Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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6
votes
1answer
64 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
7
votes
2answers
76 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
0
votes
0answers
34 views

Can This Expression Be Simplified? (Involves Square Roots)

I started with the expression $$ \frac{4mlt(1-\sqrt{1-\frac{v^2}{c^2}})c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ and have ended up at: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
-2
votes
2answers
57 views

Find all integer numbers $n$ such that $\frac{11n-5}{n+4}$ is a perfect square.

Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$ I really tried but I couldn't guys, help please.
1
vote
2answers
43 views

Fraction with numbers [on hold]

In an audience of people watching a film, 1/10 are under 16 years old, 3/5 are between 16 and 40 years old and the rest are over 40 years old. What fraction are over 40 years old? How to get the ...
-4
votes
3answers
31 views

Simplifying the fraction [on hold]

How do we solve it step by step? No shortcut please =) the result of evaluating $\dfrac{x(x+2)-3x}{x}$ is?
1
vote
1answer
35 views

Limit of a function - $x$ either rational or irrational - limit $1$ or $0$. [duplicate]

Show that: The continuous functions $f_{n,k}(x):=(\cos(k!\pi x))^{2n},0\leq x \leq 1$ satisfy the relation $\lim_{k\to \infty}(\lim_{n\to \infty}f_{n,k}(x))=\begin{cases} 1, & \textit{if ...
3
votes
1answer
37 views

How to find out the number of repeating digits of a rational number in decimal form?

Upon dividing two integers, I would like to programmatically predict the number of decimal places that repeat after the decimal point. For example in $\frac{1}{3}=0.\overline{3}$, I want to know that ...
11
votes
2answers
111 views
+50

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
1
vote
2answers
41 views

Another (in)dependence over the nonzero rationals question

About one hour ago I asked a question which at first sight looked non-trivial to me but it is really trivial. Shame on me, whether I want it or not. Now I have, solely for fun, another question which ...
0
votes
3answers
69 views

Problem with simplifying $\frac{(3+h)^2-9}{(3+h)-3}$ [on hold]

I need help simplifying $$ {(3+h)^2-9\over (3+h)-3}. $$ The answer is $6+h$. I keep getting $h$.
237
votes
13answers
29k views

Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was ...
0
votes
1answer
44 views

Jimmy got a 38.5% (± 0.05%) on his math test. How many questions did the test have at a minimum?

The answer is not "200 questions", though it would be if he got a score of exactly 38.5%. The fact that anything that rounds to the nearest decimal is allowed complicates things. I know the answer, ...
2
votes
2answers
72 views

How find this minimum of the $q$, if such $\frac{95}{36}>\frac{p}{q}>\frac{96}{37}$

let $p,q$ is postive integer,and such $$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$ Find the minimum of the $q$ maybe can use $$95q>36p$$ and $$37p>96q$$ and then find this minimum of ...
1
vote
2answers
149 views

Understanding the concepts of division and fractions

$\require{cancel}$ I'm having some issues regarding division so I will start by asking how this concept was developed throughout the ages: What was the first civilization to introduce the idea of ...
0
votes
2answers
47 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that ...
0
votes
3answers
44 views

How to calculate this expression and get an integer number?

Hello there I don't have idea how to calculate this: $$\left[\frac {116690151}{427863887} \times \left(3+\frac 23\right)\right]^{-2} - \left[\frac{427863887}{116690151} \times \left(1-\frac ...
-1
votes
3answers
34 views

Question in fraction [closed]

I find it difficult to solve this. Please help me out.... At a party, the number of men was $\frac{2}{5}$ the number of women. After $144$ women left, the number of women become $1\frac{1}{2}$ ...
4
votes
2answers
71 views

Is a continuous function $f : \mathbb{Q}\to\mathbb{Q}$ always bounded on a closed interval?

