Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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1answer
74 views

Sum of the digits of $2012^{2012}$ and sum of that sum

I attempted to solve the following problem: Let $S_1$ be the sum of the digits of $2012^{2012}$ and $S_2$ be the sum of the digits of $S_1$. Find $S_1$ and $S_2$. Here is what I've got: Let $n = ...
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0answers
23 views

Questions about cyclic numbers, repeating decimals, and full reptend primes

I have a few questions about cyclic numbers in base $b$ ($b = 2$ in particular). We are dealing here with primes $p$ such that the length of the period in the decimal (more precisely base $b$) ...
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2answers
52 views

Can TWO different points on the number line ever be represented by the SAME infinite decimal?

So I am visualizing this problem, and on the surface it seems like there is no way two individual and unique values could have the same infinite decimal representation. Any thoughts on this? It is ...
2
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1answer
58 views

The product of two repeating decimals

In An exceptional talent for calculative thinking, (IML Hunter, 1962), Professor Aitken explained how he found the decimal expansion of 1/851: 851 is 23 times 37. I use this fact as follows. 1/37 ...
4
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4answers
90 views

If $a^b$ has $b$ digits, what is the greatest value of $b$?

For natural numbers $a$ and $b$, what is the greatest value of $b$ so that $a^b$ has $b$ digits? I knew that the greatest value of $b$ is $21$, where $9^{21}=\underset{21 \text{ digits}}{\underbrace{...
9
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2answers
104 views

How many zeros in the decimal representation of $5^n$?

I'm curious about some properties of the powers of 5 $$5^2=25,\quad5^3=125,\quad 5^4=625,\quad 5^5=3125,\quad ...$$ Is it true that at least $50$% of the digits in the decimal representation of $5^n$ ...
3
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1answer
65 views

How are the Periods of the Decimal Expansions of $\frac{p}{q}$ and $\frac{q}{p}$ Related?

In an excellent post several years ago, we learn that the period of the decimal expansion of a rational number $\frac{p}{q}$ must divide the multiplicative order of $10\pmod q$ assuming that there are ...
2
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2answers
64 views

What level of infinity is referred to when talking about recurring digits?

If a digit is written as $3.\dot{3}$, what level of infinity do the dots continue on for? Can this be proven to be true, or is it just a quirk of the notation? More specifically, $1/3$ can obviously ...
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2answers
39 views

What is known about representations of numbers of the form $\frac{1}{n}$, where $n$ is odd?

I calculated some decimal representations of numbers of the form $\dfrac{1}{n}$, where $n$ is odd, and most of them had a period that began immediately after the decimal point. One example of a number ...
43
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3answers
3k views

Extending prime numbers digit by digit while retaining primality

I looked at a table of primes and observed the following: If we choose $7$ can we concatenate one digit to the left so as to form a new prime number? Yes, concatenate $1$ to obtain $17$. Can we do ...
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1answer
50 views

Asymptotic formula for “Self Numbers” $s_n$

While digging through some old notebooks today, I found a problem from a long time ago that I was never able to solve. It involves a sequence of positive integers called the “self numbers” defined ...
4
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1answer
333 views

Find the smallest positive integer such that $S(n)=10, S(n^2)=100$.

Consider the numbers: $1, 11, 111, 1111, 11111$ and so on. $S(1)^2=S(1^2)$ and, in fact, $(S(n))^2=S(n^2)$ for all the numbers where $S(n)$ is the sum of the digits of the number $n$. Edit 1: $S(n^2)...
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1answer
64 views

Does $\sqrt{2}$ contain every digit in every base?

Does $\sqrt{2}$ contain every digit in every base? This popped into my mind and I have no idea how to begin attacking it. Each digit has to occur only once. Of course, $\sqrt{2}$ is probably normal ...
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1answer
23 views

Finite Sequence in an Infinite Non-Cyclic Sequence

My friend claims, that the digits in the decimal representation of pi contains every finite sequence of digits. For example my phone number will occur eventually. He claims that this is because there ...
7
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1answer
169 views

Prove that $2^{30}$ has at least two repeated digits.

