For questions about decimal expansion, both practical and theoretical.

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0
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0answers
26 views

How to find if $n$th root of $m$ is rational?

I know that if $m$ is an integer, then the $n$th root of $m$ must be an integer, else it is irrational. But what if $n$ and $m$ both are decimals - is there a way of easily telling if the $n$th root ...
2
votes
2answers
65 views

Show that every rational number $q\in\mathbb{Q}, q\in [0,1]$ has an eventually repeating ternary expansion.

Show that every rational number $q\in\mathbb{Q}, q\in [0,1]$ has an eventually repeating ternary expansion. Recall that $q$ is a rational number provided it can be written as $q=\frac{m}{n}$ where $m\...
3
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3answers
74 views

The last two digits of $13^{1010}$.

$13^{1010}$ $13^{\phi(100)} \equiv 1 \mod 100$ $13^{40} \equiv 1 \mod 100$ $(13^{40})^{25} \equiv 1^{25} \mod 100$ $13^{1000} \equiv 1 \mod 100$ $13^{1010} \equiv 13^{10} \mod 100$ That's all I ...
5
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2answers
70 views

Last $m$ digits of a sum

What is an efficent way (not using any computer programs and such) to find last $m$ digits of some terrible looking sum, for example I don't know $$1^{1000}+2^{1000}+3^{1000}+\ldots+(10^{1000})^{1000}?...
1
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2answers
61 views

In a given range how can i find how many times a two digit number appears ?

I want find how many times a two digit number appears in a given large range , Range is 10^500 . Example : I want to find 21 in given range and the range is 15 to 240 , there are total of 12 numbers ...
0
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0answers
39 views

Proof Regarding Decimal Expansions

Just as a precursor, I'm not a mathematics major (though I've had some experience with abstract algebra and number theory), so I have absolutely no idea how to approach this problem. Hints, not ...
5
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4answers
1k views

Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
2
votes
2answers
72 views

prove that $\sqrt{2}$ is not periodic.

If I am asked to show that $\sqrt{2}$ does not have a periodic decimal expansion. Can I just prove that $\sqrt{2}$ is irrational , and since irrational numbers are don't have periodic decimal ...
14
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1answer
223 views

Is $0.248163264128…$ a transcendental number?

My question is in the title: Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$). It is possible to express $a$ as a ...
9
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0answers
134 views

Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
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0answers
26 views

Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors function?

(Note: This post is a bit related to this earlier MSE question.) The title says it all. Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors ...
25
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1answer
356 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...
1
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2answers
29 views

How do i find three successive natural odd numbers for which the sum of their squares can be written in decimal system as :$\overline{xxxx} $? [closed]

let $a, b,c$ be a successive natural odd numbers, I would like to know how do i find three successive natural odd numbers for which the sum of their squares : $a²+b²+c²$ can be written in decimal ...
8
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4answers
424 views

Decimals of the square root of $n$.

Let $a_1, \ldots, a_k$ be any sequence of digits (i.e., each $a_i$ is between 0 and 9). Prove that there exists an integer $n$ such that $\sqrt{n}$ has its first $k$ decimals after the decimal point ...
0
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1answer
18 views

Write negative decimal in binary(octal etc..) by hand

How do I convert a negative decimal number into other systems(binary, octal)? I got the decimal numbers: -22,5 , -60 and 166. I have to convert them to binary(16 bit) and octal(by hand). I know the ...
6
votes
1answer
116 views

Does every positive integer appear in the digits of $2\cdot 0.1234567891011… $?

Let $C = 0.1234567891011121314…$ the Champernowne constant. My question is : Does the real number $2 \cdot C \simeq 0.24691357820222426283032343638404244464850525456586062646668707274...$ contain ...
1
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0answers
38 views

square monotonic numbers

A monotonic number is a number in which the digits are in non-decreasing order. I found by computer that most of these numbers are squares of these numbers $$3 \ldots 34,3 \ldots 35,3 \ldots 37,3 \...
14
votes
1answer
297 views

Variation of the Kempner series – convergence of series $\sum\frac{1}{n}$ where $9$ is not a digit of $1/n$.