Can a function $f : \mathbb{Q} \to \mathbb{Q}$ that is continuous on an interval $[a,b]$ not be bounded on $[a,b]$? I'm asking this because in Spivak's Calculus, the "Boundedness Theorem", which ...
3
votes
0answers
42 views

How to estimate the number of decimal places required for a division?

Given two decimal numbers, is it possible to estimate the number of decimal places required to fit the result of their division? Provided that the division yields a finite number of decimals, of ...
-1
votes
4answers
51 views

Why do we switch the denominator and numerator when we divide fractions? [duplicate]

Why do we switch the denominator and numerator when we divide fractions? I've been trying to find out why and I've asked several people and checked many websites but none that give me a good answer. ...
-4
votes
1answer
95 views

How to approximate $0.714286$ as a fraction of $\pi$?

I'm doing an exercise that tells me that the answer must be a multiple of pi, like $12\pi$ or $\dfrac23\pi$. I need to approximate $0.714286$ as a fraction of $\pi$. How do I achieve this?
1
vote
1answer
25 views

for which values of $\theta$.does this equation: $x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1$ have rational solutions for all $n$?

I'd surprised if this equation:$$x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1 $$ have rational solutions for all $n$ and for all values of $\theta$. My question here is: for which values of $\theta$ ...
3
votes
1answer
70 views

Multiplying and adding fractions

Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b \times d},$$ And why $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ ...
18
votes
9answers
814 views

Distributive property on fractions

I'm in seventh grade and my teacher wasn't able to explain this to me. why is $\frac{1}{a+b}$ not equal to $\frac 1b +\frac 1a$? I'm sorry if this is obvious. EDIT: thank you to everyone who ...
2
votes
1answer
343 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
2
votes
4answers
153 views

How to get the given equality?

I have the following sum ($n\in \Bbb N)$: $$ \frac {1}{1 \times 4} + \frac {1}{4 \times 7} + \frac {1}{7 \times 10} +...+ \frac {1}{(3n - 2)(3n + 1)} \tag{1} $$ It can be proved that the sum is equal ...
1
vote
1answer
41 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
2
votes
1answer
62 views

Is it possible to express $\Gamma\!\left(\tfrac{1}{50}\right)$ through values of the $\Gamma$-function at rational points with smaller denominators?

Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. ...
1
vote
4answers
133 views

Wrong Answer - Rewrite Rational Number as a Fraction.

This number 2.962962 can be rational $$x=2.962962$$ $$10x=29.62962$$ $$100x=296.2962$$ $$1000x=2962.962$$ $$1000x-10x=\frac{990x}{990}=\frac{2933}{990}$$ why is this wrong? That way of getting the ...
2
votes
3answers
45 views

Why are there opposite rules for dividing positive numbers and negative numbers?

I'm in confusion from some time about division of negative numbers. When we divide a positive number with a positive number, for example $$5/3 = 1.66 $$ we see what is biggest multiple of 3 which is ...
0
votes
3answers
42 views

Math trigonometry transformation

Hi, I haven't done math in a while, and stumbled upon this thing. The angle ($\arccos 7/25) is given, and i have to calculate the cosine of it's half. I've used the basic formula for cosine of an ...
2
votes
0answers
48 views

Fixed Points of Function from Rationals to Reals

Consider a function $f$ from the positive rationals to the reals such that $f(x)f(y)\ge f(xy)$ and $f(x+y)\ge f(x)+f(y)$. Further assume this function has a fixed point greater than $1$. Prove this ...
7
votes
6answers
185 views

I was wondering, shouldn't the fraction $\frac {-2}{-1}$ be less than 1?

