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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4
votes
2answers
42 views

Why the remainder is uniformly distributed when 1,2,3,… are divided by an irrational number?

Let remainder $r$ be defined as $$ r = n - pq $$ where $n \in \mathbb{N}$ is the dividend , $q \in \mathbb{R}$ is the divisor, and $p = \mathrm{floor}(n/q)$. I calculated the remainders by dividing ...
2
votes
1answer
57 views

Simplified rational distance problem

① Is there a point on a square with sides of rational length that is a rational distance from each vertex? Note that this is a very specific case of the Rational Distance Problem, which can be ...
-2
votes
7answers
90 views

$0.333333$ - a recurring or non-terminating decimal?

I have read like, 1.All terminating and recurring decimals are RATIONAL NUMBERS. 2.All non-terminating and non recurring decimals are IRRATIONAL NUMBERS. if the statements are right, then here ...
0
votes
2answers
69 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
1
vote
2answers
68 views

Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...
2
votes
1answer
352 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 ...
0
votes
1answer
48 views

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ for all $x,y\in\mathbb{Q}^+$. Before this problem, I have solved few similar ...
2
votes
2answers
55 views

If $a \in \mathbb{I}$ , how is $\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$

If $a \in \mathbb{I}$ , how is $$\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$$ It says in my notebook that this set in dense in $\mathbb{R}.$ How do I prove this density? With say $\mathbb{Q}$ and ...
4
votes
5answers
444 views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
2
votes
2answers
35 views

Can a non-rational polynomial be rational at all integers?

Is there a polynomial $f \in \mathbb{R}[X]$ such that for every $x \in \mathbb Z,\>\> f(x)$ is rational but at least one of the coefficients of $f$ is irrational?
2
votes
0answers
66 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
1
vote
1answer
96 views

(Ir)rationality of Real Numbers

I am working on this proof and this is what I got so far, can someone help me verify if what I have done is right? For all real numbers $x$ and $y$, if $x+y$ is rational and $x-y$ is irrational ...
1
vote
1answer
65 views

Prove $\log(x)$ is transcendental

What is a proof that $\ln(\alpha)$ is transcendental for $\alpha$. I believe I heard somewhere that the natural logarithm of any rational number is transcendental. Do you guys have any proofs of that ...
0
votes
3answers
87 views

Prove that rational numbers (not just positive) are countable without using axiom of choice.

Prove that rational numbers (not just positive) are countable without using axiom of choice(since it is controversial). I have seen proofs that use the fact that union of countable sets is countable, ...
6
votes
1answer
133 views

Can set of integers form a vector space over field of rationals?

As field of reals $\mathbb{R}$ can be made a vector space over field of complex numbers $\mathbb{C}$ but not in the usual way. In the same way can we make the ring of integers $\mathbb{Z}$ as a ...
1
vote
0answers
34 views

Procedure converting decimals to rationals.

Suppose I have been given a rational number in decimal format (since decimals of rationals repeat, finite precision presentation suffices), what is the most effective way to write it in form of ratio ...
4
votes
2answers
522 views

Length of period of decimal expansion of a fraction

Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
1
vote
1answer
111 views

IMC 2008 first problem first day. Finding continuous functions so $x-y\in \mathbb Q \implies f(x)-f(y)\in \mathbb Q$

I would like an alternate solution and proof verification for the following problem: Find all continuous functions $f:\mathbb R \rightarrow \mathbb R$ so that if $x-y$ is rational then $f(x)-f(y)$ is ...
0
votes
1answer
75 views

Are $x,y$ rational if $x+y$ is rational and $x-y$ is rational? [closed]

Are $x,y$ rational if $x+y$ is rational and $x-y$ is rational? This question was given in maths class, and I don't know where to start. I would be happy if the answer was included in the proof.
1
vote
1answer
37 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
38 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 ...
14
votes
2answers
139 views

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
44 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
1answer
52 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
73 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 ...
0
votes
3answers
45 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 ...
4
votes
2answers
74 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
46 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
55 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
104 views

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

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
72 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 $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ ...
19
votes
9answers
853 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
4answers
157 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
42 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
63 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
49 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 ...
8
votes
6answers
196 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
37 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
359 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
33 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
45 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: ...
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
236 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 ...