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Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

8
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0answers
53 views

Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?

While discussing prime powers and divisors, I came up with the following problem. Examples $\to$ prime $p=3$ digit sum (in base ten) of $p^3=27$ is $p^2=9$, a power of $p$,. $\to$ prime $p=7$ ...
11
votes
3answers
2k views

Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?

I conjecture that for irrational numbers, there is generally no pattern in the appearance of digits when you write out the decimal expansion to an arbitrary number of terms. So, all digits must be ...
8
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1answer
72 views

What reason is there to conjecture that every finite string is really in the decimal expansion of $\pi$?

One of my students asked me this, and it occurred to me that I had never really questioned it. Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ ...
0
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0answers
10 views

Given S, count numbers A such that A - reverseA = S

Given integer $S$ up to $10^9$ count all numbers $A$ such that $A - A' = S \text{ and } A' < A$ and $A'$ is reversed form of $A$ (e.x. $A = 18, A' = 81$). For example if $S = 9$ the answer is $8$, ...
9
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1answer
326 views

Hopping to infinity along a string of digits

Let $s$ be an infinite string of decimal digits, for example: \begin{array}{cccccccccc} s = 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 & \cdots \end{array} Consider ...
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0answers
39 views

Decimal digit extraction of $\pi$

I've seen many folks asking for this so I thought I'd take a shot @ answering it: This function $\pi[d]:\mathbb{N}\rightarrow\left\{\mathbb{W}<10\right\}$, based on the BBP closed form expression, ...
1
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2answers
28 views

Replacing a natural number containing a certain digit with the sum of two without that digit

A question in Google Code Jam 2019 qualification round wanted a positive integer n which contains at least one digit 4 to be represented as a sum of two positive integers a and b, neither containing 4....
2
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0answers
9 views

Normal vs disjunctive vs lexicon

Apologies for lack of rigour but I'll attempt to phrase this in an answerable way. In this question, @Charles writes: [Being a normal number] (or even the weaker property of being disjunctive) ...
-1
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1answer
95 views

The “special” number $8263$

Prime $8263\equiv 1\pmod {17}$ and $8\cdot 2\cdot 6\cdot 3\equiv -1\pmod {17^2}$. Are there other odd primes $p$ without digit $0$ such that: $p\equiv 1\pmod q$ and the product of the digits of $p$ ...
0
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0answers
50 views

Can we expect infinite many primes $p$ equal to the start of $frac(\sqrt{p})$?

The following routine searches for prime numbers $\ p\ $ equal to the beginning of the fractional part of the decimal expansion of $\ \sqrt{p}\ $ : ...
2
votes
3answers
103 views

Hopeless Numbers

Beatriz Viterbo has called a positive integer which is not divisible by any of the ($2^n$, where $n$ is the number of its digits) numbers that result by introducing a plus or minus sign to the left of ...
1
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1answer
42 views

The number 37 trick - generalization

Suppose we have a number $aaa\ldots a$ composed of $k$ equal digits $a$ in base $b$. Let's divide the number by the sum of its digits. When do we get an integer result? I am reading a book which ...
1
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2answers
53 views

Are there any non-trivial examples of this decimal-binary property

My birthday is 10th of October or 1010 in MMDD format. I just realized that 1010 contains two copies of the number 10 and if spelled out in binary, $1010_2=10$ I was wondering how many other ...
-1
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1answer
44 views

Find the digit in hundred-thousandth place of sum

Sum: $1 + 3 + 9/2 + 27/6 + 81/24 + \ldots$ This is a problem on a competitive mathematics test, and I am trying to master the concept so I can understand when similar problems show up in future ...
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2answers
51 views

How many even numbers could be formed [closed]

How many even numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, 7 and 8?
1
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1answer
87 views

Questions about 0.999… equals 1 [closed]

Being 0.999... = 1, I expect that they have the same behaviour when applying the same algorithm/operation, but: If we define >, <, =, as checking digit by digit two number, we have that 0 < 1 ...
0
votes
1answer
90 views

Primes with digits only 1

Let $Y(k)$ be the number consisting of $1$, repeated $k$ times. We know that $Y(2) =11$ is prime. It so happens that $Y(19)$ and $Y(23)$ are also prime. Are there any more? Regards, David
2
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1answer
52 views

Digits in product of two numbers

When we multiply a $m$ digit number with a $n$ digit number, the product will have either $m+n$ digits or $m+n-1$ digits. I want to get some condition on the numbers so that we can predict about these ...
0
votes
1answer
44 views

Show that decimal addition on real numbers is well defined

I am currently self-studying Hubbard's textbook on Vector Calculus but am stuck on the logic for this question: 0.4.1(a): "Let $x$ and $y$ be two positive reals. Show that $x + y$ is well defined by ...
0
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2answers
136 views

Count the 8-digit integers with 1s and 0s satisfying a digit sum property

This question appeared in a contest in Indonesia in 2011, this is #10. Find the number of positive integers which satisfy the following conditions: It contains 8 digits each of which is 0 or 1. The ...
0
votes
1answer
18 views

Translating binary plaintext into alphabetic plaintext using an $18$-digit base-$26$ integer system

I'm working on a cryptography problem and went through the long process of decrypting a sent message to get a $27$ digit number. It then says: Plaintext blocks have $18$ letters and that such an ...
18
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2answers
423 views

Sum of digits of $a^b$ equals $ab$

The following conjecture is one I have made today with the aid of computer software. Conjecture: Let $s(\cdot)$ denote the sum of the digits of $\cdot$ in base $10$. Then the only integer ...
3
votes
3answers
209 views

Why 0.33… is the only expression of 1/3? [duplicate]

I am an undergraduate math student who loves mathematics very much, and I am confused by a math problem. Given $1/3$, we know that $0.33...$ (there are infinite $3$s) is the decimal expression. But ...
2
votes
2answers
48 views

Can we prove that infinite many primes begin with any given digitstring?

With Dirichlet's theorem, we can easily prove that infinite many primes end with a given digitstring with final digit $1,3,7$ or $9$. Can we also prove that infinite many primes begin with a given ...
0
votes
1answer
20 views

Continuity of the reals in terms of decimal expansions

I was wondering about how we could prove the completeness of $\mathbb R$ when this set is defined to be the set of all decimal expressions of the form : $$\underbrace{-}_{\text{sign}}\underbrace{317}_{...
1
vote
4answers
786 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
105 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
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2answers
44 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
65 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
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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 ...
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
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1answer
117 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 ...
5
votes
3answers
343 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
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0answers
37 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
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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, ...
1
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2answers
72 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 $n \geq 1$, ...
12
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3answers
754 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
156 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
20 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
41 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
96 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 ...
0
votes
1answer
19 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
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1answer
58 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
29 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
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1answer
114 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 ...