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3
votes
1answer
110 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.
2
votes
4answers
470 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
35 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 ...
13
votes
3answers
552 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
42 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
151 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 ...
0
votes
0answers
68 views

$1.99999999…$ periodic number is really $2$? [duplicate]

Someone told me yesterday that is mathematically demonstrated that $1.99999999999999...$ and so is exactly $2$, as every periodic number is equal to his nearest integer. He wrote: x = 1,99999999... ...
1
vote
0answers
24 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
172 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
231 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
73 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 ...
4
votes
0answers
74 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 ...
2
votes
2answers
41 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
13 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
24 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
43 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 ...
6
votes
2answers
152 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 ...
22
votes
1answer
512 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
50 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
137 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
225 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
22 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
53 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
109 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
46 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
34 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 ...
-1
votes
1answer
131 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
237 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
36 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
82 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
vote
0answers
58 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
81 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
vote
0answers
108 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 ...
2
votes
1answer
56 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
3
votes
1answer
182 views

$\displaystyle \frac{6}{2n-1} - \frac{1}{n} = \frac{p}{2^i5^j}$

$\displaystyle \frac{6}{2n-1} - \frac{1}{n} = \frac{p}{2^i5^j}$ For which $n$ is this expression true. $n$ and $p$ are integers. $i$ and $j$ are positive integers or zero.
4
votes
3answers
49 views

Graph of the last digit of $x^n$ - why is it symmetric when $n$ is even, and not when $n$ is odd?

I have discovered this fact: "The graph of the last digit of $x^n$ (where $x$ is positive) is asymmetrical if $n$ is odd, and symmetrical if $n$ is even." What is the logic behind this? For ...
0
votes
1answer
41 views

Finding number of primes of the form $1010\ldots 101$ (in base 10) [closed]

How many primes among the positive integers, written as usual in the base $10$, are such that their digits are alternating $1$’s and $0$’s, beginning and ending with $1$?
-1
votes
2answers
37 views

11th form to 7th form

I can't solve this one: Consider an integer expressed in the 11-based form. In the 11-based form, digits 0 to 9 correspond to their decimal values, and A corresponds to 10. For example (15)_11 = (1 * ...
1
vote
1answer
110 views

What are the last 20 digits of mega?

What are the last 20 digts of the number mega, which is "2 in a pentagon" in steinhaus-moser-notation ? In contrary to power towers or tetration, the ending digits are not stable. I found out that ...
7
votes
0answers
77 views

Which digit occurs most often?

Is there any method to calculate, which digit occurs most often in the number $$4 \uparrow \uparrow \uparrow \uparrow 4\ ,$$ the fourth Ackermann-number ? Or would it be necessary to calculate the ...
0
votes
1answer
89 views

do you know another Magic Square with this property?

with the repeating digits of $\frac{1}{19} = 0.052631578947368421$ we can construct an exceptional magic square : The number 19 is a cyclic number with a period of 18 before the digits start to ...
4
votes
3answers
468 views

Dividing Decimals.. But remainders?

So, I understand how to do long division with decimals. So let's consider this problem: $10.5$ divided by $5.5$ (I chose this problem because it will OBVIOUSLY have a remainder) So we will look at ...
1
vote
1answer
172 views

How to calculate decimals of the fractional number 1/49?

I find this tricky one. How to calculate the first 50 digits/decimals of the fractional number 1/49? Two of my calculators and MatLab gives different answers so I'm curious, how this is calculated ...
4
votes
1answer
79 views

First digits of extremely large binomial coefficients

Can the first digits of a binomial coeffecient $$\binom{n}{k}$$ be calculated, if n and k are very large numbers ? For example Calculate the first ten digits of $$\binom{10^{85}}{10^{23}}$$ Any ...
2
votes
0answers
70 views

First digits of extremely large numbers (Generalization of “first digits of Graham's number”)

I found a question about the first digits of Graham's number and would like to generalize it : We want the first n digits of the number $a\uparrow^b c$. Which method is the most effective to do ...
8
votes
0answers
92 views

Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$ Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation. Is there a largest $n$ ...
5
votes
3answers
214 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
2
votes
2answers
84 views

Which real numbers have two representations?

Are they only numbers that end with 9999... and 0000... after the dot or some other too? If so, can you give an example?
0
votes
2answers
69 views

Decimal expansion of a Cauchy sequence

In one of the construction of $\mathbb{R}$ we make each real number an equivalence class of Cauchy sequences in $\mathbb{Q}$. More precisely, two Cauchy sequences $a_n$ and $b_n$ are equivalent iff ...