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Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

4
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3answers
45 views

How many 4 digit natural numbers are there, such that sum of their digits is not bigger than 31?

How many 4 digit natural numbers are there, such that sum of their digits is not bigger than 31? How to approach this problem? I tried applying stirling numbers, but with no success.
0
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1answer
22 views

Decimal representation of the set [0,1)

I have encountered the next statement in statistics lecture (translated from german): "From the analysis you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal ...
1
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1answer
52 views

Understanding Normal Numbers

I am trying to understand what normal numbers are. Just for simplicity I want to talk about base 10. I understand that a number is normal in base 10 if there a probability of $\frac{1}{10 } $ such ...
0
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3answers
26 views

Converting Decimal with 3 decimal places to Octal

I know how to convert from $769_{10}$ to base 8 that is $1401_8$ through dividing and remainder method but what is the method for $769.513_{10}$ to convert to octal?I know that it is to separate the ...
6
votes
1answer
92 views

Real irrational algebraic numbers “never repeat”

An oft-used phrase describing irrational numbers is that their (decimal) expansions "never repeat". The sense of "never repeating" intended is, of course, that their expansions don't repeat forever. ...
37
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10answers
4k views

How are the known digits of $\pi$ guaranteed?

When discussing with my son a few of the many methods to calculate the digits of $\pi$ (15 yo school level), I realized that the methods I know more or less (geometric approximation, Monte Carlo and ...
1
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1answer
40 views

Proving a decimal expansion is bijective

I'm trying to see if a function $f : (0, 1] × (0, 1] → (0, 1]$ is bijective, where $0.a_1a_2a_3 . . . $is the decimal expansion of $x ∈ (0, 1]$, and $0.b_1b_2b_3 . . .$ is the decimal expansion of $y ∈...
1
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1answer
23 views

What is the generic name for “decimal” type fractions?

A number in the form: 1.234 is often loosely called decimal, though the name really refers to the fact that its base is 10, and has nothing to do with the ...
4
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7answers
212 views

$n$ has digit sum 100; $2n$ has digit sum 110

My question is: A $n$-digit number is given whose digit sum is $100$, the number when doubled gives digit sum as $110$ then what is this $n$-digit number? My approach: I assumed $n$-digits to be $...
0
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1answer
29 views

Decimal expansions of rational numbers.

One can use modular arithmetic to find the decimal expansion of a rational number. (see 1: https://i.stack.imgur.com/kw4Gk.png). Using the same method I have run into a couple of problems. $x = \...
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1answer
36 views

decimal expansion of an integer

Can someone be so kind as to explain what is meant by the decimal expansion of an integer? I saw the following at this link but I don't know what decimal expansion of an integer refers to: https://...
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2answers
26 views

Maximum Period of Decimal Expansion

My question is similar to (but different from) the one here. I came across this sentence on Wikipedia: "The decimal expansion of a rational number always either terminates after a finite number of ...
5
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5answers
114 views

Divisibility 1,2,3,4,5,6,7,8,9,&10

Tried: Seems the ten-digit number ends with $240$ or $640$ or $840$ (Is not true, there are more ways the number could end) $8325971640,$ $8365971240,$ $8317956240,$ $8291357640,$ $8325971640,$ $...
2
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1answer
36 views

Converting a decimal number to a number having all 1s in another base

I am actually trying to solve the problem: "Beautiful Numbers", asked in Google Kickstart $2017$ Practice Round $2$ (Link: https://code.google.com/codejam/contest/12254486/dashboard#s=p2). The problem ...
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0answers
13 views

Frequency of digits in repeating decimal expansions

If I have a function f(x) which is defined on the rational numbers where x can be represented as $\frac{a}{b}$ where a and b are mutually prime positive integers and b > a. If x can can be ...
1
vote
1answer
26 views

Arithmetic way to get the number of decimal digits in a number [closed]

There is any general formula to get the number of decimal digits in a decimal number? For example in 8.888, there are 3 decimal digits. Thanks for any reply!
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2answers
99 views

A super weird quality of numbers that is hard to explain. [duplicate]

First off, I know that 0.9… = 1 and I'm not trying to prove or debate that, but my discussion about it is necessary for understanding the question. I was talking to my brother about whether 0.9… ...
1
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1answer
34 views

Does a real number with this decimal expansion for $r$ and $r^2$ exist?

Does there exist a real number $0< x <1$, such that the decimal expansions of $x$ and $x^2$ are the same, starting from the millionth term, and neither expansion has an infinite ...
0
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2answers
38 views

Number having sexagesimal expansion end with infinitely many zeros?

