Questions tagged [irrational-numbers]

Questions about real numbers not expressible as the quotient of two integers. For questions on determining whether a number is irrational, use the (rationality-testing) tag instead.

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Irrational combinations of rationally independent polynomials.

Let $p_1,...,p_k: \mathbb{N} \to \mathbb{Z}$ be rationally independent polynomials with zero constant term. If $t_1,...,t_k \in [0,1)$ are not all rational, is it true that the polynomial $$p(n)=\...
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What are some techniques to prove that a set of numbers is rationally independent?

I am solving a math problem for fun and it amounts to proving that a specified (finite) set of numbers, each defined by an infinite series involving polynomials and the factorials, is a rationally ...
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Formal definition of Real Numbers

I was recently getting started with mathematic formal notation by defining sets of numbers. In Wikipedia, there are some of them definited as: $\mathbb{Z} = \{\ldots -3,-2,-1,0,1,2,3\ldots\}$ $\mathbb{...
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4 votes
1 answer
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Is integer part of $e^n$ infinitely often even and odd?

Let $a_n:=\lfloor e^n\rfloor$. Are $\{n\mid 2|a_n\},\,\{n\mid 2|a_n-1\}$ both infinite sets? More generally, for any irrational number $\alpha>1$, is each set of the form $\{n\mid p|\lfloor\alpha^n\...
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-2 votes
0 answers
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Are the non trivial zeros of Riemann's Zeta function rational, irrational or we don't know? [duplicate]

I've read and watched from several sources that all non trivial zeros are hypothesized to be on the critical line (vertical line at x=1/2), and that, so far, nobody has been able to find one outside ...
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2 votes
2 answers
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Understanding a 'geometrical proof' of irrationality of √2

I had been having trouble understanding a proof of the irrational nature of √2. I found this proof in the first page of the foreward to 17 theorem provers of the world where a 'geometrical proof' (is ...
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Does a set of all decimal expansions of $\pi$ contains $\pi?$ [duplicate]

Let's say there is a set containing all finite decimal expansions of $\pi$: $$A = \{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... \}$$ Does this set contains $\pi$? I see that it is probably not true ...
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Proving that for any irrational number there is a irrational numbers and rational numbers sequence converging to it [duplicate]

I am struggling with how to prove that for any irrational number there are sequences of rational numbers that converge to it but also a sequence of irrational numbers converging to it. That is a ...
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-1 votes
0 answers
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Showing that sequence of rational numbers converges to an irrational number [duplicate]

I am struggling to understand one problem. I have to show that for every $ n $, and numbers $ a_n = i - 1/2^n $ and $ b_n = i + 1/2^n $, where $ i $ is ANY irrational number, there exist a rational ...
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Let $m,n\in\mathbb{Z}^+$ & $n$ odd. Let $f:\mathbb{R}→\mathbb{R}$ by $f(x)=(x^{1/n})^m=x^{m/n}\quad\forall x$. Show $f$ is continuous on $\mathbb{R}$ [closed]

Problem Let $m$ and $n$ be positive integers and $n$ is odd. Define the function $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=(x^{1/n})^m=x^{m/n}\quad\forall x$. Show that $f$ is continuous on $\...
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Why is this proof wrong? ( I proved the irrationality of $\pi$ using the most basic techniques so I suspect that it must have gone wrong somewhere)

So it starts off by way of contradiction, supposing $\pi\in \mathbb Q$, then by De-Moivre's theorem for rational powers: $$\left((\cos(2k\pi)+i\sin(2k\pi)\right)^{\pi}= \cos(2k\pi^2)+i\sin(2k\pi^2)=1^{...
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1 vote
2 answers
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Solve for reals $[2022+x(2+\sqrt{x})][111-x(2+\sqrt{x})]=-52^3$