It is easy to argue that the Kempner series converges: $$ \sum\limits_{\substack{n \text{ s.t. 9 is}\\\text{ not a digit} \\\text{ of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ ...
1
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1answer
47 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
0
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2answers
64 views

What points in $[0,1)$ will have two binary expansions?

What points in $[0,1)$ will have two binary expansions? I know that $\frac{1}{2}$ has the two expansions $0.1\bar{0}$ and $0.0\bar{1}$ $0.1\bar{0}$ is found by starting with $\frac{1}{2}$ and ...
3
votes
1answer
45 views

Measurability of set of numbers with infinite number of digits in decimal expansion equal to 8

Say A is the set of all Real numbers on $[0,1]$ whose decimal expansion contains an infinite number of 8s. I am trying to prove the measurability of this set. I realize that this is the set of ...
0
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0answers
115 views

3-digit chopping vs 3-digit rounding and relative error

3-digit chopping, 3-digit rounding, relative error: Would my calcs below be correct: given 4/5 * 1/3: Exact value: 0.2666666666666667 3-digit chopping: 0.266, its relative error: 0.0025 ...
2
votes
2answers
64 views

Is the last digit of this number :$ {{4^4}^n}+1 $ always $7 $ for $n>1$ and could this be prime?

Some computations in wolfram alpha for $n=2,3,4,5 ,6$ showed that the last digit of this number $ {{4^4}^n}+1 $ for $n>1$ always $7$ . My question here :How do I know if it's last digit always ...
3
votes
1answer
66 views

Repeating decimal notation of 1/53 on WolframAlpha vs notation on Wikipedia

WolframAlpha shows for 1/53 $0.0\overline{1886792452830}$ as the repeating decimal. Why is it not $0.\overline{0188679245283}$ instead? For example, Wikipedia shows for 1/81 $0.\overline{...
2
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1answer
35 views

What is the growth relationship of the number of digits a number has as numbers increase?

To clarify the question, since I'm sure the wording is awkward: In the decimal number system, to get from 1 digit to 2, it takes n=10 numbers. To get from 2 to 3, it takes 90 more numbers added to n. ...
115
votes
14answers
15k views

Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
1
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1answer
35 views

Equivalence of Repeating Decimal

GRE exam question asks what is greater $.\overline{717}$ or $.\overline{71}$ I believe both are equal, but GRE says that $.\overline{717}$ is greater. But why? If they repeat for infinity, isn't ...
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3answers
66 views

why $10$ in any base number system written as $10?$

I am a student trying to write an article in number system can same one give me an idea why $10$ in any base number system written as $10$ $?$
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2answers
105 views

Does $[0.9999…]=1$? [duplicate]

We all know that $0.99999...=1$ So does that imply $[0.99999...]=1?$ Or do we consider it as $0?$ My doubt is: any gif of the form $[0.xyz...]=0$. If $[0.99999...]=1$ won't that be contradicting? ...
1
vote
1answer
39 views

Definition of real by infinite series instead of their Cauchy limits

Looking at Wikipedia´s definition of real numbers I choose a variant one of the alternative definitions, using Cauchy limits. However, Instead of taking a limit I choose the number to be represented ...
8
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3answers
118 views

Last $500$ digits of $2015!-1$

As the title says, I'm looking for the last $500$ digits of $2015!-1$. I assume it's a repetition of zeroes from the factorial, so the final result is a lot of $9$-s, but I can't formulate a solution ...
0
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0answers
58 views

How many trailing zeroes does 4617! contain? [duplicate]

I am getting $1151$ as answer on continuous division by $5$. Is it right? On each division by 5, some remainder is generated...doesn't that count? Example: 4617/5 + 923/5 + 184/5 + 36/5 + 7/5 ...
2
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7answers
746 views

Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. My question is, Does it make any sense to prove this equality? Can one give any "meaning" of the symbol $0.999\ldots$ ...
3
votes
2answers
82 views

Proof that for any $x$ there is a $y$ such that $xy$ is a palindrome

I'm wondering how I would prove For any $x$ there exists at least one $y$ such that $xy$ is a palindrome. For example: 91*99=9009
3
votes
1answer
77 views

Compute Lebesgue measure of set of all real numbers in $[0,1]$ whose decimal representations don't contain the number 7

