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.

Filter by
Sorted by
Tagged with
0
votes
0answers
14 views

Is the sum of partial sums $\frac{2\pi}{k}-{k \mod 2\pi}$ bounded?

Suppose, we are given sequence $a_k= k\mod 2\pi$ and sequence $y_{k,n}$, where $y_{k,n},0\leq n\leq k$ is just sequence $x_0,\dots,x_k$ sorted in increasing order for a given integer $k$. What can we ...
1
vote
0answers
27 views

What are some ways we measure irrationality?

I am wondering in what ways we may quantify an irrational number's approximability. This came up as I was reading about badly approximable numbers, which are those numbers $x$ such that $$\liminf_{q ...
1
vote
2answers
40 views

Is the following constructed number always irrational?

Given $\ $ $ a, b, c \in \mathbb{N} $, $\ $ $ a,b,c > 1 $, $\ $ $ b > c $, $\ $ $b$ and $c$ coprime, $a$ and $c$ coprime, is $ a^{\frac{b}{c}} $ irrational? Examples: $2^{\frac{4}{3}}$, $13^{\...
0
votes
2answers
26 views

Given the following clues, is this number irrational? [closed]

Given $ a, b \in \mathbb{N} $ and $ c \in \mathbb{Q}_{+}$, if $ a^c $ and $ b^c $ are irrational, with $ a $ and $ b $ coprime, is $ \frac{a^c}{b^c} $ irrational too?
1
vote
0answers
27 views

What is known about the approximation constant?

A badly approximable irrational is one whose continued fraction denominators are bounded; equivalently, if $\alpha$ is badly approximable then there is a $c(\alpha) > 0$ such that $$c(\alpha) = \...
3
votes
1answer
68 views

Do the decimal digits of $\pi$ and $\sqrt{2}$ coincide infinitely often?

I suspect the answer to the title question is "yes," but can it be proved? One would expect digits match in one tenth of cases.
-1
votes
4answers
58 views

Show that the following theorem is true or false: If $x\notin\Bbb Q$ then $x^2\notin\Bbb Q$. [closed]

In my approach, I attempted a proof by contrapositive, where it is assumed that $X^2$ is rational and $X$ is also rational. Therefore, we can represent $X^2$ with a ratio of two integers, $a$ and $b$...
7
votes
1answer
99 views

Distribution of $\{n^p\alpha\}$ for irrational $\alpha$

Let $\alpha$ be an irational number. Consider sequence $x_n=\{n^p\alpha\}$, $n\in\mathbb{N}$ (it's the fractional part of $n^p\alpha$), where $p$ is a nonzero real number. Question. For which values ...
1
vote
2answers
73 views

How to prove the irrationality of this number.

Prove that $\cos\frac{1}{\sqrt{2019}}$ is irrational. Would someone give me a hand on this question? Even a hint is appreciated. I don't have any clue.
-1
votes
1answer
45 views

Prove that this number is irrational it's involve the exponential [duplicate]

Prove that $2^{\sqrt{3}}$ is irrational or even it is transcendental? I am wondering if there exists the method that describes its property and its irrationality of these kinds (or even with some ...
-1
votes
2answers
32 views

Rational or Irrational?

Here is my question: can an Irrational number; like $e$, be equal to a second irrational number, $\pi$, times an integer, then divided by a second rational number? Such as: $e = \frac{\pi a }{b} $ , ...
0
votes
0answers
45 views

Irrationality of the real number based on non-zero bits in the binary representations of primes

Assuming that $n$ is an odd natural number greater than $2$, let $B(n)$ denote a sequence of bits between the leftmost and rightmost non-zero bits of the base-$2$ representation of $n$. The ...
0
votes
3answers
49 views

Why the first $x$ decimal places of $(5+\sqrt{26})^{x}$ are following a pattern?

$\sqrt{26}$ is irrational number, so the decimal places should show no pattern. But $(5+\sqrt{26})^{x}$ has these values: ...
-5
votes
2answers
69 views

$\pi$ is defined as the ratio of circumference to diameter, so why is it called “irrational”? [closed]

By definition, $\pi$ is the ratio of circumference and diameter of a circle. So, by definition itself it is a rational number. Why it is called irrational?
2
votes
0answers
91 views

Copeland-Erdős constant: does the cumulative even digit count ever reach or overtake the cumulative odd digit count, and if so, at what prime?

Arthur Herbert Copeland and Paul Erdős proved in 1946 that the Copeland-Erdős constant is a normal number. Since all prime numbers other than 2 are odd, all prime numbers other than 2 end in an odd ...
1
vote
0answers
76 views

Proving $\pi$ is irrational without arguing by contradiction

I know few proofs to prove that $\pi$ is irrational, but in all proofs one thing is common: it is done by the method of contradiction. So, I am looking for a proof without using contradiction. Is ...
1
vote
1answer
55 views

Polynomial with natural coefficients at irrational points

I thought a lot at this problem: a polynomial $p(x) = a_{n}x^n + a_{n - 1}x^{n - 1} + ... + a_1x + a_0$, where $\forall i : a_{i} > 0$ (so that means that $p(x)$ includes all non-negative integer ...
0
votes
1answer
22 views

Rationality of $\log_b a$ when at least $a$ or $b$ is irrational and $a,b>1$

I am looking at the rationality of $\log_ba$ when at least $a$ or $b$ is irrational such that $a,b>1$ and neither of $a$ or $b$ is of the form $\left(\frac{e}{f}\right)^{(g/d)}$, where $d,e,f,\text{...
1
vote
1answer
56 views

Proving $\sin\left(\frac{2\pi}{p}\right)$ is irrational where $p\geqslant3$ is prime

Now I have proven that $\cos\left(\frac{2\pi}{p}\right)$ is irrational by taking $\left(x^2-2x\cos\left(\frac{2\pi}{p}\right) +1\right)$ as the minimal polynomial dividing $\frac{(x^p-1)}{(x-1)}\cdots ...
0
votes
1answer
33 views

Is it possible to convert an irrational number into rational by using a different number system?

It is certainly possible to convert a repeating number into a non repeating number by changing its number system. For example, 1÷3 is a repeating number in decimal as it equates to 0.3333333333......
1
vote
4answers
46 views

Square root of prime is irrational. Is this a valid proof?

I know similar questions exist, but I want to know, if this is a valid proof. A prime number has 1 and it self as divisors. So the subset of a prime number is $ D(p) = \{1, p\}$ Now I want to prove $...
1
vote
0answers
16 views

Are some irrational questions more irrational than others? (A-Level EPQ)

I'm currently researching the question 'Are some irrational questions more irrational than others?' for an extended project qualification I aim to write. My research has led me to seeing how spokes on ...
4
votes
2answers
113 views

Proving the density of a function in an interval.

I am reading Steven G. Krantz's Real Analysis and Foundations and came across this problem. Problem: Let $\lambda$ be a positive irrational real number. If $n$ is a positive integer, choose by the ...
3
votes
2answers
57 views

Is there an example of $\mathbb{R} \to \mathbb{R}$ function that produces only rationally independent numbers?

Is there a function $f(x): \mathbb{R} \to \mathbb{R}$ such that $$nf(x) \neq mf(y) \quad \forall\ x,y \in \mathbb{R},\ n,m \in \mathbb{Z},\ x\neq y ?$$
0
votes
0answers
21 views

Is Inducing Limits in Irrationality proof valid?

first of all, I want to say that I am pretty new to math. So, if there are any mistake, please told me. I attempt to proof that $e^{e}$ is irrational. $e^{e} = e^{1}*e^{1}*e^{1/2!}*e^{1/3!}*...$ ...
0
votes
0answers
28 views

A question about the supremum of a rational set

If we have a set $A$ that is defined as a non empty set in rational numbers. Then we use it's $\sup(A)$ to define real numbers as it is not always in the rationals. My question is, does this mean ...
14
votes
0answers
198 views

If $r\in\mathbb{Q}\setminus\mathbb{Z}$ is it possible that $r^{r^{r^r}}\in \mathbb{Q}$?

It's straightforward to prove that $r^r\notin\mathbb{Q}$, which furthermore allows us to use the Gelfond-Schneider theorem to prove that $r^{r^r}\notin\mathbb{Q}$. Is it true that $r^{r^{r^r}}\notin\...
3
votes
5answers
248 views

“If $a+a^3$ is irrational, then $a+a^2$ is also irrational”.

Given $a\in\mathbb{R}$, define $p=a+a^2$ and $q=a+a^3$. Show that the following statement is false: "If $q$ is irrational, then $p$ is also irrational". My approach: I just was lucky to find a ...
0
votes
1answer
62 views

Multiplication of two irrational to give rational

I know that we can obtain any rational $r\in\mathbb{R}\setminus\{0\}$ by the multiplication of two irrational numbers. There are many beautiful answers to this here. But I want to know that if there ...
0
votes
2answers
152 views

What is the solution of $\frac{x^3}{y^3+z^3}+\frac{y^3}{x^3+z^3}+\frac{z^3}{y^3+x^3}=1$ in rational?

I have tried to solve the following equation in rational($x,y,z$ are rational numbers) $$\frac{x^3}{y^3+z^3}+\frac{y^3}{x^3+z^3}+\frac{z^3}{y^3+x^3}=1$$ , I can't find such pairs of rational numbers $(...
0
votes
0answers
38 views

Is it possible to put pi into terms of rationals [duplicate]

I don't know of that is the correct terminology but by this I mean in the way that $\phi$ can be put into the equation $(1+\sqrt 5)/2$. I have heard that it has never been achieved but I was wondering ...
3
votes
0answers
47 views

A Question about rationality , irrationality or Transcendence of definite integral

( this is my first question on the site so please forgive any possible mistake ) Consider integral of the form : $$\int_0^\infty f(x)dx$$ Can we have a set of conditions of $f(x)$ such that ...
0
votes
1answer
51 views

proving $\sqrt{a}+\sqrt[3]{b}$ where $a$ and $b$ are integers is rational only if $a$ is a perfect square and $b$ is a perfect cube.