Solve in real numbers: $$\sqrt[3]{2022+x(2+\sqrt{x})}+\sqrt[3]{111-x(2+\sqrt{x})}=9$$ Cubing both sides, we get$$ 2022+111+3\sqrt[3]{2022+x(2+\sqrt{x})}\sqrt[3]{111-x(2+\sqrt{x})}\Big(\sqrt[3]{2022+x(...
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2 votes
2 answers
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Find $\sqrt{\frac{1}{2}-f(1)}+\dots+ \sqrt{\frac{1}{2}-f(99)}$

Define $f(n)=\sqrt[2]{n^4+\frac{1}{4}}-n^2.$ Find $$\sqrt{\frac{1}{2}-f(1)}+\dots+ \sqrt{\frac{1}{2}-f(99)}$$ I tried to simply $f(n).$ So rationalising, we get $$\sqrt{\frac{1}{2}-f(n)}\sqrt{\frac{1}{...
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For all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational.

How can I show that for all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational? I have tried to use a proof by contradiction to no avail. I have gotten decently far with a proof by ...
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1 answer
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Position of specific value

Let's assume a have an arbitrarily long number, take π for example. Since we know π is infinite, there will at some point be a group of numbers like "2015201620172018...", correct? My ...
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1 vote
1 answer
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If $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$ then $p/q$ is necessarily one of the convergents : Extend the proof to irrational $x$

Prove that, if $x$ is any irrational number, and if $p/q$ is a rational fraction in lowest terms, with $q\geq 1$, such that $$\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$$ then $p/q$ is necessarily one ...
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1 answer
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A pattern that leads to regular continued fractions of quadratic irrationals [closed]

The following expression can be obtained by converting the continued fraction of quadratic irrationals to single fraction. $$ \sqrt{N} = \frac{b\sqrt{N}+aN}{a\sqrt{N}+b} $$ The equation holds for any ...
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1956 AHSME Question 23 on the quotient of an irrational number with a real number

I was solving some problems from the American High School Math Exam of 1956 (AHSME) from the Art of Problem Solving Website. Question 23 Problem 23 About the equation $ax^2 - 2x\sqrt {2} + c = 0$, ...
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1 vote
1 answer
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Does every line through the origin come arbitrarily close to some other lattice point?

Given positive real numbers $x$ and $\epsilon$, do there necessarily exist positive integers $p$ and $q$ such that $\left| qx - p \right| < \epsilon$? Or geometrically: in a two-dimensional ...
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22 votes
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What is meant, exactly, by nonrepeating when talking about irrational numbers?

My question is referring to the exact definition mathematicians use when describing the decimal expansions of irrational numbers as "nonterminating and nonrepeating." Now, I understand, at ...
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-4 votes
1 answer
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The set of irrationals numbers is countable?

I tried to prove this using statement using the difference of sets $\mathbb{R}-\mathbb{Q}$ and the fact that $\mathbb{R}$ is not countable and $\mathbb{Q}$ is countable. In general, is it possible to ...
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Decimal expansion of irrational numbers?

Apologies if this is a trivial question. Is there any particular study which deals with the properties of decimal expansions of irrational numbers? Studies that deal with patterns and so on. Are there ...
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0 answers
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Why does $e/\pi$ (or $1/2$) arise in this limit?

(Related to this prior post.) Let $f(n)$ for $n\in\mathbb N$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $1$. e.g. $f(20)=f(2^2\cdot 5)=f(p_1^2\cdot ...
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2 votes
1 answer
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Given irrational numbers $0<\gamma_1<\gamma_2<1,$ real numbers $0\leq a<b\leq 1,$ does $\exists n$ such that $\{n\gamma_1\}, \{n\gamma_2\}\in(a,b)?$

For this question, $\{x\}$ means the fractional part of the real number $x.$ Given irrational numbers $0<\gamma_1<\gamma_2<1,\ $ real numbers $a,b$ with $0\leq a<b\leq 1,$ does there exist ...
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How is it that the quotitient of two rationals is a common divisor for both?

