For questions about decimal expansion, both practical and theoretical.

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First digit inequality [on hold]

If $d_n$ is the first digit of $n$ and $f(k)$ is the number of squares $(n+1)^2$ and $n^2$ of $k+1$ digits that hold $d_{(n+1)^2}-d_{n^2}\le1$, then find the sum of the digits of $\sum_{k=1}^{1008} ...
3
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0answers
53 views

Partitioning positive integers using digital rivers

I stumbled on a very simple computer science question from the British Informatics Olympiad for schools and colleges. Embedded in it is a very interesting numbers theory problem. Here is the ...
3
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3answers
27 views

Is there a binary fraction with finite decimal expansion that does not end in $5$?

I'm trying to come up with the finite decimal fraction not ending with $5$ which can be finitely expressed in binary. At the moment, I don't see how's that possible. Since decimal fractions can only ...
2
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3answers
355 views

WolframAlpha function that returns the 'decimal' part of a number [closed]

Is there a function or command in Wolfram Alpha for getting only the decimal part of a number? Something like this: DecimalPart(3.4231) = 0.4231 I will be using ...
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3answers
99 views

$1000$th decimal digit of $(8+\sqrt{63})^{2012}$

Find the digit at the $1000$th position at the right of the decimal point of the number $(8+\sqrt{63})^{2012}$ I took this problem from a Mexican Math Olympiad called Galois-Noether. It's the ...
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0answers
10 views

Least upper bound property of decimal representation of reals

This is my attempt at a proof that real numbers represented by infinite decimals satisfy the least upper bound property, i.e. every upper bounded set has a least upper bound. I am not sure it is ...
2
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4answers
79 views

Is $1.0000…$ ( $1$ with infinite zeros) greater than $1.0$? [closed]

Given that $0.3333...$ is greater than $0.3$ and similarly $0.777...$ is greater than $0.7$, does it follow that the sum of $0.33...$ and $0.77...$ is greater than sum of $0.3$ and $0.7$?
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0answers
18 views

Get first digits of a very large quotient

Is there a method to get the first $n$ digits of a quotient (ex. a thousand digit number divided by a 5 digit number) without dividing all the way through? I suppose long division until $n$ digits are ...
3
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1answer
29 views

Theorem on Repeating Decimals

So I am wondering if anyone recognizes the following theorem: Given a prime $p$, and a base $b$ (natural number $>1$), the period of $\frac{1}{p}$ expressed in base $b$ is the unique $d$ that ...
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1answer
45 views

Last digit of a number

I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...
2
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1answer
40 views

Decimal expansions and topological connectedness

I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold: One may think of ...
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0answers
15 views

Is there any difference between fixed point and decimal point?

Source: Introduction to Computers' 1999 Ed.1999 Edition Fixed point number 774.3675 is just a decimal number with a decimal point to show a fractional part 3675/10000. I see no difference in the ...
2
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1answer
111 views

How to prove $\limsup_{n \to \infty} |\sin(n)| = 1$?

Does decimal expansion of $\pi$ contain blocks of zeroes of any integer length? I.e. $0$, $00$, $000$, $\ldots$ I discovered this question, when trying to prove $$\limsup_{n \to \infty} |\sin(n)| = ...
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1answer
60 views

a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
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2answers
108 views

How to prove that every power of 6 ends in 6?

Yesterday I had the traditional math matriculation exam, and in it there was a question "In what digit does the number $2016^{2016}$ end in?" After the test The Matriculation Examination Board ...
3
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2answers
60 views

Who is Petrick from Petrick's method?

I would like to ask your help. I think this is the best place for this. In my language -as well as English- I haven't found anything about Petrick yet. His method okay, but I would like to know about ...
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6answers
208 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...
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4answers
32 views

Why does dividing a number with $n$ digits by $n$ $9$'s lead to repeated decimals?

For example, $\frac{1563}{9999} = 0.\overline{1563}$. Why does that make sense from the way the number system works? I can vaguely see that since the number $b$ with $n$ $9$'s is always greater ...
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0answers
24 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
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2answers
60 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 ...
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3answers
68 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 ...
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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 ...
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2answers
53 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 ...
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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
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2answers
65 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
199 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 ...
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0answers
128 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
25 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 ...
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1answer
337 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 ...
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2answers
27 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
391 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 ...
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1answer
17 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 ...
5
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1answer
108 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 ...
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0answers
37 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
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1answer
275 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 $E \subset \Bbb N_{>0}$ ...
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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 ...
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2answers
61 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
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1answer
44 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
73 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 ...
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2answers
63 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
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1answer
64 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 ...
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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. ...
114
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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 ...
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1answer
34 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
59 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
93 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? ...
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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 ...
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3answers
114 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
57 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 ...