Because technically, the numerator is smaller than the denominator as $-2 < -1$ I know it's an extremely stupid question. I mean I know that I can just multiply $-1$ to the numerator and the ...
0
votes
1answer
26 views

Linear (in)dependence of roots over the nonzero rational numbers

I was reading some question on this site and stream of thought led me to the creation of another question that could be trivial for someone but I am unable even to start solving it. I wanna share this ...
-2
votes
3answers
36 views

Recurring duodecimals fractions [closed]

I get the idea about duodecimals from what I read till I reach the fractions point where: $\frac{1}{8}=0.16$ instead of $0.15$ $\frac{1}{9}=0.14$ instead of $0.13333333$ ...
8
votes
4answers
354 views

Find the sum of reciprocals of divisors given the sum of divisors

Let $d_1, d_2, \cdots d_k$ be all the factors of a positive integer '$n$' including $1$ and $n$. Suppose $d_1 + d_2 + d_3+\cdots+d_k = 72$. Then find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\cdots + ...
1
vote
1answer
41 views

Online math question didn't do a square of a square root properly?

I'm currently taking an online college math course, and I recently came across something that I can't make any sense out of. $(\frac{5x \cdot \sqrt{3}}{6})^2 = \frac{25x^2}{12}$ It looks like ...
0
votes
0answers
32 views

Differential equation - fractions, circular answer?

Hi this might seem like a really stupid question but then hopefully someone can asnswer it quite easily :) I have function $P{_t}$$=(E{_t}$ $(P{_t}{_+}{_1}+$ $δ{_t}{_+}{_1}$$ )-γΩx$${^*})/$$(1+rf+ψ_t ...
-1
votes
5answers
104 views

How to find the limit $\lim_{x\to0}\frac{x\tan x-x\sin x}{x\sin^2x/\cos x}$ [closed]

Here is the limit I'm struggling with: $$\lim_{x\to0}\cfrac{x\tan x-x\sin x}{x\sin^2x/\cos x}.$$ Worked so hard to find it, but couldn't.
3
votes
2answers
44 views

Simplifying $\frac{1/(\frac{1}{z_1}(1-t)+\frac{1}{z_2}t) - z_1}{(z_2 - z_1)}$

This drives me mad! I am not very good in math but thought I could at least do basic things like this one, but can't figure it out and I spent a day on it. I am trying to simplify: ...
61
votes
14answers
11k views

Express 99 2/3% as a fraction? No calculator

My 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is ...
3
votes
1answer
28 views

How Many Rational Slopes?

Given an $N$ by $M$ grid with integer coordinates (e.g. like pixels in an image), how many slopes are defined by the set of lines passing through the each grid point pair? Note that because the ...
1
vote
2answers
235 views

How is N^2/3 equivalent to 1/(N^1/3)?

I've tried to look for similar things on StackExchange and elsewhere on the net, but can't seem to find anything, so thought I'd just ask for some help on here... Someone has kindly helped me with a ...
3
votes
2answers
46 views

Representing rational numbers on a number line

Though the cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers, when one looks at a number line this fact seems counterintuitive, since between ...
0
votes
1answer
56 views

why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?

When I was comparing proof for $\sqrt{2}$ and sum of two rational numbers, I found that the proof of two rational number did not mention anything about common factor in the ratio. one proof I found ...
1
vote
3answers
68 views

How do I simplify $\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$?

How do I simplify the following equation? $$\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$$ I have no idea where to start. If I multiply either fraction by its denominator I will still end ...
21
votes
3answers
851 views

“Least trivial” function preserving rationality

Is there a "non-trivial" function $f(x,y)$ such that $$f(x,y) \in \mathbb{Q} \iff x,y\in \mathbb{Q}?$$ An example of a "trivial" function would be $$f(x,y) = \begin{cases} 0 & x,y\in ...
2
votes
1answer
39 views

How to simplify a diabolical expression involving radicals

A friend and I have been working on this problem for hours - how can the following expression be simplified analytically? It equals $\frac{1}{2},$ and we have tried the following to no avail: ...
39
votes
0answers
2k views

If the decimal expansion of $a/b$ contains “$7143$” then $b>1250$

I recently stumbled upon this really interesting problem: If we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical ...