Questions tagged [rational-numbers]

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.

Filter by
Sorted by
Tagged with
0
votes
1answer
109 views

Tensor product of finite group with group of rational number.

The tensor product of the finitely generated abelian group G with a rational number is zero if and only if G is finite. is it true for any finite group? For example, we take the tensor product of the ...
1
vote
1answer
163 views

Prime and rational numbers

I came across the following question while studying that is stumping me. Can anyone please help me solve it? Let "$a$" be a prime number greater than $10,000$ and let $x=\sqrt{a}$. Which of the ...
2
votes
1answer
115 views

Proof for rationality of a number

Let $a, b, c\neq 0$ be rational numbers such that $\sqrt[3]{ab^{2}}+\sqrt[3]{bc^{2}}+\sqrt[3]{ca^{2}}\neq 0$ is a rational number. Prove that $$\sqrt[3]{\frac{1}{ab^{2}}}+\sqrt[3]{\frac{1}{bc^{2}...
0
votes
2answers
32 views

Question about the validity of a group of rational numbers

The question asks the following: I know that I need to prove the four criteria of a group (closure, associative, identity, and inverse), but I'm having a hard time with closure. When you input ...
0
votes
1answer
133 views

Can you multiply a rational by an irrational and get a rational number?

For two numbers $a,b$, if $a,b\in\mathbb{Q}$, then $ab\in\mathbb{Q}$, but if $a,b\in\mathbb{I}$, then $ab\in\mathbb{I}$ OR $ab\in\mathbb{Q}$, depending on the particular values of $a$ and $b$, where $\...
0
votes
2answers
83 views

Set of natural and rational numbers

Just a quick question: Is it correct to say that the set of rational numbers cannot be a subset of the set of natural numbers? Certainly, we know these two sets have the same cardinality and there ...
2
votes
3answers
79 views

Determine the greatest common divisor of polynomials $x^2+1$ and $x^3+1$ in $\Bbb Q[X]$.

Exercise: Determine a gcd of the polynomials $x^2+1$ and $x^3+1$ in $\Bbb Q[X]$.. Write the gcd as a combination of the given polynomials. Is it correct that I keep using long division until the ...
0
votes
2answers
29 views

At least one rational is within interval (A, B)

I'm reading a book and there is some strange proof (strange for me) of the theorem that within each interval, no matter how small, there are rational points. Proof: we need only take a denominator $n$ ...
2
votes
1answer
76 views

If the sum of a sequence of positive rational numbers is transcendental, is the sum of every subsequence that is irrational, also transcendental?

Assume we have a sequence of positive rational numbers $(a_n)$, and $\sum_{n=1}^\infty a_n = x$ and $x$ is transcendental. If we have a subsequence of $(a_n)$, $(b_n)$ and $\sum_{n=1}^\infty b_n = y$ ...
4
votes
1answer
155 views

Continuous function that takes rationals to irrationals and vice-versa?

In this question - https://www.quora.com/Can-you-create-a-continuous-function-that-takes-rational-numbers-to-irrational-ones-and-vice-versa How $|f(\Bbb{Q})| \leq |\Bbb{Q}|$ ? in the first answer, I ...
-1
votes
1answer
39 views

A little question on rationals and open sets in $\mathbb{R}$.

Let $A$ be an open set of real numbers, and $D$ be the set of rationals in $A$. For every $d\in D$, let $J(d)$ be an (arbitrary) open interval such that $d \in J(d) \subseteq A$. Is it true that $\...
0
votes
5answers
66 views

How would you prove that $\mathbb{Q} - B$, where $B$ is a finite subset of $\mathbb{Q}$, is dense in $\mathbb{R}$?

I had a problem in my book asking to prove if various sets were dense in $\mathbb{R}$. This set I was working with ended up being equivalent to $\mathbb{Q} - \{0\}$ , and I realized $\mathbb{Q}$ ...
0
votes
1answer
88 views

Is uncountable transcendental-additions enough to make $\mathbb{Q}$ into $\mathbb{R}$?

Consider $\mathbb{Q}$ and then consider "adding" a transcendental $\zeta$ to it, while still retaining the field axioms (i.e. $\mathbb{Q}(\zeta)$). We could add another transcendental in an obvious ...
2
votes
1answer
62 views

Fields isomorphic to $\mathbb{Q}$

We know that certain fields has an isomorphic copy of $\mathbb{Q}$ (for example, every ordered field). But, is there an no trivial explicit example of a field that is isomorphic to $\mathbb{Q}$?
0
votes
3answers
43 views

Solving complex algebraic equation with radicals

How do I solve this math question? If $x$ and $y$ are rational numbers and $(3 + 4\sqrt{3})(x + y\sqrt{3}) = 26$, find the sum of $x$ and $y$. I tried solving for $x$ and $y$ individually to add them ...
2
votes
2answers
64 views

Prove that $(b^m)^{\frac{1}{n}} = (b^p)^{\frac{1}{q}}$

The exercise is: Fix $b>1$. If $m, n, p, q$ are integers, $n>0$, $q>0$ and $r=m/n=p/q$, prove that: $$(b^m)^{\frac{1}{n}} = (b^p)^{\frac{1}{q}}$$ Hence it makes sense to define $...
2
votes
4answers
64 views

Rational Roots of a Quadratic Equation

If $a,b,c$ are non zero, unequal rational numbers then prove that the roots of the equation $$(abc^2)x^2+ 3a^2cx+b^2cx-6a^2-ab+2b^2=0 $$ are rational. Use theory of equations and basic ...
0
votes
1answer
33 views

Proof of rational + algebraic is algebraic and rational + transcendental is transcendental?

I've heard of and seen some proofs that the product and sum of two algebraic numbers is algebraic, however many of them are quite complex and require a variety of machinery (from matrix eigenvalues to ...
0
votes
1answer
58 views

Counting such 3-digit numbers

Count the number of 3-digit natural numbers N with the property that the sum of the digits of N is divisible by the product of the digits of N. Let abc be the number then (abc)k = a + b + c k= 1/bc+ ...
0
votes
3answers
139 views

Quotient vs Ratio vs Fraction vs Rational

I can't seem to differentiate between a Quotient and a Ratio and a Fraction and a Rational. From what I know a rational is a number like $2/3$ or $5.4/7$ whereas quotient, ratio and fraction all are ...
0
votes
6answers
107 views

Is there any proper way through which we can check whether a number is rational or not

Is there any proper way through which we can check whether a number is rational or not because finding the decimal expansion at times is not viable. For example the number $48/121$ shows no sign of ...
0
votes
1answer
197 views

Proving that the union of two infinite disjoint sets with same cardinality is equipotent with either one

This question is almost a duplicate to the question Q, but I would like to prove it in a more personalised manner. The fact, that the individual sets, say $A$ and $B$ are countable, as well as ...
1
vote
1answer
73 views

Rational point at optimum

$$\begin{array}{ll} \text{maximize} & \displaystyle\prod_{i=1}^r P_i\\ \text{subject to} & x_1 + x_2 + \dots + x_n = 1\\ & x_1, x_2, \dots, x_n \geq 0\end{array}$$ where each $P_i$ is a ...
11
votes
3answers
3k views

Conjugate of real number

I'm slightly confused on the subject of conjugates and how to define them. I know that for a complex number $ a - bi $ the conjugate is $ a + bi $ and similarly for $ 1 + \sqrt 2 $ the conjugate is $ ...
1
vote
1answer
65 views

What identities collapse number systems?

I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=\bar{c}$, in order to show that $c\in\mathbb{R}$. I ...
1
vote
2answers
229 views

Show that any right angled triangle with hypotenuse 1 may be approximated arbitrarily close by one with rational sides

Taken from Mathematics and Its History Book by John Stillwell, page 8. Please check if there are any flaws in my answer (have I actually understood the meaning of "arbitrarily close" and also, if you ...
0
votes
1answer
42 views

How would I simplify this rational expression?

The original expression: $\frac{(x + 2)^{2} - (x + 2) - 20}{x^{2} - 9}$ I was taught to completely factor the numerator and the denominator before stating any non-permissible values, then to cancel ...
1
vote
2answers
65 views

Find an upper-bound for the denominator of a given expression

I have the following variables: USD EUR CNY JPY ...
4
votes
0answers
142 views

Representation of rationals as finite continued fractions with restricted coefficients

This question and its answer incidentally show that every non-zero rational number $q$ can be written as a finite generalised continued fraction of the form: $$ \dfrac{2^{n_0}}{1- \dfrac{2^{n_1}}{1 -...
2
votes
2answers
125 views

Is the set $\{(x,y)| x^2+y^2 = \frac{1}{n^2}, n \in \Bbb{N}, x\in \Bbb{Q}$ or $y \in \Bbb{Q}\}$ countable?

I was thinking about the set $\left\{(x,y)\,\middle|\,x^2+y^2 = \frac{1}{n^2}, n \in \Bbb{N}, x\in \Bbb{Q}\text{ or }y \in \Bbb{Q}\right\}$ Certainly, this set is non-empty as I can find a pair $\...
3
votes
1answer
143 views

Find rational numbers $a,b,c$ satisfying $(2^{1/3}-1)^{1/3} = a^{1/3}+b^{1/3}+c^{1/3}$.

$\textbf{Problem} $ Find rational numbers $a,b,c$ satisfying \begin{align*} (2^\frac13-1)^\frac13 = a^\frac13+b^\frac13+c^\frac13 \end{align*} My Attempt: I try to $2^\frac13-1 = (a^\frac13+b^\...
1
vote
1answer
72 views

defining the value of real numbers raised to rational exponents [closed]

if $b^{1/n}$ where $b$ is a real number and $n$ even is defined as the positive real solution to the equation $x^n=b,$ how did they suddenly decide to define the value of $b^{m/n}$ where $m$ and $n$ ...
2
votes
1answer
63 views

For $x^3+px+q=0$, one of the solutions is $\sqrt 3-1$. Find the value of rational number $p$ and $q$, and other two solutions of the equation.

I cannot find any error from my answer. What’s wrong?
1
vote
1answer
356 views

Proving that there is no rational number solution of the equation $x^2-3x+1=0$, extrapolating to a more general problem.

I would like to know whether I have correctly proved the following statement and have correctly extrapolated out a general situation. We're asked two things: a) Prove there is no rational number ...
0
votes
2answers
190 views

Show that the closure of $\Bbb{Q}$ is equal to $\Bbb{R}$

Proof (Show:$\Bbb{\overline{Q}}=\Bbb{R}$): Consider that $\Bbb{\overline{Q}}:=\{r\in\Bbb{R}:r\in\Bbb{Q}\quad \lor\quad \forall\delta>0\exists q\in\Bbb{Q}(\vert r-q\vert<\delta)\}$. Certainly, $\...
1
vote
1answer
62 views

Proof Irrational number $x$, given $N\in{\mathbb{N}}$ exists $\epsilon>0$ so all rationals $\in{V_\epsilon{(x)}}$ have denominator larger than N.

I am trying to show that Thomae's function is continuous at every irrational point (I have already shown discontinuous at every rational point), and for that - following this answer Prove continuity/...
0
votes
3answers
63 views

Equalities of a rational numbers

I have the equalities $$\frac a {1 + a + b } = \frac x {1+x+y}$$ $$\frac b {1 + a + b } = \frac y {1+x+y}$$ How can I show that $a = x$ and $b=y$?
0
votes
1answer
39 views

Need the name of algebraic structure suitable for this

If I want to prove that addition and multiplication are internal operations in the rational domain Q. What algebrac structure I need to prove that (Q,+, *) will be? ...
3
votes
2answers
78 views

