Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
4
votes
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
votes
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
vote
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
votes
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
votes
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
vote
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
vote
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
votes
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
votes
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 = \...
1
vote
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://...
0
votes
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
votes
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
votes
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 ...
1
vote
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!
-3
votes
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
vote
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
votes
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?
0
votes
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
votes
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 ...
1
vote
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)?
0
votes
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
vote
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......
1
vote
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
votes
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
votes
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 ...
-1
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
votes
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
votes
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 ...
-2
votes
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
votes
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
votes
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 ...
-1
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
votes
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, ...
1
vote
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
votes
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 ...
0
votes
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 $0.\bar3$ ...
-3
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 $6 \mod 7$?
0
votes
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 ...