I am looking for all the real numbers whose sexagesimal expansion (base $60$) ends in infinite tail of zeros. Does they really exist? It seems absurd to me or mm thinking it in a wrong manner?
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1answer
24 views

Decimal to octal transformation

52.8 div 8 = 6.6 mod 4.8 6 div 8 =0.75 mod 6 The result is 64.8 Is that correct? I'm quite confused with 4.8
0
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1answer
31 views

Why converted values from Decimal to binary isn't the same? [closed]

the professor told us today about binary and decimal and how to convert them , and give us example of a decimal number (13) and we converted it to binary which is (1101) . Now when I'm trying to do ...
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0answers
29 views

How many primes are there on the form $100\cdots 0 1$? [duplicate]

For example 11 and 101 are primes, but apart from them, can we determine how many primes on the form $100\cdots 00 1$ there exist (in decimal number system)?
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4answers
40 views

Expressing $1.24\overline{123}=1.24\;123\;123\;123\;\ldots$ as the ratio of two integers

So I'm supposed to express: $$1.24\overline{123}=1.24\;123\;123\;123\;\ldots$$ as the ratio of two integers. So I got $$1+\frac{24}{100}+\frac{123}{10^5}+\cdots$$ I don't know if this is correct, ...
1
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1answer
30 views

A fraction whose digits are the same as it's decimal representation [closed]

Do there exist any numbers such that their fractional representation $$\frac{\overline{a_1a_2a_3a_4...}}{\overline{...a_{n-3}a_{n-2}a_{n-1}a_{n}}}$$ can be represented as $$ \overline{0.a_1a_2a_3......
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3answers
1k views

How many numbers are there which only contain digits $4$ and $7$ in them? [closed]

I wanna know how many numbers $n$ are there which only contain digits $4$ and $7$ in them, where $1 ≤ n ≤ 10^9$. Ex: $4, 7, 44, 47, 74, 77, ...$ I am trying to find a general equation to compute the ...
0
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1answer
27 views

Two whole numbers n and k . Print k decimal digits of 1 / n .

I have to print these decimal numbers in C++ . But first i need to understand this question mathematically .
2
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1answer
57 views

Example of an irrational number of this form.

Let $b:=\overline{0,b_{1},...}$ such that $b_{k} \in \{1,2\}$ and $b$ is irrational. Is $0,121122111222...$ a good example? Can you give me another example of number like this?
2
votes
1answer
64 views

Difference between a non-increasing number and it's mirror (digit reversal)

Consider a subtraction between a non-increasing number $N$ and its mirror $M$ (a non-decreasing number) where N has half or more digits equal to $0$. Example:$\qquad N-M = 76620000-00002667 = D$ I ...
0
votes
3answers
43 views

Show set of all real number in ($0,1$) with base $10$ decimal expansion contains no $3$s or $7$s is uncountable

here is the question: Show set of all real number in ($0,1$) with base $10$ decimal expansion contains no $3$s or $7$s is uncountable My thoughts: to show it's uncountable, we should map it to an ...
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votes
2answers
55 views

Infinite numbers of decimals for a finite point in a line

Recently I started studying real analysis. In the beginning itself I was introduced to numbers which can't be represented as ratios of other natural numbers. But before studying them I had doubts ...
2
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0answers
67 views

About the proof that every real number in the unit interval is the limit of a sequence of dyadic numbers

Given $x \in (0,1)$, show there exists a sequence $(x_n) \subset \{0,1\}$ such that $x = \sum_{n=1}^\infty \frac{x_n}{2^n}$. After running into difficult in trying to solve this problem, I found this ...
0
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1answer
42 views

Simplify binary expansion

Let $Bin[n]$ denote the binary expansion of integer $n$. Does there exist a simplification of the formula $Bin[\sum a_i 2^i]$ ? Clearly when $a_i \in \{0,1\}$, then the $a_i$ already represent ...
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2answers
36 views

Cardinality of number of digits [closed]

What is the cardinality of the number of digits (in decimal form) of an irrational number like $ \pi $?
16
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5answers
395 views

Multiples of $999$ have digit sum $\geq 27$

How could we prove the following claim? The sum of the digits of $k\cdot 999$ is $\ge 27$ I checked $k = 1$ up to $9$. And I found that if it's true of $d$ it's also true of $10\cdot d$. I also ...
5
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1answer
103 views

