For questions about decimal expansion, both practical and theoretical.

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2
votes
2answers
22 views

How many numbers are in a numbering system with the basis 15 and 4 digits, where the digit sum equals 15

First of all, my question is very similiar to this one: How many numbers between $100$ and $900$ have sum of their digits equal to $15$? but i didn't quite understand how to adapt it to my problem, so ...
35
votes
1answer
585 views

Numbers $n$ such that the digit sums of $n, n^2,\cdots,n^k$ coincide.

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. When I was playing with numbers, I noticed the followings : ...
0
votes
1answer
33 views

Multiplying numbers splitting the number into 4 digit numbers

Doing some programming exercise how to sum big numbers,I split the numbers into $n$ numbers of $4$-digit numbers $1240135981395813958$ I split into $1240$,$1359$,$8139$,$5813$,$958$ and summing with ...
14
votes
0answers
239 views

Numbers $n$ such that the digit sum of $n^2$ is a square

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. About a month ago, a friend of mine taught me the followings : $$S\left(9\color{red}{^2}\right)=S(81)=8+1=3\color{red}{^2}$$ ...
4
votes
2answers
80 views

Prove: for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$. [closed]

In the following $n,m$ are natural numbers. I need to prove that for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$. Any ideas? Thanks.
25
votes
1answer
424 views

Does my “Prime Factor Look-and-Say” sequence always end?

I'm trying to create a challenge for PP&CG where the object will be to find the longest sequence in a given time, but I'm worried that there may be an infinite sequence that will ruin things. The ...
1
vote
4answers
238 views

How many digits are there in 100!? [closed]

How many digits are there in 100 factorial? How does one calculate the number of digits?
0
votes
0answers
24 views

Having problems understanding this algebraic expression

So I have a solution where $B = 0.1959552$ and $A = 0.00048515$. The problem asks, $A$ is $10$ times more likely than $B$. The teacher wrote (i.e. $A = 10 \cdot B$). Is this the right notation or ...
0
votes
1answer
44 views

$Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all convergent numerators and denominators. True?

Where $x = A001203$, $Pi = A002486$, $A002485$ $Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all $Pi > n+1 $ Hypothesis: This relation evaluates true for all $A002486$ and $A002485$. Lemma: All ...
1
vote
1answer
61 views

Given $x,y\in\mathbb R$ is there a “formulaic” way to obtain a $q\in\mathbb Q$ with $a<q<b?$

Is there an assignment of reals $x,y$ to a rational number $q(x,y)$ for which $$\forall_{\mathbb R} x.\forall_{\mathbb R}(x<y).\left(x<q(x,y)<y\right)\hspace{.2cm}?$$ For computable reals, ...
0
votes
1answer
42 views

How to round percents

Suppose there is election such that $n$ votes are given to $m$ candidates. I would like to express the results of elections in two decimal places, like ...
10
votes
1answer
179 views

Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

For example, when I divided $1$ by $98$, I got an amazing result of $0.0102040816326530612244897...$ where so many numbers get multiplied by two every time in the right pattern with some carrying. ...
5
votes
5answers
2k views

How many even 3 digit numbers contain at least one 7.

How many even 3 digit numbers contain at least one 7. I got 126, but it was not an answer choice for the problem. Can anyone help?
2
votes
2answers
248 views

Real numbers as decimals

I'm looking for a book that develops the theory of real numbers in a rigorous way in terms of their decimal expansions. The exposition should be concrete and preferably aimed at mathematically ...
0
votes
2answers
108 views

Prepend a 9 or append a 0?

Given a positive integer $x$, will $x$ always be larger if one prepends a 9 in comparison to appending a 0? For x = 1, prepending is largest because $91 > 10$ For x = 9, prepending is largest ...
2
votes
1answer
47 views

Irrationality of Decimal Expansion of Primes

I've heard the proof that this number is irrational is accessible to even a novice to number theory: $\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$ The proof may utilize that a number is ...
0
votes
1answer
25 views

Decimal expansion__Represent numbers as $x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$?

