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.
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Real Analysis Inequality Proof Involving Reals and Rationals $0 < |r - q| < \varepsilon$
I'm having difficulties making progress in proving:
$$\forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$
To clarify, $r$ is a real number and $q$ is a ...
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There is no smallest rational number greater than 2 [closed]
I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why.
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Prove that $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$.
Prove that the function : $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$.
To prove the continuity
it is necessary to prove that for every ...
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Why do we still do symbolic math?
I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one.
Numerical computations, to my understanding, never ...
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Division by $0$ and its restrictions
Consider the following expression:
$$\frac{1}{2} \div \frac{4}{x}$$
Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$.
But if we rearrange the ...
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Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?
A student asked me the following today :
Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?
I am quite perplexed by it. Clearly, the only non-trivial part is to check
For any $x, y\...
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Total distance traveled when visiting all rational numbers
My students found an old problem given in my school in 2007 (probably from a Honor Calculus class) and had been trying to solve for some time. Here is the problem:
Prove or disprove: there exists a ...
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Axiomatic characterization of the rational numbers
We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this).
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Solutions to $f'=f$ over the rationals
The problem is as follows:
Let $f: \mathbb{Q} \to \mathbb{Q}$ and consider the differential equation $f' = f$, with the standard definition of differentiation. Do there exist any nontrivial solutions?...
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Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?
Obviously:
$$\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=0.1111\dots=\frac{1}{9}$$
is a rational number.
Now, if we make terms with demoninators in the form:
$$q_n=\sum_{k=0}^{n} 10^k$$
Then ...
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Do all rational numbers repeat in Fibonacci coding?
In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, $\frac{1}{3}$ in base 10 is $0.33333...$, in ...
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How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?
Open problem in Geometry/Number Theory. The real question here is:
Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational?
The ...
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Will every rational number eventually be in this set?
Let $A_0=\{0\}$. For every $n\ge 0$, let $B_n=\bigcup_{k\ge 0}A_n+k$, and let $A_{n+1}=f(B_n)$, where $f:[0,\infty)\to[0,1):x\mapsto\frac{x}{x+1}$.
Let $q\in\Bbb Q\cap [0,1)$. Can we prove that $q\in ...
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An interesting way of expressing any real number using the harmonic series.
I recently saw the identity $$ \frac{1}{1} - {1 \over 2} +{1 \over 3} - {1 \over 4} + {1 \over 5} - {1 \over 6} \dotsb = \log(2) $$
which I found rather interesting. I was intrigued by the way a ...
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Are there any bases which represent all rationals in a finite number of digits?
In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in ...
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An amazing integral with a typo
This question Concerns a certain integral
$$f(p)=\int_0^1\frac{x^p(1-x)^p}{1+x^2}\mathrm dx$$
Which, as proven by Calvin Khor, has the property that $f(4k)-4^{k-1}(-1)^k\pi\in\Bbb Q$ when $k\in\mathbb ...
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Remarkable/unexpected rational numbers
Consider the Riemann $\zeta$ function. We know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (in particular is transcendental). We also know that $\zeta(3)$ is irrational, and we expect $\zeta(...
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A set with measure $0$ has a translate containing no rational number.
Suppose $E$ is a set with measure $0$. Show there exists $t\in \mathbb{R}$ such that $E+t$ contains no rational number.
My idea is to find an interval in $E$, then we can get a contradiction. I try ...
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Is a non-repeating and non-terminating decimal always an irrational?
We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats).
$.0303$ $\cdots$ tends to $\frac{1}{33}$.
So,I was wondering this:
In the decimal representation, if we start writing the $10$...
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Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$
Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ?
$$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
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Integral of rationals
Define $f(x)$ as
$$f(x)=\begin{cases}0,&\text{if }x\in \mathbb{Q}\\
1,&\text{if }x\notin \mathbb{Q}\;.
\end{cases}$$
Considering the fact that there is a countable infinity of rationals yet an ...
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Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.
I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$."
This is my attempt of proving it:-
Assume that $x=p/q$...
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Irreducibility of $f(x)=x^4+3x^3-9x^2+7x+27$
Question at hand is:
Is $x^4+3x^3-9x^2+7x+27$ irreducible in $\Bbb Q$ and/or $\Bbb Z$.
This is for an exam, reasoning is trivial, but no calculators in hand. Clearly, if there is a rational root, ...
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Proof that continued fractions are finite for rationals?
How does one prove that the continued fraction representations of rational numbers are finite?
For every $x\in\mathbb{R}$, the (simple) continued fraction representation of $x$ is:
$$
x = [a_0; ...
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For what algebraic curves do rational points form a group?
For what real algebraic curves do rational points form a group ?
How does this relate to Jacobian Varieties ?
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Prove that the square root of any irrational number is irrational.
The problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning.
Theorem:
Prove that the square root of any irrational ...
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Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?
Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$?
Motivation : We know that
$$\zeta (2k)=(-1)^...
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Is there a bijection between $\mathbb{Q}$ and $\mathbb{Q}_{>0}$?
