Questions tagged [fractions]

Questions on fractions, i.e. expressions (not values) of the form $\frac ab$, including arithmetic with fractions. Not to be confused with the tag (rational-numbers): fractions denote rational numbers, but the same rational number may be written in different ways as a fraction.

9
votes
0answers
89 views

Largest Numerator of Sum of Egyptian Fractions

What is the largest possible numerator when put in reduced form over all sums of the form $$\sum_{k=1}^n\frac{c(k)}{k}$$ where $c(k)\in\{-1, 0, 1\}$? An easy bound is to consider what happens when we ...
5
votes
0answers
190 views

Multi-dimensional Integration

Let $C=\sum_{s=1}^tx_sx_s'+aD^{-1}$, which is symmetric positive definite, vector $x,w\in\mathbb{R}^n$ and a scalar $y\in\mathbb{R}$. The integral I am trying to solve is as follows: $$\sqrt{\frac{a\...
5
votes
0answers
327 views

Origin/history of mixed number notation with misleading hyphen, e.g. 1-1/2

So there is a system of writing mixed numbers (that is, a combination of whole number and fraction, used instead of an “improper” fraction) used in cases where typing vulgar fractions (e.g. ½) ...
5
votes
0answers
112 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
4
votes
0answers
99 views

Geometric explanation as to why $\sum_{n=1}^\infty \frac{1}{x^n}$ converges to $\frac{1}{x-1}$?

Is there any simple, preferrably geometric, explanation as to why the sum $$\sum_{n=1}^\infty \frac{1}{x^n}$$ converges to $\frac{1}{x-1}$ ?
4
votes
0answers
509 views

Sum of First k binomial coefficients divided by 2^n

I've been solving a complex problem, in which at one step, I have to calculate the following $$ \frac {{n \choose 0} + {n\choose 1} + {n\choose 2} + ... + {n\choose k}}{2^n} $$ The values of n and k ...
4
votes
0answers
211 views

Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

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 $$\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $$ Why is that so ? ...
4
votes
0answers
88 views

Representation with distinct egyptian fractions with “small” denominators

Suppose, I have a rational number $r$ with $0<r<1$, for example $$r=\frac{53143}{274851}$$ The goal is to write $r$ as a sum of DISTINCT egyptian fractions (fractions with numerator $1$). The ...
3
votes
0answers
636 views

Why is $\inf g \sup g = \frac{9}{16} $?

Consider this question here : Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $? Call that conjecture about $\frac{5}{4} $ conjecture $1$. Let $g(n) = \prod_{i=0}^n (\sin^2(n) + \frac{9}{16}) ) $ ...
3
votes
0answers
43 views

Dirac delta from poles of a function

Suppose we are given the simple expression $$ F(k) = \frac{1}{E^2-E(k)^2} $$ which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
3
votes
0answers
75 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies |S(...
3
votes
0answers
950 views

Four candle problem: Using candles as timers

The candles each take one hour to burn completely. Cutting off bits of the candles is forbidden, but the candles are placed on a raft of fork handles so they may be burnt at both ends (e.g. to time $1/...
2
votes
0answers
65 views

Can I put a fraction of the form, $\frac{A}{(Bx^2+C)^2}$ into partial fractions?

For example, $\dfrac{1}{(x^2+1)^2}$ This may be a dumb question as I'm pretty sure you can't, but then why not?
2
votes
0answers
60 views

Find the inverse Laplace transform of a function

Find the inverse Laplace transform of: $$\frac{\frac{1}{1+s^2}}{1+s+\frac{s}{1+s+\frac{s}{1+a+s}}}\tag1$$ Where $a\in\mathbb{R}^+$ I've no idea how to find this.
2
votes
0answers
99 views

General term and closed form of a fraction of fraction of a fraction of a …

I've been trying to solve this series of fractions and find the closed form of this sequence. The problem is that new ramifications of fractions are growing exponentially in each term and I have to ...
2
votes
0answers
122 views

How to select best k fractions out of n fractions (k<=n) so as to have (numerator sum / denominator sum) maximum?

For ex. Given are 4 fractions. 4/2, 2/3, 1/2, 10/20 I have to select 3 fractions out of these 4 so that the value of (num sum)/(den sum) is maximum. As in this case, selecting 4/2, 2/...
2
votes
0answers
171 views

Expressing a Fraction in Simplest form

I have given a fraction in the form of $p/q$ , i want to express the given fraction as sum of series which is in the form of $1/x$ and $x$ is odd. i.e $$ p/q = 1/x + 1/y+ 1/z+1/l/s $$ such that ...
2
votes
0answers
46 views

How can we write the following expression for the general case $\frac{1}{m_{1}^{k_1} m_{2}^{k_2} … m_{r}^{k_r}}$?