Proving that 1/3 has no finite decimal representation

There is a problem where i need to prove that 1/3 has no finite decimal representation Here's my proof, can someone tell me if its valid? Proof Lets assume there is a decimal representation for $\...
-1
votes
1answer
45 views

Convert a fraction to whole number [closed]

Lets say that there are a few fractions: x = 0.584592145015 y = 0.443242244323 How do one convert these fractions to whole ...
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votes
2answers
45 views

recurring decimals - base 10 to base 12

I came across duodecimal (base 12) numbers. In base 10 system 10/3 = 3.333.... i.e repeating decimal. But in base 12 system where "t" represents 10 of base 10 - t/3 = 3.4 which is a non repeating ...
0
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3answers
165 views

Find all real number(s) $x$ satisfying the equation $\{(x +1)^3\}$ = $x^3$ , where $\{y\}$ denotes the fractional part of $y$

Find all real number(s) $x$ satisfying the equation $\{(x +1)^3\}$ = $x^3$ , where $\{y\}$ denotes the fractional part of $y$ , for example $\{3.1416\ldots\}=0.1416\ldots$. I am trying all positive ...
3
votes
1answer
96 views

What is the first digit of $-10$?

Generally, what is the first digit of a negative whole number? In case of $-10$, it can not be $-1$, because a digit is defined to be from $0$ to $9$. Is it $1$? Or does it not have the first digit ...
6
votes
0answers
124 views

Weird sum that is almost definitely not $\sqrt 2$

I have not the ability to compute more than four digits of $$\sum_{n=1}^\infty \frac{1}{n^2 H_n^{(\ln n)}}$$ $H_n^{(m)} = \sum_{k=1}^n \frac{1}{k^m}$ is the generalized harmonic number. I know ...
2
votes
1answer
37 views

Number Theory - Decimal Representation Question. Difficulty understanding a given solution.

This question is from the 1995 Hungarian Mathematical Olympiad. Let $k, n$ be positive integers such that $(n+2)^{n+2}, (n+4)^{n+4}, (n+6)^{n+6}, ..., (n+2k)^{n+2k}$ end in the same digit in decimal ...
0
votes
1answer
52 views

Convert into a decimal number

How can we convert in $\mathbb{Q}$ the inverse of $1{,}2^2$ (i.e the number $\frac{1}{ 1{,}2^2}$) into a decimal number? Also how could we show that with the complex multiplication $\cdot$ then $(G, ...
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2answers
53 views

Ternary Negatives

If, according to 2's Complement any binary string can be converted to it's negative counterpart by flipping each digit to it's opposite number (eg. $1001_2 \to 0110_2$) and then add 1, then how ...
2
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0answers
16 views

Generating decimal representation of continued fractions without ever-increasing terms?

I'm trying to generate a very large number of digits in the decimal representation of a continued fraction, but am running into a scaling problem. The more digits I compute, the larger my ...
4
votes
1answer
103 views

Functional Square Root of Digit Sum?

Define $\text{sdig}(n)$ to be the sum of the decimal digits of $n$, where $n$ is a positive integer. My question is as follows: Does there exist a function $h:\mathbb Z^+\mapsto\mathbb Z^+$ such ...
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5answers
152 views

Why aren't repeating decimals irrational but something like $\pi$ is?

We use closest representations for both of them, but they are not completely true. $\frac{22}7$ and $3.14$ are not exactly $\pi$ but we use them as the best option available. $\frac13$ is $0.\bar3$ ...
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votes
2answers
104 views

Logical fallacy that suggests $3/9=3/10$

I have seen that $3/9 = 1/3$ can be written as $0.3$. However, $0.3=3/10$. Does this mean that $3/9=3/10$, or am I confused?
6
votes
1answer
890 views

A square integer has more than a million digits. What is the least amount of even digits the square can contain?

Thank you for your help. NB. I would very much appreciate if you only gave tips and not the whole solution.
0
votes
1answer
81 views

Primes formed by concatenating $n \;\text{and}\; n+1$ [closed]

$23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081$ These are primes formed by concatenating $n$ and $n+1$. Is it possible to prove that none of these primes is congruent to $6 \mod 7$?
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2answers
53 views

how to confirm that for $n<1000$, each number $0,…,9$ appears exactly $300$ times

How can we know that among all numbers with $3$ or fewer digits (i.e. a number $n<1000$), each digit (from $0$ to $9$) appears exactly $300$ times? I'm trying to convince myself, but I can't seem ...