Is there a bijection between $\mathbb{Q}$ and $\mathbb{Q}_{>0}$?
For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\mathbb{...
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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 ...
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Finding irrational entries such that the determinant will never be zero
Context.
The main goal is to find whether or not a subspace of $\mathbb R^5$ of dimension $3$ intersects a rational subspace of dimension $2$. By rational subspace, we mean a subspace of $\mathbb R^5$ ...
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Please help me check my proof that the transcendental numbers are dense in $\mathbb{R}$
I need to prove that the set of all transcendental numbers is dense in $\mathbb{R}$, and to that end, I have written the following proof:
Let $\mathbb{T}$ denote the set of transcendental numbers ...
user100463
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Rational analysis
I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real ...
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Polynomial bijections from $\mathbb{Q}$ to $\mathbb{Q}$
Prove or Improve: Polynomials $f\in \mathbb{Q}[x]$ which induce a bijection $\mathbb{Q}\to\mathbb{Q}$ are linear.
The question of existence of a polynomial bijection $\mathbb{Q}\times\mathbb{Q}\to \...
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Prove that $x^3 + x^2 = 1$ has no rational solutions?
Is this enough for a proof?:
$$x^3+x^2 = 1$$
I would factor and get: $x^2(x+1) = 1$
I would show that $x = \sqrt1$, which is rational but then what else would I have to show? $x+1=1$ which gives me ...
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Visualizing rational numbers as multiplication graphs
It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication graph $n/m$ ...
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Prove that $\sqrt{2} + \sqrt[3]{3}$ is irrational [duplicate]
$\sqrt{2} + \sqrt[3]{3}$ is irrational ?
These are my steps -
$\sqrt{2} + \sqrt[3]{3} = a$
$3 = (a-\sqrt{2})^{3}$
$3 = a^{3} -3a^{2}\sqrt{2} + 6a -2\sqrt{2}$
$3a^{2}\sqrt{2}+2\sqrt{2} = a^{3}+6a-...
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Does there exist a nonzero ring homomorphism from the ring of square rational matrices to the ring of rational numbers?
I am wondering if it is possible to construct a nonzero ring homomorphism from $M_n(\mathbb{Q})$ to $\mathbb{Q}$.
So far, I've been unsuccessful in constructing such a nonzero ring homomorphism. Is ...
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Uses of vector spaces over $\mathbb Q$
I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$:
Hilbert's third problem; and
The Buckingham pi ...
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subtraction of two irrational numbers to get a rational [duplicate]
Say you have a number like $\pi$ or e. Is it possible to subtract another number from it and end up with a rational number? I mean I guess you could write an equation like $\pi-x=3$ But could there ...
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The period of decimal expansion of $1/m$
I am trying to prove that for a composite, positive integer $m$ such that $2 \nmid m$ and $5 \nmid m$, the period of the decimal expansion of $1/m$ is equal to $\text{ord}_{m}(10)$, where $\text{ord}_{...
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Why is epsilon not a rational number?
I was wondering why epsilon, the smallest positive number, isn't a rational number. I was watching a video a few days ago about surreal numbers, and I've learned that, in the field of surreal numbers, ...
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Field containing all square roots of rational numbers
What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
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Is there a well-behaved finitely additive “measure” on $\mathbb{Q }$?
There exists no nontrivial measure on the set of rational numbers for which the measure of singletons is zero. That’s because the rational numbers are countable, so any set of rational numbers is a ...
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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 ...
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Is the number 0.2343434343434.. rational? [duplicate]
Consider the following number:
$$x=0.23434343434\dots$$
My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...
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$p$, $q$ and $\sqrt[n]{p} + \sqrt[n]{q}$ are rational, with the latter being non-zero. Are $\sqrt[n]{p}$ and $\sqrt[n]{q}$ rational?
Let $p, q \in \mathbb Q$, $n \in \mathbb Z^+$ and label $a = \sqrt[n]p, b=\sqrt[n]q$.
Conjecture: If $a + b$ is a non-zero rational, then both $a$ and $b$ are rational.
(Preliminary question: is ...
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What exactly are those "two irrational numbers" $x$ and $y$ such that $x^y$ is rational? [duplicate]
It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
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Rational Trig Solutions for $n\ge 3$
Are there solutions to
$$\sin(x+y)\sin(x-y)=n\ \sin(x)\sin(y)$$
for $n\ge 3$ where $x$ and $y$ are rational multiples of $\pi$? (excluding the trivial solutions when both sides are $0$).
Known ...
user377597
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Why must the decimal representation of a rational number in any base always either terminate or repeat?
Wikipedia makes the following statement about rational numbers.
The decimal expansion of a rational number always either terminates
after a finite number of digits or begins to repeat the same ...
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Rational solutions of $x^3+y^3=9$
An old alchemist had two sphercial flasks, one with a circunference of $12$ inches and the other one with a circumference of $24$ inches. He desired to transfer their contents into two other spherical ...