By using the concept of $$\frac{1}{pq}=\frac{1}{q(p+q)}+\frac{1}{p(p+q)}$$ we can write $$\frac{1}{p^nq^m}=\sum_{i+j=m+n, i,j>o} \left[ \binom{j-1}{n-1}\frac{1}{q^i(p+q)^j}+ \binom{j-1}{m-1}\frac{...
2
votes
0answers
32 views

What types of fractional tetrations are possible to calculate without using extensions of tetration?

For example, any number tetrated to a positive integer can be calculated by just doing repeated exponentiation from top to bottom. Similarly, it is possible to calculate values of $^{0.5}2$, $^{3}3$, $...
2
votes
0answers
58 views

Prove or disprove this inequality involving fractions

Let $n$ be a positive integer. Assuming $x_1, x_2, x_3,...,x_{2n}$ are all positive real numbers, we need to prove or disprove: $$\frac{x_1}{x_2}+\frac{x_2}{x_3}+...+\frac{x_{2n}}{x_1} \geq n+\frac{...
2
votes
0answers
62 views

Which fractions have no representation with egyptian fractions?

I would like to have a criterion for an extension of the egyptian-fraction representation (Egyptian fractions have numerator $1$). I allow negative fractions, but the occuring denominators have to be ...
2
votes
0answers
53 views

Can every rational number in $(0,1)$ be expressed as a sum or a difference of special fractions?

In a paper, I found the claim that every real number can be expressed as a sum of two so-called $F(4)$-numbers. These are numbers , for which the entries of the simple continued fraction (except the ...
2
votes
0answers
116 views

Pull constant out of a summation of fractions

General problem $$ \sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0 $$ Is it possible for solve for $x$? Some context I've hit a road block in my derivation... At this point, I need to pull the model ...
2
votes
0answers
72 views

How to do this ratios problem without algebra?

Dinesh had some fiction and nonfiction books. The number of fiction was $4/9$ of the total number of books. After he had donated $80$ fiction books and $25$ nonfiction books, there were $20\%$ as many ...
2
votes
0answers
123 views

Trigonometric functions of rational fractions of pi

Consider rational numbers $\frac{m}{n}$ and $\frac{m'}{n'}$, where $0<\frac{m}{n}, \frac{m'}{n'} <1$. Then $$\sin^2 (\tfrac{m}{n} \pi) = 2 \sin^2 (\tfrac{m'}{n'} \pi)$$ When $\frac{m}{n} = \...
1
vote
0answers
56 views

Unit fractions pairing

(I have asked this question on stackoverflow and received a suggestion to try posting here so hey guys!) I have been given a problem where fractions between 1/2 - 1/1000 have to be added to create ...
1
vote
0answers
42 views

If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if ...
1
vote
0answers
25 views

Multiplying an inequality with an argument

I have a question about the following inequality: $a - \frac{2}{a} + 1 > 0$ This is obviously a polynomial of the second degree. Is this the simplest way of solving the given equation? $\frac{a}...
1
vote
0answers
25 views

Number of muffins baked

Mr Ali baked some muffins. (3/5) of the muffins wee chocolate and the rest were vanilla. After she sold (2/9) of the chocolate muffins and (1/2) of the vanilla muffins, 230 muffins were left. How many ...
1
vote
0answers
60 views

Expected value of Rayleigh quotient

I want to compute the expectation of a generalized Rayleigh quotient, i.e., $$\mathbb{E}_{\mathbf{x}} \bigg[ \frac{\mathbf{x}^{\mathrm{H}} \mathbf{A} \mathbf{x}}{\mathbf{x}^{\mathrm{H}} \mathbf{B} \...
1
vote
0answers
30 views

Will this equation ever have an positive integer value?

Imagine we have some set of positive integers $\{k_1,k_2,...k_n\}$. We are given this formula $$ x = \dfrac{\sum_{i=1}^{n}3^{n-i}2^{\sum_{j=1}^{i-1}k_j}}{2^{\sum_{i=1}^nk_i}-3^n} $$ and need to ...
1
vote
0answers
32 views

Economics Math: Optimal allocation of free time from Core Econ website. MRT = MRS

CONTEXT: I have the following equation for MRT=MRS (marginal rate of transformation = marginal rate of substitution): $\frac{αy}{24-t} = \frac{ay}{bt}$ I am asked to "multiply through" by $\frac{t}{y}...
1
vote
0answers
43 views

A quick way for decomposing fractions

The complete method for decomposing fractions is obvious . For example $y = \frac{2x+1}{(x-1)(x+3)} = \frac{a}{x-1} + \frac{b}{x+3} = \frac{a(x+3) + b(x-1)}{(x-1)(x+3)} \Rightarrow $$ \left\{ \begin{...
1
vote
0answers
43 views

Name of maths technique?

I'm currently looking at runs of identical outputs, and looking at the fractions I get back. When I was running through it, I had a feeling I'd come across something similar before, and it would be ...
1
vote
0answers
24 views

the same convergents among 3 irrotations

Let a, b, c be irrational numbers with a < b < c. If a and c have identical convergents $\frac{p_0}{q_0}$, $\frac{p_1}{q_1}$, . . . , up to $\frac{p_n}{q_n}$, how to prove that b also has these ...
1
vote
0answers
81 views

What type of relationship intercede between fractions and division?

