For questions about decimal expansion, both practical and theoretical.

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3answers
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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 ...
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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 ...
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2answers
54 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)} = ...
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1answer
36 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 ...
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0answers
18 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 ...
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6answers
117 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
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2answers
146 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 ...
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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 ...
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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
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2answers
84 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
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3answers
126 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
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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 ...
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1answer
33 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 ...
3
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0answers
69 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
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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.
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2answers
21 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
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1answer
78 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
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3answers
58 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 ...
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0answers
72 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
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0answers
45 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
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1answer
21 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
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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
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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
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1answer
359 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 : ...
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2answers
565 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
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1answer
49 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 ...
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2answers
177 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
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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
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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, ...
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2answers
165 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 ...
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3answers
142 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
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1answer
64 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
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4answers
39 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 ...
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2answers
42 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 = ?`
2
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1answer
68 views

What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?

Warning: This problem requires a bit of setting. Fix a finite set $A \subset \Bbb{Z}$ and consider an infinite non (ultimately) periodic sequence $\mathbf{a}=(a_i)_{i \geq 1}$ of elements of $A$ such ...
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0answers
17 views

Monograph about periodic representations of numbers in non-integer bases

I'm looking for a monograph (book, article, lecture notes, whatever) about the representation of numbers (real or complex) in non-integer bases. I am especially interested in results about algebraic ...
3
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2answers
89 views

Can a number have both a periodic an a non-periodic representation in a non-integer base?

Fix an algebraic number $\beta$ and consider a complex number $\alpha$ which admits multiple representations in base $\beta$. If one representation of $\alpha$ is ultimately periodic, must every other ...
2
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1answer
46 views

How to get the amount of digits after the decimal point

How can I easily find the amount of digits after the decimal point? For example: ...
9
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2answers
177 views

Does π start with two identical decimal sequences?

Let d$(x,y)$ be the sequence of decimals of π from the x:th one to the y:th one. My question: is there a number $n$ such that d$(1,n)$ = d$(n+1, 2n)$? I.e., does π start with two identical decimal ...
0
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2answers
39 views

Repeating Decimal in different base

I've come across the following question. Find $0.\overline{204}_6$ as a base ten fraction. I understand that is the question asked the repeating decimal in base $10$, I would then say that: $$x ...
2
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0answers
27 views

Question about repeating decimal?

For simple fraction, we can easily convert it to repeating decimal by calculator. Ex 1/3 = 0.33333..., 1/7=0.(142857),... But some fraction fraction like 10/29, 1/97,... The repeating part of them are ...
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1answer
74 views

Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that ...
8
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3answers
130 views

nonzero digits in decimal representation of $\sqrt{2}$

let $1,d_1d_2d_3\dots$ be a decimal representation of $\sqrt{2}$. Prove that at least one $d_i$ with $10^{1999}<i<10^{2000}$ is nonzero. I have no idea how to solve it. I think that the given ...
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2answers
46 views

Find the fraction that creates a repeating decimal that repeats certain digits

Is there any way to find the fraction $x/y$ that, when converted to a decimal, repeats a series of digits $z$? For example: ${x}/{y} = z.zzzzzzzz...$ or with actual numbers, $x/y = 234.234234234...$ ...
3
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4answers
64 views

Proof that $2^{0.5}$ will not touch the $1.5$ mark on the number line when we try to mark it exactly?

I know this would be kind of silly, but then this has been troubling me for the past few days. All of us know $2^{\frac{1}{2}}$ is irrational. Let us try to mark this on the number line "exactly", as ...
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3answers
131 views

Why 6? Is this meant to be intuitive?

What is this question asking? Do you just simply count to number of 1's and 0's and then choose that integer, 6? $$y=10^k+10^{k+1}+10^{k+2}$$ In the equation above, $k$ is an integer. For ...
0
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3answers
79 views

Formal notation for representing decimal numbers

What is the formal mathematical notation for representing a decimal number using variables as it's digits? I am honestly surprised that this has not been asked yet on MSE. Let me clarify a ...
2
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1answer
144 views

What is the biggest fivedigit $abcde$ number, thats divisible by $bcde$, $cde$, $de$ and $e$?

What is the biggest fivedigit $abcde$ number, thats divisible by $bcde$, $cde$, $de$ and $e$? I was trying to find it but I couldnt. Can you help me with a step by step answer?
3
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2answers
106 views

Determine whether the decimal expansion of a rational number is infinite

This may be a naive question but I would like to know whether we can determine if a fraction (say $1/3$) will produce a rational number with an infinite number of digits after the decimal when ...
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2answers
311 views

Convert decimal to Binary Floating Point - 8 Bit [closed]

I am trying to convert +3.5 to binary floating point, but im struggling to find the exponent. (8 Bit) Where 1st bit is the Sign, 3 bits for Exponent and 4 bits for Mantissa. Hope somebody can explain ...