When $2$ number's ratio can not be expressed as a rational number then we call these number incommensurable. But I also read that incommensurable also has the meaning of being able to measure two ...
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0 answers
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Does $a_1\sqrt{1}+a_2\sqrt{2}+...+a_n\sqrt{n}=0$ have no solutions over the rationals? [duplicate]

I had recently met a question that asked to show that $a\sqrt{2}+b\sqrt{3}+c=0$ has no solutions over the rationals (except for $0$) for $a,b$ and $c$. To show this I moved $c$ to the other side then ...
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2 votes
6 answers
231 views

How do we choose/find the rational sequence that is asymptotic to an irrational one?

I read that to any irrational sequence we can assign at least one rational sequence which is asymptotic to it. The fact that it is asymptotic I think implies that the rational sequence approximates ...
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8 votes
3 answers
737 views

Is $5^\pi$ an integer?

I saw this very short math problem on Twitter: Is $5^\pi$ an integer? It isn't (it's 156.992545309…), but is there some technique to prove this without a calculator? My first and only idea so far is ...
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4 votes
1 answer
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formulas for binary expansion of irrational number between $0$ and $1$

One can write any irrational number between $0$ and $1$ composed as closed expression of popular known numbers, such as, for example, the expression $$\frac{1}{\sqrt{2}}$$ in binary by successively ...
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1 vote
3 answers
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Prove that $Ae + B/e$ is irrational.

Prove that $Ae + B/e$ is irrational. Here A and B are non-zero integers. We know that the sum of two irrational numbers need not be an irrational number. For example $\sqrt2+1$ and $1-\sqrt2$ are ...
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2 votes
1 answer
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Does $(\ln(a))^b$ ($a\in\mathbb Z_{>1}$; $b\in\mathbb Z_{\ne0}$) have a unique representation in this form?

I just want to deal numbers in a form of $(\ln(a))^b$ ($a\in\mathbb Z_{>1}$; $b\in\mathbb Z_{\ne0}$), but I wondered if each number in this form has a unique form. In other words: is a function $f$ ...
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2 votes
0 answers
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The first $0$ in a base $b$ expansion

Consider the following function $Z \colon \Bbb{Z}_{\ge2} \times [0,1) \to \Bbb{N} \cup \{\infty\}$. If $b \in \Bbb{Z}_{\ge2}$ is an integer greater than $1$ and $x \in [0,1)$, let $Z(b,x) \in \Bbb{N} \...
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-1 votes
1 answer
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Prove that the sum of two irrational numbers can be irrational [closed]

The question title itself is self explanatory. We need to prove that the sum of two irrational numbers can be irrational (it necessarily doesn't need to be always, rather i am trying to prove that, ...
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2 votes
1 answer
72 views

Irrationality Result

I have recently derived the following result: for any odd positive integer $k$, one of $\zeta(k)$ and $\zeta(k,2/3)$ is irrational. I cannot find this result anywhere. Is it already known? Thank you ...
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6 votes
1 answer
547 views

Correlation between $\sqrt{1/10}$ and length of powers of integers.

I'm searching information about this simple problem involving square roots and length of powers. It's very simple but it seems interesting, at least for me. I'm not a mathematician. Description By ...
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0 votes
1 answer
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Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number?

Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number? I suppose it would depend one what would be defined as "circular curves", as ...
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0 votes
1 answer
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Proving that a certain set is dense using a useful characterization of the irrational numbers. (Based on Chapter 4 Exercise 25 (b) Baby Rudin)

The following is exercise 25 (b) Chapter 4 in Rudin's Principle's of Mathematical Analysis: Let $\alpha$ be an irrational real number. Let $C_1$ be the set of all integers, let $C_2$ be the set of all ...
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Spivak, Chapter 8 "Least Upper Bounds", Problem 5c: for $r<s$ rational numbers, prove there is irrational number between $r$ and $s$?

