For questions about decimal expansion, both practical and theoretical.

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14
votes
1answer
290 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
vote
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
votes
2answers
63 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
votes
0answers
100 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
65 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
votes
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
vote
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 ...
-3
votes
3answers
64 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 $10$ $?$
-1
votes
2answers
104 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
38 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
votes
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
votes
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
votes
7answers
735 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
204 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
147 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
votes
1answer
121 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
vote
2answers
92 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?
1
vote
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
votes
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
45 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 ...
-1
votes
2answers
112 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
votes
6answers
789 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
51 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
votes
1answer
37 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
34 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{...
2
votes
2answers
89 views

If $n$ is a square, can $n$ consist of only odd digits?

The question is: If $n$ is a square, can $n$ consist of only odd digits? I have a feeling that the answer is no, with the only exceptions being $n=1,9$. I am not sure how to go about proving this ...
6
votes
5answers
410 views

Prove that in every sequence of 79 consecutive positive numbers written in decimal system there is a number whose sum of the digits is divisible by 13

Prove that in every sequence of $79$ consecutive positive numbers written in decimal notation there is a number the sum of whose digits is divisible by $13$. I tried to take one by one sets of $79$ ...
3
votes
0answers
92 views

Is there any kind of known pattern to $\sqrt 2$ in base 2?

Is there any kind of known pattern to $\sqrt 2$ in base 2? Is there any classification categories for decimal digits of numbers that for example would put $\sqrt 2, \sqrt 3 \cdots \sqrt n$ into ...
2
votes
2answers
88 views

Maximum length of repeating digits in a decimal

Total math novice here. I'm wondering if (as I think should be the case) there is no maximum length of a repeating decimal period, for example: ...
0
votes
2answers
46 views

Are there methods to recursively calculate the decimal expansion of real numbers?

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence, $$(1) \quad C_0=0.1, \ C_1=0.101, \ ...
1
vote
0answers
40 views

What formulas are available to find the nth digit of a number?

Imagine that'd I'd like to investigate the digits of $\sqrt{2}$, or of any real number. If I want a formula for the nth digit of a real number $x$, we have, $$(1) \quad \operatorname{d_n}(x)=\lfloor ...
6
votes
2answers
137 views

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

I was wondering at which decimal place $\pi$ first repeats itself exactly once. So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is. To clarify, this ...
5
votes
2answers
79 views

Repeating decimals linked to reciprocals of primes

Now this question is base dependent, I will assume base 10 but feel free to generalize. I was noticing that for small primes that are not factors of the base ($2$ and $5$ terminate) the reciprocal of ...
0
votes
1answer
22 views

Series of digits in a number

Given a certain number $\mathcal{N}$ with infinite digits, for instance $\pi$ or $\sqrt2$, it's possible to know if this number has a L-string of repeated numbers, but without actually computing the ...
3
votes
2answers
116 views

Write $0.2154154\overline{154}$ as a fraction

Let $x = 0.2154154\overline{154}$ , I have to prove that it is a rational number just by writing it as a fraction with the proper steps. I note that the repeating part, $154$, is composed by 3 digits....
1
vote
0answers
63 views

Should we say $0.\bar9$ *can* equal $1$ also in the hyperreals?

Consider the sequence $R_n$ of repunits, defined as $\displaystyle\frac{10^n-1}{9}$. We have $$\frac{R_{n+1}}{R_n}=\frac{9}{9} \frac{10^{n+1}-1}{10^n -1}=\frac{10^{n+1}-1}{10^n -1}=\frac{\overbrace{99\...
1
vote
2answers
69 views

Calculating last digit of a number using binomial theorem

How to calculate the last digit of a number say like $$\large 3^{4^{5}}$$ using binomial theorem? P.S:I know how to solve it using modular arithmetic. I saw this one but its not of much use in ...
8
votes
1answer
113 views

Finding the $1000$-th decimal of $\sqrt{1111…111}$

As I was cleaning up my desk, I found my Calculus exam from almost a year ago. I remember there was only a bonus task that required either a tad more wit, either a bit more time. It goes like this : ...
1
vote
1answer
63 views

How can i count number of digits in tetrated numbers?

As you read in the title, I need a technique for counting number of digits in tetrated numbers. For example: ${3^{(3)}}^3 = 7625597484987 $(13 digits) ${7^{(7)}}^7 = $How many digits (...
1
vote
2answers
37 views

creating a fraction or decimal using only addition or subtraction

how do i create a decimal or a fraction by only using addition or subtract? I have the numbers 1 and 2, and I want to end up with .5 -- have been stuck on this for quite a bit! I cannot just do 1/2, I ...
0
votes
1answer
27 views

Significant figures.

Calculate how many gram of p-nitrophenol you require to prepare 250 mL of a 11 mM solution. (Answer to 3 significant figures) I worked it out as $$250/1000 L * (11*10^{-3}\text{ M})$$ = $$2.75\times ...
0
votes
3answers
247 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 or ...