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Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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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 $...
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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)\...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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 ...
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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, ...
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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 $n \geq 1$, ...
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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 ...
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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 ...
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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 ...
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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 $...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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
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 ∈...
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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 ...
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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 $...
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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 = \...
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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://...
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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 ...
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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) $8325971640,$ $8365971240,$ $8317956240,$ $8291357640,$ $8325971640,$ $...
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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
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 ...
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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!
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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… ...
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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 ...
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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?
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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
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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 ...
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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)?
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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, ...
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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......
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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 ...
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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 .
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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
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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 ...
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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 ...
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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
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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 ...