes in $100!$

Is there a way to find all the $0$s in $100!$? (Including zeroes that come between two non-zero numbers) I know that to find the $0$s at the end we can use the greatest integer method. I was just ...
9
votes
4answers
46 views

A 10-digit number whose $n$th digit gives the number of $(n-1)$s in it

There is a ten-digit number $X$ such that its first (left-most) digit is equal to the number of $0$s in $X$, the second digit gives the number of $1$s in $X$, and so on. The last (right-most) digit ...
1
vote
2answers
43 views

Do numbers higher than the base number have any meaning?

For instance, in base 5, does this have any meaning 6 + 7 + 8 = ? or in base 2 2 + 3 + 4 = ?`