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0
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2answers
49 views
24
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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
22 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
74 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?
51
votes
10answers
5k 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
0answers
37 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
60 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 $4900$, $490000$ ...
-1
votes
1answer
46 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 ...
-6
votes
4answers
105 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
22 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: ...
4
votes
1answer
27 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
76 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
49 views

Calculating decimal range for two's complement

Given this question : ...
0
votes
1answer
58 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
51 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 ...
0
votes
0answers
15 views

Lemma. $\exists N\in\mathbb{N}\forall n>N: x_n=0\Leftrightarrow x=\frac{q}{p^N}$ for some $q\in\mathbb{N}.$

https://proofwiki.org/wiki/User:J_D_Bowen/Math710_HW1 in the lemma of exercise 5 in direction $\Leftarrow$: Why $S_{N-1}=x$? I dont see! Thanks!
-1
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4answers
40 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
338 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
516 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
51 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
3answers
650 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
50 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
263 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
179 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
247 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
84 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
127 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
49 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
14 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
118 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
46 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 ...
8
votes
2answers
189 views

Regularities when $n$ and $2n$ contain the same digits

Suppose we would like to find positive integers $n$ such that the base-10 representations of $n$ and $2n$ contain precisely the same digits. $142857$ is a well-known example, and computer search ...
21
votes
1answer
586 views

Is $0.1010010001000010000010000001 \ldots$ transcendental?

Does anyone know if this number is algebraic or transcendental, and why? $$\sum\limits_{n = 1}^\infty {10}^{ - n(n + 1)/2} = 0.1010010001000010000010000001 \ldots $$
0
votes
2answers
103 views

Mixed repeating decimals

How can be proven that a fraction having at the denominator a multiple of both 2 and 3 is transformed to a mixed repeating decimal number? I thought to bring the denominator to the form of ...
0
votes
0answers
140 views

How prove $|S-10^k\cdot AB|\le 9k$

Let $0 \leq a_k, b_k \leq 9$ and let $A,B$ be numbers with decimal expansions $$A=0.a_{1}a_{2}\cdots a_{k}>0 \hspace{0.5in} B=0.b_{1}b_{2}\cdots b_{k}>0.$$ Let $S$ be the number of possible ...
2
votes
3answers
263 views

Why irrational implies having an infinite decimal expansion?

Why irrational means having an infinite decimal expansion?
0
votes
2answers
23 views

Equal number of 1s and 0s in number of n digits

How many ways could one create a binary number of n digits where the number of 1s and 0s are equal? For example, if n was 8 then we could have: 10101010 or 11110000 In addition to this, I may ...
1
vote
1answer
48 views

Conversion from decimal to unknown number system

If we have the number $(387)_{10} \rightarrow (762)_n$ , how do we calculate the $n$? Thanks in advance.
1
vote
1answer
65 views

Digits difference of numbers between 10 and 99

I was asked the question, how many numbers between 10 and 99 have digits that differ EXACTLY by 3? I didn't understand the question, I thought it meant numbers like 11,44,77 then 22,55,88 and ...
3
votes
2answers
137 views

How do you calculate how many decimal places there are before the repeating digits, given a fraction that expands to a repeating decimal?

If you have a fraction such as $$\frac{7}{26}=0.269230\overline{769230}$$ where there are a number of digits prior to the repeating section, how can you tell how many digits there will be given just ...
0
votes
1answer
56 views

If 0.99…=1 What about 0.89…=0.9?

I notice the general pattern is that ?.??999... equals to 0.??1 more than the repeating 9 part. Is it true?
2
votes
1answer
46 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...
0
votes
1answer
147 views

What is $\tau$ in base $12$?

I'm a big fan of both $\tau$ and the duodecimal system. And while I can find information for $\pi$ on both, I can't seem to find the number of $\tau$ in base $12$. $\tau$ is given as $\tau = 2\pi$. ...
8
votes
2answers
253 views

Computing the last non-zero digit of ${1027 \choose 41}$?

I am working on the following problem: Let $x_n$ be a sequence of positive odd numbers. If $N$ is the number of ordered pairs $(x_1, x_2, x_3, \dots, x_{42})$ such that $$x_1 + x_2 + x_3 + \dots + ...
0
votes
0answers
38 views

Metric for precision of a decimal number

I am working on cleansing a cities database and trying to increase the precision of the latitudes and longitudes of these cities. I am comparing what I have against an outside datasource and want to ...
2
votes
0answers
89 views

Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
1
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0answers
83 views

Rudin: Supremum of Finite Decimals

I am reading Rudin. Note the following construction: Let $x>0$ be real. Let $n_0$ be the largest integer s.t. $n_0 \leq x$. Then, having chosen $n_0, \ldots, n_{k-1}$, let $n_k$ be the largest ...
1
vote
1answer
82 views

For which $n \in \mathbb{Z^+}$ is $(4n+9)/(2n^2+7n+6)$ a terminating decimal?

I saw this problem here. My approach: Let $A = (4n+9)/(2n^2+7n+6) = \frac{6}{2n + 3} - \frac{1}{n + 2}$ If $\frac{6}{2n + 3}$ and $\frac{1}{n + 2}$ are terminating decimals, then so is $A$. A ...
1
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0answers
112 views

Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...