5c) Suppose that $r<s$ are rational numbers. Prove that there is an irrational number between $r$ and $s$. Hint: as a start, you know there is an irrational number between $0$ and $1$. There is no ...
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-1 votes
1 answer
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Alternative proof that $\sqrt{2}$ is irrational. [duplicate]

If we set $\sqrt{2} = \frac{a}{b}$ where $a,b$ are positive and coprime, we get $2 = \frac{a^2}{b^2}$. However, as $a^2,b^2$ must also be coprime, this fraction can only be an integer if $b^2 = 1$, so ...
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0 votes
0 answers
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Non-periodic continued fraction with explicitly known convergents?

Is an irrational number with non-periodic continued fraction expansion known, for which one can give explicit formulas for the convergents $p_n/q_n$ or at least for the denominators $q_n$ (similar to ...
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4 votes
4 answers
110 views

Simplification of $\frac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$

Simplify $$\dfrac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$$ Final solution should have rational denominators. Suppose the solution is $X$, I have tried to make up an equation for $...
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2 votes
1 answer
50 views

Prove that any real non negative number can be expresed in golden ratio base using only 1 and 0 [closed]

I´ve been thinking about this problem for a long time but I don´t reach any conclusion and I don´t find any related information. I would be grateful if someone could help me. Here is the problem: ...
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5 votes
1 answer
85 views

Proof that four values have to be rational

Following is defined: $$z=\frac{cx+dy}{ax+by}$$ I have these four terms: $$v_1=b+az$$ $$v_2=a-bz$$ $$v_3=d+cz$$ $$v_4=c-dz$$ The requirements are: $x,y\in\mathbb{Z};v_1,v_2,v_3,v_4\in\mathbb{Q}$. ...
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0 votes
1 answer
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Proof that either one of two expressions is always irrational

I have these two terms: $$b+a\frac{cx+dy}{ax+by}$$ $$a-b\frac{cx+dy}{ax+by}$$ The requirements are: $x,y\in\mathbb{Z};a,b,\frac{cx+dy}{ax+by}\in\mathbb{R}\backslash\mathbb{Q};c,d\in\mathbb{R}$. So ...
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1 vote
1 answer
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Show that it is impossible to find $a,b \in \mathbb{Q}$ such that $ \frac{\sqrt{5}}{10}=a+b \frac{\sqrt{3}}{7} $

Show that it is impossible to find $a,b \in \mathbb{Q}$ such that $$ \frac{\sqrt{5}}{10}=a+b \frac{\sqrt{3}}{7} $$ We have $\frac{1}{20}=a^{2}+\frac{2 \sqrt{3}}{7} a b+b^{2} \frac{3}{49}=\left(a^{2}+b^...
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0 votes
1 answer
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Seeking clarification about another question concerning irrational numbers.

The post A question about decimal representation of irrational numbers. asked: "Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the ...
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0 votes
1 answer
25 views

Question about irrational numbers and finite subsequences of their decimal places.

A more general question than in "A question about decimal representation of irrational numbers.": Since there is an infinite amount of irrational numbers could you always find one that ...
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0 votes
1 answer
51 views

Limit of a piecewise function defined by x being rational or irrational

If $$ f(x)=\begin{cases} x² \quad \quad \text { if x is rational} \newline 0 \ \ \ \ \ \ \ \ \ \ \text {if x is irrational} \end{cases} $$ prove that $\lim_{x \to 0}f(x) = 0$
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8 votes
2 answers
156 views

Given $\varphi$ is golden ratio, how do I prove this $\sum \limits_{j=1}^{\infty}\frac{(1-\varphi)^j}{j^2}\cos{\frac{3j\pi}{5}}=\frac{\pi^2}{100}$?

Given $ \varphi$ is golden ratio, how do I prove this: $ \displaystyle \tag*{}\sum \limits_{j=1}^{\infty}\frac{(1-\varphi)^j}{j^2}\cos{\frac{3j\pi}{5}}=\frac{\pi^2}{100}$ My approach: We can reduce ...
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2 votes
0 answers
62 views

irrationality of a decimal expansion

Consider the real number in $(0,1)$ having the decimal expansion $${\alpha} = 0.{a_1}{a_2}{a_3}\cdots $$ where $a_j$ is obtained by adding up the digits in the decimal expansion of the positive ...
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