Prove that $2^{30}$ has at least two repeated digits. I assume that the question is asking me to prove that $2^{30}$ has at least one digit that appears twice. Correct me if I'm wrong. (I later ...
2
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4answers
108 views

Find the least whole number only consisting of the digit 1 such that it is divisible by 3333…3.(100 3's)

Find the least whole number only consisting of the digit 1 such that it is divisible by 3333...3.(100 3's). My approach: we see that 111 is divisible by 3. Hence 100 3's would divide 300 1's. Is my ...
2
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1answer
40 views

What symbol means continues irrationally?

Typically we use an ellipsis (...) to show that a number continues on; e.g.: 1/3 = 0.33333... Even in the English language, it is used to indicate that something ...
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3answers
85 views

The integral of $f(0.x_1 x_2 \cdots)= 0.\sigma(x_1)\sigma(x_2)\cdots$

Given $\sigma:\{0,1,2,\cdots,9\} \to\{0,1,2,\cdots,9\}$ being a bijection. Given $f:[0,1]\to[0,1]$ satisfying $$f(0.x_1 x_2 \cdots)= 0.\sigma(x_1)\sigma(x_2)\cdots$$ where $0.x_1 x_2 \cdots$ ...
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1answer
31 views

How can I prove that if $(n)_{10}$ ends in $k$ zeros, then $(n)_5$ ends in at least $k$ zeros?

If I have to be completely rigorous, I believe I would have to prove that if $n=2^{e_1}3^{e_2}5^{e_3}...$ is $n$'s prime factorization, then $(n)_{10}$ ends in exactly $k = \min\{e_1,e_3\}$ zeros. ...
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1answer
25 views

Infinite decimal expansions

In the infinite (non recurring) decimal expansion of a number, such as pi, can you be sure that any given digit appears an infinite amount of times in the expansion?
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1answer
53 views

How do we know that a non recurring number will not repeat after many digits?

We are told that there are rational numbers that either terminate or repeat and irrationals that neither terminate nor repeat. But how are we so sure that a non terminating non recurring number will ...
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0answers
34 views

How do I find out which check digits algorithm was used to generate these check digits

199917310179 199957410250 199935910137 200025902253 199960710304 199836610072 199904610305 199911310180 199957710108 199957510123 The above are numbers which the ending number is a check digit for ...
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19 views

Is there a formula for the reverse sequence of a repetend?

@DavidK Did you ever expand on the reverse sequence/ backward sequence from this thread? Doubling sequences of the cyclic decimal parts of the fraction numbers "I'll give some thought to what can be ...
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1answer
34 views

Find the binary of decimal numbers given with powers of 10 [closed]

Convert to binary : 46.5 * 10^(-24) Or something like 46.5 * 10^(24) I have to find the binary equivalents here, for the purpose of representation in IEEE 754 floating point representation. But I ...
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1answer
47 views

Power of 2 with equal number of decimal digits?

Does there exist an integer $n$ such that the decimal representation of $2^n$ have an equal number of decimal digits $\{0,\dots,9\}$, each appearing 10% of the time? The closest I could find was $n=1,...
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0answers
67 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 $p=7$ ...
12
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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
100 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 $\pi$ ...
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0answers
12 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 $8$, ...
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1answer
337 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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2answers
31 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....
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0answers
13 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) ...
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1answer
98 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 $p$ ...
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0answers
55 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}\ $ : ...
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3answers
127 views

Hopeless Numbers [closed]

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 ...
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1answer
45 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 ...
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2answers
55 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 ...
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1answer
45 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
55 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?
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1answer
95 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 ...
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1answer
117 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
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1answer
63 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 ...
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1answer
45 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 ...
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2answers
164 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 ...
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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 ...
19
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2answers
484 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
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3answers
225 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 ...
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2answers
58 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 ...
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1answer
24 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}_{...