Here's what I thought: Let's start with a fraction, it is just a number which represent a value by telling us how many pieces of a certain size we have. Now, what about another way to represent this ...
1
vote
0answers
44 views

How does one prove the following inequality is true?

I can prove that if $\frac{a_{i}}{b_{i}} \geq \frac{a_{j}}{b_{j}} $, it follows that $\frac{a_{i}}{b_{i}} \geq \frac{a_{i} + a_{j}}{b_{i} + b_{j}} $. However, I'd like to generalise this result by ...
1
vote
0answers
68 views

Simplified Fraction.

My son has been set homework with half-completed fractions along a line of $30$. He has to complete the missing half. For example, there is an indicator at the $10$th line and the fraction is half ...
1
vote
0answers
51 views

What is the reduced ratio for $\frac{\left\{\begin{array}{c}n\\k\end{array}\right\}}{\binom n k}$?

Looking at the table of $\frac{\left\{\begin{array}{c}k+m\\k\end{array}\right\}}{\binom{k+m}k}$, $1\leqslant k,m\leqslant10$, where $\left\{\begin{array}{c}k+m\\k\end{array}\right\}$ is the Stirling ...
1
vote
0answers
33 views

Inequality involving irreducible fractions and constants

Let $\frac m n$ be an irreducible fraction such that $1 \gt \sqrt 2 + \sqrt 3 - \frac m n \gt 0$. Show that there is a constant $c \gt 0 \space$ such that $$\sqrt 2 + \sqrt 3 - \frac m n \gt \frac {1} ...
1
vote
0answers
70 views

Given a fraction a, what's the terminology for a fraction b, which when added to a equals 1?

As the title says... Given a fraction x, what's the terminology for a fraction y, which when added to x equals 1? I'm assuming there must be a name for this. Something similar 'reciprocal' in ...
1
vote
0answers
338 views

Splitting a fraction

Question: How do you split$$\dfrac {1}{\left\{1+\dfrac {x^2}{a^2}\right\}\left\{1+\dfrac {x^2}{(a+1)^2}\right\}\left\{1+\dfrac {x^2}{(a+1)^2}\right\}\cdots\left\{1+\dfrac {x^2}{(a+n-1)^2}\right\}}\tag{...
1
vote
0answers
291 views

What is the difference between a fraction, ratio, quotient, rational number?

They are mixed everywhere and I can't find a single article that treats them right and with the required rigor they should get. Can someone explain them?
1
vote
0answers
54 views

Is this fraction a natural number only in case ${n_1}^2+{n_2}^2+{n_3}^2={m_1}^2+{m_2}^2+{m_3}^2$?

Suppose that $m_1,m_2,m_3,n_1,n_2,n_3 \in \mathbb N$ and $m_1<m_2<m_3$ and $n_1<n_2<n_3$. If $\dfrac {{n_1}^2+{n_2}^2+{n_3}^2-3}{{m_1}^2+{m_2}^2+{m_3}^2-3}$ is a natural number, ...
1
vote
0answers
139 views

Simplify simple fractions after Hospital's rule

I'm having trouble simplifying the result of complicated limits which contain mutliple fractions. I understand this is basic math level, but i find it difficult to find exercices to practice that ...
1
vote
0answers
76 views

unique ringhomomorphism from the Field of fractions to another field

$R$ is a ring, $L$ a field and $K$ the fraction field constructed from $R$. For any injective ring homomorphism $f=R \rightarrow L$, there is a unique ring homomorphism $\tilde{f}:K \rightarrow L$ ...
1
vote
0answers
48 views

How much information is missing?

If we know the value of $\frac{(a-b)}{(c-d)}$, can we calculate the value of $\frac{(a-d)}{(c-b)}$ That is : Let $\frac{(a-b)}{(c-d)}=k$ , can we calculate $\frac{(a-d)}{(c-b)}$ in terms of $k$ And if ...
1
vote
0answers
523 views

quotient of two differentiable functions is differentiable

I have two functions $k(t)$ and $l(t)$ in a certain closed interval $[a,b]$ both functions are continuous and differentiable in the interval. In addition we have: Both functions are increasing with ...
1
vote
0answers
141 views

three fractions between π and 22/7

three fractions between π and 22/7 π=355/113 =3.14159 22/7=3.1428 using a/b 355/113<22/7 then a+b/c+d 1) 355+22/113+7 =377/120~ 3.14166 2) 377+22/12+7 =399/127~3.14173 3) 399+22/127+7 =421/134~3....
1
vote
0answers
642 views

How many elements are in the following set?

The set is $$\{ x \in Q:x^2 =64/25 \} $$ I thought the answer was $\{ \frac{8}{5}, -\frac{8}{5} \}$ but I am told there are in fact 4 distinct elements: $$\{ \frac{8}{5}, \frac{8}{-5}, \frac{-8}{5}, ...