$n$ as an integer and a real number

Since $\mathbb{Z} \subset \mathbb{R},$ all integers $n$ are also a real number. However, $\mathbb{Z}$ and $\mathbb{R}$ is defined by Peano's axiom and Dedekind cut, two very different methods. Does ...
0
votes
1answer
144 views

Irrationality of the Euler–Mascheroni constant

The definition of the Euler–Mascheroni constant is the limit of $$H_n - \log(n)$$ as n approaches infinity. So, why is it so hard to prove the irrationality of this constant? $H_n$ is defined only for ...
0
votes
2answers
34 views

Is there a way to form a sequence of intervals s.t they cover all the rational numbers in $(0,1)$ in a way that $C_n \subset Int(C_{n+1}),$

Is there a way to form a sequence of intervals $C_i \subset \mathbb{R} $ such that they cover all the rational numbers in $(0,1)$ in a way that $$C_n \subset Int(C_{n+1}),$$ where $C_i$ is closed. ...
0
votes
2answers
107 views

find all rational points on $x^2+7 y^2 = 1$

I am familiar with the solution of "how to find all rational points on $x^2+y^2=1$ ?". I would like to know if i can solve: "how to find all rational points on $x^2+7y^2=1$ ?" using the same ...
4
votes
1answer
62 views

(Dis)continuity of $f(x)=\sum_{j: q_j < x} 2^{-j}$

Consider an indexing $q_1,q_2,\dots$ of the rationals in $(0,1)$ and define $f:(0,1)\to (0,1)$ by $f(x)=\sum_{j: q_j < x} 2^{-j}$. Prove that $f$ is discontinuous at all rationals and continuous ...
7
votes
0answers
265 views

Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
9
votes
8answers
4k views

How to explain irrational numbers to laymen? [duplicate]

I am trying to describe how irrational numbers, which are all modeled as a series of fractions, can themselves not be fractions, and are instead part of a unique group of "decimal numbers" outside of ...
5
votes
1answer
472 views

Does there exist non trivial group homomorphism from $S_3$ to ( $\mathbb Q $,+)

Let $G = S_3$ be the permutatiin group of 3 symbols.Then $G$ is isomorphic to a subgroup of a cyclic group There exists a cyclic group $H$ such that $G$ maps homomorphically onto $H$. $G$ is a ...
0
votes
2answers
87 views

Is any subsequence of a sequence of rational numbers dense in $\mathbb{R}$

We know that $\mathbb{Q}$ is countable so let $\{q_n\}_{n \in \mathbb{N}}$ a sequence of all rational numbers. My Question is: Is any subsequence $\{q_{n_k}\}_{k \in \mathbb{N}}$ dense in $\mathbb{R}...
4
votes
1answer
157 views

Why does the law of distributivity use the distributive property to prove it exists?

I was looking at the Law of Distributivity and disagreed with how it was proven. It proved that $$\frac ab\bigg(\frac cd + \frac ef\bigg) = \frac ab\cdot \frac cd + \frac ab\cdot \frac ef.$$ It ...
2
votes
0answers
42 views

Prove or refute that $\sum_{n=1}^\infty\frac{1}{n^2+A\varphi(n)+B}$ will be irrational for some choice of integers $A,B\geq 1$

We denote the Euler's totient function as $\varphi(n)$ for integers $n\geq 1$, that is a multiplicative function that counts the number of integers $1\leq k\leq n$ up to the given integer n that ...
0
votes
1answer
81 views

find $f(99)$ where $2f(x-\frac{1}{x}) + f(\frac{1}{x}-x) = 3(x+\frac{1}{x})^2.$

Suppose $f(x)$ is defined for all positive numbers $x$, and $2f(x-\frac{1}{x}) + f(\frac{1}{x}-x) = 3(x+\frac{1}{x})^2.$ Find $f(99)$. Plugging in $x=1$, I was able to get $3f(0) = 3(1+1)^2 = 3\cdot ...