Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
1
vote
4answers
78 views
How can $\frac{4}{3} \times 3=4$ if $ \frac{4}{3}$ is $1.3$? [duplicate]
Ok use your closest calculator, and type $\frac{4}{3}$, which is $1.3333333333$,and then multiply it with $3$ which is $3.9999999999$ but then type $\frac{4}{3} \times 3=4$ how?. How can it be $4$ if $...
2
votes
4answers
97 views
Find $21^{1234}\pmod{100}\equiv \ ?$
The I'm having trouble to do this only by hand (no software or calculator). I tried the following:
\begin{align}21^{1234}(\text{mod} \ 100) &= 21^{1000}21^{200}21^{20}21^4(\text{mod} \ 100)\...
0
votes
0answers
16 views
Expansion in power of $\frac{1}{Z}$ and $\frac{ln(Z)}{Z}$
When I read the paper I met the problem in the step expansion in power. We have
\begin{align}
s(\epsilon)=\frac{A\epsilon^{a}}{bB|\dot\epsilon|}e^{-Be^{b}} \left[1+\frac{a}{bB}\epsilon^{-b}+\frac{a(a-...
0
votes
0answers
17 views
When do a real $x$ have many decimal representation? How to prove there is a maximum of two decimal representation? [duplicate]
I was asking myself when a real number has two decimal representation. Looking around, it seemed this was true if and only if one of its decimal representation end with only zeros $x_D=...0000000$ (I ...
0
votes
2answers
33 views
Every real number has a decimal representation.
I was reading this answer explaining why all real number have a decimal representation. I think it is really a nice explanation but I don't really see were (I think it is a little hidden) we use the ...
-1
votes
3answers
57 views
Can the product of two rational numbers be an irrational number? (Kindly see the example in description)
I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = \frac{1}{7} ; \;\;\; b = \frac{22}{1} ;$$
both a, b are individually rational (either ...
1
vote
2answers
51 views
Mathematical representation of each digit? [closed]
First than nothing, sorry for my english, I'm not native.
I was wondering how I could represent mathematically, each digit of a number.
Example: 172
...
1
vote
1answer
16 views
Closed Form Addition of BCD numbers
Binary Coded Decimal (BCD) number representation is a 4-bit encoding which maps numbers 0-9 to their counterpart binary codes. Addition of BCD numbers can be formulated as follows:
z = a + b (If z &...
0
votes
1answer
47 views
Find all 3-digit numbers divisible by a sum of groups of its digits
How to find all three-digit number which are divisible by a sum of specific digit groups explained below?
The original number should have only non-zero and non-repeating digits.
example:
$301$ has a ...
4
votes
3answers
318 views
Find all three digit numbers which are divisible by groups of its digits [closed]
How can I find all three-digit numbers which:
Do not contain a $0$ digit
Have different digits
Are divisible by below described groups of its own digits
The number passing first two conditions ...
1
vote
0answers
33 views
Writing down consecutive natural numbers until a certain number of digits $k$ is reached.
A person starts writing consecutive natural numbers from $5$ until $k$ digits are reached. For some values of $k$, this will be impossible, for example $6$ or $8$ are impossible as then after writing ...
20
votes
6answers
3k views
Repeatedly dividing $360$ by $2$ preserves that the sum of the digits (including decimals) is $9$
Can someone give me a clue on how am I going to prove that this pattern is true or not as I deal with repeated division by 2? I tried dividing 360 by 2 repeatedly, eventually the digits of the result, ...
1
vote
2answers
67 views
Apostol Proof for Finite Decimal Approximations to Real Numbers
I'm self-learning real analysis, and I am trying to understand a part in the proof for the following theorem in Mathematical Analysis by Apostol:
Let $x\geq 0$. Then for every integer $n \geq 1$, ...
13
votes
4answers
710 views
Numbers such that they equal the product of their own digits
For the sake of simplicity, I've been only looking in base $10$ numbers, though I've wondered how this might work in other bases - easier to stick with the familiar, though, since this is purely ...
7
votes
1answer
150 views
$\sum\limits_{m\geq1}\sum\limits_{n\geq1}\frac{(-1)^n}{n^3}\sin\left(\frac{n}{m^2}\right)=\frac{\pi^6}{11340}-\frac{\pi^4}{72}$ Numerical evidence
I am looking for numerical evidence that
$$\sum_{m\geq1}\sum_{n\geq1}\frac{(-1)^n}{n^3}\sin\left(\frac{n}{m^2}\right)=\frac{\pi^6}{11340}-\frac{\pi^4}{72}$$
I have proven it, but I just want to be ...
0
votes
1answer
16 views
Round off to decimals
I m not sure about this problem. Pl help. 1. Roundoff this number to tenths place 87.952
2. Round off this number to hundredths place 75.195
As per me answer should be 88.0 and 75.2 for ...
0
votes
0answers
37 views
Finding messages in decimals
A puzzle asked if there is message in the decimal expansion of $\pi$, starting after the decimal point.
The answer, with $\pi$= 3.1415, using A1-Z26 conversion, split 14|15 is no.
Trying this for $...
0
votes
4answers
80 views
How do you add, subtract, multiply, and divide infinite decimals?
In elementary school, we are taught how to add, subtract, multiply, and divide two terminating decimals. My question is, what are the corresponding algorithms in the case of non-terminating decimals, ...
-1
votes
1answer
62 views
Finding sum of digits of $m$ [closed]
If the sequence of 5 positive integers (a,b,c,d,e) satisfy:
$$abcde\leq {a+b+c+d+e} \leq 10m$$ then find the sum of digits of m.
I don't know how to approach this question. I know it's not a good ...
0
votes
1answer
18 views
Equation Involving Digit Sum Function
Define the digit sum $S$ of a number as the sum of its digits. For example, $S(456)=4+5+6=15$. Given positive integers $a_1, \cdots, a_n$ and $Q$, I'd like to ask how to obtain the nonnegative ...
0
votes
1answer
24 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
53 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
27 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
96 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
48 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
26 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 ...
3
votes
7answers
282 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
36 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
43 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
36 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
147 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
14 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
28 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
106 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
43 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
34 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
34 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
32 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
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4answers
43 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
32 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
59 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
67 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
51 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
60 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
2answers
101 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 ...