If $b>1$ is an integer, is well know that the numbers $x\in (0,1]$, can be written as $$x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$$ for some integers $a_k \in \{0,1,\ldots ,b-1\} $. My problem is ...
4
votes
3answers
163 views

Subtraction of two repeating decimals

When I was looking at the proof that every repeating decimal is rational, I came across this example: $x=5.33333333\ldots$ ($3$ repeat indefinitely) $10x=53.3333333\ldots$ ($3$ repeat indefinitely) ...
0
votes
3answers
119 views

How do irrational numbers lie on the number line?

If we construct a square with side length 1, take its diagonal length : $\sqrt{2}$ However I still don't understand HOW it can lie on the number line. Imagine another irrational number $\pi = ...
0
votes
2answers
67 views
29
votes
9answers
3k views

Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?

Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
1
vote
1answer
32 views

Does decimal fraction has hex value?/can hex be fraction?

I was wondering if a decimal fraction could be converted into a hexadecimal fraction? I have seen it many times ? but I have been also told that decimal or binary fraction has no meaning in hex. ...
1
vote
3answers
80 views

How would I find the decimal expansion of $1/99^2$

I want to find the repeating decimal expansion of $1/99^2$. All I know is that $1/99 = 0.010101\cdots$. How would I continue?
53
votes
10answers
6k views

Can a number have infinitely many digits before the decimal point?

I asked my teacher if a number can have infinitely many digits before the decimal point. He said that this isn't possible, even though there are numbers with infinitely many digits after the decimal ...
1
vote
1answer
76 views

The digit 3 and 2 digit number question

The digit 3 is written at the right of a certain 2-digit number forming a 3-digit number. The new number is 372 more than the original 2-digit number. What is the sum of the digits of the original ...
4
votes
1answer
82 views

Perfect squares formed by two perfect squares like $49$ and $169$.

Let $a$ be a perfect square number whose decimal representation is the concatenation of two perfect squares, for example $49$ (from $4$ and $9$), $169$ (from $16$ and $9$), ... and $4900$, $490000$ ...
0
votes
1answer
69 views

Usage of decimal expansion

I learned about the rigorous construction of rationals as a set of equivalence classes of ordered integers with operations defined on this set. I understand that the decimal expansion is another way ...
-5
votes
4answers
125 views

Computing $0.0625^{-2.25}$ without a calculator

It is quite easy to see that $0.0625^{-2.25} = 512$ by plugging this into a calculator. Of course, mathematics existed for millennia before the invention of the calculator; is there a way to compute ...
1
vote
1answer
28 views

How to interpret fractional number of bits of precision

In double-precision floating-point format there're effective $53$ bits of mantissa stored. This lets us estimate maximum number of decimal digits of precision available: ...
5
votes
1answer
39 views

Numbers $n$ such that there is some digit occuring in each power of $n$

If a positive integer $n$ is congruent to $0$, $1$, $5$ or $6$ modulo $10$, there is some digit occuring in each of the powers of $n$. If the decimal expansion of $n$ ends in a $0$, $1$, $5$ or $6$ ...
4
votes
2answers
99 views

Find the 1005th digit after the decimal point expansion of the square root of N.

Let $N$ be the positive integer with $2008$ decimal digits, all of them $1$. That is, $N=1111...1111$, with $2008$ occurrences of the digit $1$. Find the $1005th$ digit after the decimal point ...
0
votes
2answers
441 views

Calculating decimal range for two's complement

Given this question : ...
1
vote
1answer
83 views

How to be unambiguous about a number's base's base?

Say you want to note down a number to another person, and want it to be unambiguous (perhaps the other person is an alien and has more than 10 fingers or something). So if you say 12345, base 42 ...
0
votes
1answer
62 views

Is there a 10-digit emirp?

Does a 10-digit emirp exist? Unfortunately, the lists of emirps I could find on the Web are quite small and my programming skills aren't good enough to write a program to check all the primes up to ...
-1
votes
4answers
65 views

Find all whole numbers such that the number increased by the sum of its digits equals 73.

