Questions tagged [fractions]
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10 questions
1 vote
1 answer
95 views
Sum of terms after partial fraction decomposition
I am facing the following problem (all $a_i$ being positive and all different) $$F=\int_0^1 \,dx\prod_{i=1}^n \frac 1{x+a_i}=\int_0^1\sum_{i=1}^n \frac {A_i}{x+a_i}\,dx=\sum_{i=1}^n A_i \log \left(1+\...
- 1,465
13 votes
1 answer
472 views
Egyptian fraction of a number in the interval (0.5,1)
An Egyptian fraction is a finite sum of distinct unit fractions, such as $$\frac{43}{48} = \frac{1}{2}+\frac{1}{3}+\frac{1}{16}.$$ Does there exist a number in the range $(0.5, 1)$ that when written ...
- 273
0 votes
0 answers
113 views
Permutation using irreducible fractions
Let $$f(n,k)=n\operatorname{mod} k, g(n,k)=\left\lfloor\frac{n}{k}\right\rfloor$$ Let $T(n,k)$ be A072030, i.e., array read by antidiagonals: $T(n,k)$ = number of steps in simple Euclidean algorithm ...
user231922
13 votes
1 answer
4k views
Is there another controversial statement by Grothendieck apart from 57 being prime?
There is a well-known story about Grothendieck being asked to explain concretely some result involving prime numbers and of his answering "You mean an actual number? All right, take 57". ...
- 48.3k
4 votes
0 answers
949 views
What fraction of fractions does Cantor's famous sequence enumerate?
Cantor's famous sequence $\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$ provides a ...
user112109
10 votes
2 answers
1k views
Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
This question is an old question from mathstackexchange. Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $ And let $ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $ It appears that we have $$\sup ...
- 799
-1 votes
1 answer
339 views
A simple matrix multiplication query [closed]
The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
- 1
0 votes
1 answer
255 views
About the sum of the first half and the latter half of the cyclic numbers of a repeating decimal
Let us call the sum of the first half and the latter half of the cyclic numbers of an irreducible fraction 'a division sum' when the period of a repeating decimal is even. Also, let $\lambda(l)$ be ...
- 4,807
-4 votes
2 answers
2k views
How do you calculate/prove the length of n, the number of non-repeating digits preceeding a periodic sequence of a fractional repeating decimal [closed]
Is there a way to calculate the number of non-repeating digits that precede the periodic repeating portion of a decimal expansion? For example: 1/6 = 0.1666.... (there is 1 non repeating digit) **(...
- 3
5 votes
4 answers
955 views
Reconstructing a fraction from its first digits
It is not difficult to see that any reduced fraction $\frac{p}{q}$ where $0 < p < q $ and both $p$ and $q$ have at most $N$ digits (where $N$ is a fixed integer) can be reconstructed from its ...
- 3,695