Consider measure space $(S, \Sigma, \mu) = (\mathbb R, \mathscr B(\mathbb R), \lambda)$. Let $V^C \subseteq S$ denote the set of all numbers in $[0,1]$ whose decimal representations don't contain the ...
8
votes
3answers
205 views

Least positive integer $n$ such that the digit string of $2^n$ ends on the digit string of $n$

What is the least positive integer $n$ such that the digit string of $2^n$ ends on the digit string of $n$: $$ (2^n)_{10} = d_m \, d_{m-1} \cdots d_{q+1} \, (n)_{10} \\ (n)_{10} = d'_{q} \cdots d_1'...
13
votes
3answers
149 views

Generating numbers by repeated doubling and digit reversal

Let $S$ be the smallest set of positive integers satisfying the following conditions: $1 \in S$, If $n \in S$ then $2n \in S$, If $n \in S$ then the digit reversal of $n$ is also in $S$. We assume ...
3
votes
1answer
103 views

Showing this function on the Cantor set is onto [0,1]

The excerpt below is taken from Rosenthal's A First Look at Rigorous Probability. $K$ refers to the cantor set. My question refers to the statement "It is easily checked that $f(K) =[0,1]$. I am ...
15
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1answer
124 views

Are there more than 2 digits that occur infinitely often in the decimal expansion of $\sqrt{2}$?

The other day I got to thinking about the decimal expansion of $\sqrt{2}$, and I stumbled upon a somewhat embarrassing problem. There cannot be only one digit that occurs infinitely often in the ...
1
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2answers
100 views

A number with 6 distinct digits which get multiplied by 5 if we move the last digit to front [closed]

There is a number with 6 different digits, if we pick the last digit of that number and place before that number we got $5$ times our number. How to find such a number?
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0answers
46 views

Representing negative numbers with an infinite number?

Motivation We all know that: $$ .\bar{9} =.999 \dots= 1$$ I was wondering if the following (obviously not rigorous) statement could be defined on the same footing? Question $$ x = \bar{9} $$ $$ \...
7
votes
5answers
671 views

Math induction problem with large numbers

I am trying to figure out how to prove $17^{200} - 1$ is a multiple of $10$. I am talking simple algebra stuff once everything is set in place. I have to use mathematical induction. I figure I need ...
0
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0answers
20 views

Normal numbers and equidistribution

Does anyone know how to prove the following criterion? (Due, 1949) $x \in (0,1)$ is normal for base $10$ $\iff \{10^nx\}_n$ is equidistributed. "$\Leftarrow$" is quite easy using definition of ...
5
votes
1answer
47 views

Does every finite digit-sequence appear in some factorial?

Suppose, some finite digit-sequence is given. Can we prove or disprove, that there is always some number $n$, such that the digit-sequence appears in the decimal-expansion of the number $n!$ ? If ...
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votes
2answers
130 views

Irrational numbers are non-terminating/non-repeating decimals [closed]

Why is it true that all irrational numbers are non-terminating/non-repeating decimals? By definition, an irrational number is one that can't be expressed as a ratio of integers.
10
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6answers
795 views

Can I guess an irrational number formula from its digits?

Let us say I have 10,000 digits started from some point (lets say the 16th digit) of the decimal expansion square root of some arbitrary number, like 13. Is there any way I can get back the original ...
5
votes
2answers
52 views

Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string

Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string? For example, given the string 123456, can I find a perfect square with leading digits ...
0
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1answer
39 views

Conversion base $r$ to base $10$ (decimal) Algorithm

There's another algorithm for converting from base r to base 10? The only one I know is the following one: For example 20 (base 5) to base 10 is: $2X5^1 + 0x5^0 = 10.$
2
votes
0answers
65 views

Can it be proven that infinite many primes can be formed only using two distinct digits?

It seems obvious that infinite many primes can be formed only using two distinct digits. $67776767776667777777$ is an example for such a prime. Even if we allow only the digits $0$ and $1$, there ...
1
vote
1answer
35 views

Average of decimal digits

I define the function $d_{\mathrm{avg}} : [0, 1]\to [0, 1]$ such that for $0.x_1x_2x_3\cdots$ the decimal expansion of $x$ (defined such that $\nexists N : x_k = 9$ for all $k \geq N$), $$d_{\mathrm{...