I'm really lost on how to figure this out. Work shown would help.
4
votes
2answers
406 views

An arithmetic sequence whose members do not contain the digit ‘9’

There is a non-constant arithmetic progression made of natural numbers only; none of them contains the digit $9$. Prove that such an arithmetic progression has no more than $72$ terms.
3
votes
4answers
594 views

Numbers whose digits sum to 7

Let $S$ be the sequence of all positive integers whose decimal digits add to exactly 7, in increasing order: $$S = \langle7, 16, 25, \ldots, 70, 106, 115, 124, \ldots 160, 205, \ldots, 10230010, ...
2
votes
2answers
59 views

binary and floating point representation

suppose that we have following binary digits $00011001.110 $,we can do following thing $00011001.110=1\cdot2^4+1\cdot2^3+1\cdot2^0+1\cdot2^{-1}+1\cdot2^{-2}=25.75$ then what does means? We then ...
14
votes
4answers
788 views

Are there numbers that describe themselves in some base but not according to the pattern 6210001000?

Call the first digit of a number digit 0. The digit after that digit 1, and so on and so forth. In base 10, the number 6210001000 describes itself, because digit 0 is 6 and it has 6 0s. Digit 1 is 2 ...
1
vote
2answers
61 views

Decimal expansion of a fraction [duplicate]

I was reading something which I found really special. It goes like this : Imagine we have a line with unity division (0,1,2,etc.) Now, we have a point on this line. The point can be on a point of ...
3
votes
2answers
734 views

How many digits are there in $2^{17}\times 3^2\times 5^{14}\times 7 ?$

How many digits are there in $$2^{17}\times 3^2\times 5^{14}\times 7 ?$$ Question added: I agree with the fellow who asked that if one cannot have 2 and 5 in the number above how we will calculate ...
1
vote
0answers
27 views

Non-repeating decimals in 1/n [duplicate]

Here's a conjecture I made: The number of non repeating decimal places in the base-ten representation of the fraction 1/n, where n is an integer, is equal to whichever is higher: the exponent of 2 in ...
5
votes
2answers
185 views

A question about decimal representation of irrational numbers.

Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ? Thanks !
2
votes
2answers
261 views

Compute and find 2009th decimal(2009th digit after the point), without automation, the following sum

Compute and find 2009th decimal of (2009th digit after the point), without automation, the following sum $$\frac{10}{11}+\frac{10^2}{1221}+\frac{10^3}{123321}+ \cdots ...
1
vote
1answer
89 views

Prove that in any base the number of digits composing the repetitive mantissa of the reciprocal of a prime $p$ never exceeds $p-1$.

I was trying to find bases where the reciprocals of primes have a short repetitive mantissa. Here is what I found: http://imagizer.imageshack.us/a/img835/7738/c7gb.png The bases are on the left. The ...
5
votes
1answer
174 views

patterns in the decimal expansions of adjacent square and cube roots

For fun I made a table in Excel which evaluated the square and cube roots of whole numbers in ascending order. Then of the result, I extracted the first, second and third decimal place digits, then ...
3
votes
2answers
79 views

Cannot find length of repeating block in decimal expansion for $\frac{17}{78}$

I am trying to find the length of of the repeating block of digits in the decimal expansion of $\frac{17}{78}$. On similar problems, that has not been an issue. Take for instance $\frac{17}{380}$. My ...
0
votes
1answer
17 views

Two orthonormal vectors in space with finite decimal representation

I'm trying to formulate an exercise related to linear algebra, and for that I need two vectors in $\mathbb{R^3}$ which have unit length, are orthogonal to one another, don't have any zero ...
0
votes
1answer
259 views

Binary expansion

I am trying to get my head around the left and right shift for binary expansion. The rules are: Shifting to the right ...
1
vote
1answer
53 views

Strange notation for a decimal expansion of a transcendental number

I am checking page proofs for one of my papers right now and an editor changed $\zeta(3)=1.202$$\ldots$ to: $\zeta(3) = 1.202,...,$ I find this latter notation very strange and think it ...