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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1answer
99 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
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1answer
71 views

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

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.
-4
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3answers
33 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
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2answers
44 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 ...
1
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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
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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 ...
0
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1answer
46 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, ...
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2answers
115 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 ...
0
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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
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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 ...
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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. ...
3
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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
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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 $\theta$ ...
-4
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1answer
96 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?
2
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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
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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 ...
18
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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 ...
0
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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
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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 ...
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6answers
186 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 ...
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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
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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 ...
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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 $0.13333333$ ...
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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 ...
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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
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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
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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 ...
3
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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 ...
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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
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2answers
49 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
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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
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1answer
36 views

How do I show that the equivalence relation defining the rational numbers is transitive?

I apologize if this is a super easy question, but there is something fishy about my proof. I was to show: $$(p,q) \sim (m,n) \wedge (m,n) \sim (a,b) \implies (p,q) \sim (a,b) $$ under the ...
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0answers
28 views

For which values of $\theta $ does this claim true?

According to the conjecture of , L. J. Lander, T. R. Parkin, and John Selfridge (1967):I suppose this claim : claim :let : $$\sum_{i=1}^n a^{\cos\theta}_i =\sum_{j=1}^m b^{\cos\theta}_j ,$$ for ...
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2answers
55 views

Presentation of the additive group of the rational numbers

We know that $\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as $$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$ Then ...
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2answers
40 views

Rational number with rational exponent becomes rational

I'm looking for a proof to show when $p^q$ for $p,q \in \mathbb{Q}$ is again in $\mathbb{Q}$, without factoring. I'm not sure, if it's possible, given these two numbers to say if the result is again ...
21
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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 ...
10
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1answer
179 views

What is known about this well-ordering of the rationals in a finite interval?

Given any interval $I=(a,b) \subset \mathbb R^+$, we may order the rationals in $I$ with a denominator-first lexicographic order, as follows: First, we list, in increasing order of numerator, all $q ...
0
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2answers
35 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
3
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2answers
95 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
5
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1answer
129 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
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3answers
106 views

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0<L<U<1$, find rational number $M=a/b$ such that $L \le M<U$ and $(a\times b)$ is as small as possible---$a$ and $b$ are integers. For example, If ...
3
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1answer
36 views

Real numbers and rationals - Decimal Expansion

How would one endeavor to show that A real number is rational if and only if its decimal expression ends in recurring digits?
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2answers
51 views

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to ...
3
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1answer
85 views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
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3answers
84 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
2
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1answer
36 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
1
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1answer
46 views

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
5
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0answers
80 views

Prove that the square root of any irrational number is irrational.

The problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irrational ...
1
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1answer
28 views

rational numbers as upper limit of a summation?

a quick question: Is it a legit way to use a fraction as the upper limit of a summation? Given is a frequency $f$ and a sample rate $f_s$. I want to use a sum like this: $\sum_{k=1}^{\frac{f_s}{2f}} ...
1
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1answer
37 views

Compare density of rationals to the density of integers

Is is possible to somehow quantitatively compare the density of rational numbers to the density of integer numbers, ascribing to the both a number characterizing the density?