This question was in an oral entry exam for a university in Italy. I have tried to procede by contradiction but don't seem to get anywhere. How can a question like this one be approached?
1
vote
4answers
29 views

Given the binary operation $x*y=x^2+4xy+y^2$ show that $a*1 \in \mathbb{N}$ has infinitely many solutions, where $a$ is irrational.

Consider the binary operation: $$x * y = x^2 + 4xy + y^2$$ defined on $\mathbb{R}$. I have to show that there are infinitely many irrational numbers $a$ such that $a * 1$ is a natural number. This ...
1
vote
1answer
75 views

Intuition for understanding irrational numbers

Someone once told me that numbers such as $\sqrt{2}$ have a unique mathematical meaning: $$\sqrt{2}=\lim\{1,1.4,1.41, ... \}$$ While I understand that this might be sufficient for a formal ...
-1
votes
2answers
69 views

Proving that $\sqrt[n]{m}$ is irrational [duplicate]

I want to make a proof that $\sqrt[n]{m} \quad \forall \quad m,n \in \Bbb N$ is irrational if $m$ cannot be expressed as a perfect $n^\text{th}$ power of some other number $z \in \Bbb N$. Begin by ...
1
vote
2answers
36 views

Is it true that if $p,q \in K_{>0}$, then $[0,pq) \subseteq [0,p)[0,q)$ in any ordered field?

It seems possible that the following holds: Conjecture. Let $K$ denote an ordered field. If $p,q \in K_{>0}$, then $[0,pq) \subseteq [0,p)[0,q).$ More precisely: If $p,q \in K_{>0}$,...
10
votes
2answers
142 views

Proof $\sqrt{2}$ irrational using last digits

"This proof was found by Sergey Markelov when yet in high school. In the decimal system, a square of an interger may only end in one of the following digits: $0$, $1$, $4$, $5$, $6$, $9$ whereas twice ...
2
votes
1answer
77 views

Dirichlet approximation theorem: How “sporadic” are the good denominators?

For an irrational number $\alpha \in \Bbb{R} \setminus \Bbb{Q}$, the Dirichlet approximation theorem states that there are infinitely many irreducible fractions $\frac{p}{q}$ such that $|\alpha - \...
2
votes
1answer
57 views

Is it possible to have $\text{transcendental}^\text{transcendental} =\text{algebraic}$?

We know that it possible to have $\text{irrational}^\text{irrational}=\text{rational}$. To verify the possibility of this, we consider the expression $\sqrt{2}^\sqrt{2}$, if it is rational, then this ...
0
votes
2answers
66 views

Secret pattern of irrational numbers

Are their any pattern in irrational numbers?, I know that there's been no hint as to the appearance of their infinite digits, but I discovered a pattern within themselves $\sqrt(1) = 1$ $\sqrt(2) = ...
0
votes
1answer
15 views

Irrationality Triples, Mathematics by Experiment, Borwein & Bailey (2003)

I am stuck on the following example/problem in the book and am seeking clarification: Page 76: ...
0
votes
0answers
17 views

Can I extend this definition from rationals to reals?

A function $f$ is known to be continuous and its (simple) functional form is known only for rational numbers. Does it imply that the same functional form is valid for any real number? For example, I ...
1
vote
3answers
43 views

If $r \ne 0$ is rational and $i$ is irrational, then $ri$ is irrational [duplicate]

Prove the following: The product of a nonzero rational number and an irrational number is also irrational. I assumed the following: Let $r = c/d$ be rational, where $c$ and $d$ are integers ...
1
vote
1answer
49 views

Proof of irrationality of $\arcsin(\frac{1}{4})$

I was working to find a different approach to Niven's theorem from the one in my textbook taking a route via Chebyshev polynomials. It all comes to proving the irrationality of $\arcsin(\frac{1}{4})$ ...
-1
votes
2answers
36 views

A question about rational and irrational

The original statement is: "Suppose x and y are irrational. Prove or disprove the following statement: Either x + y or xy is irrational." So I use contraposition to prove it which is: If x + y and ...
0
votes
0answers
31 views

Number of rational points $(a,b)$ on the circumference of a circle with centre as $(\pi,e)$

What is the number of rational points $(a,b)$ (a point is considered to be rational if both $a$ and $b$ are rational) on the circumference of a circle with center as $(\pi,e)$? My line of reasoning ...
5
votes
1answer
42 views

Is this a valid proof that $e$ is irrational?

We will start with the known identity that $$e=\sum_{n=0}^\infty\frac{1}{n!}$$ and assume for contradiction that $e=\frac pq$. Then we have that $p=qe$, where $p$ is an integer. So then $q!e$ is ...
3
votes
2answers
96 views

Finding a monic polynomial with integer coefficients having $\sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{7}$ as one of its roots.

The original question is to prove that $\sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{7}$ is irrational, which already has many answers. But one of the answers to that question used this method: They found ...
-1
votes
1answer
65 views

Can there be a continuous function $f:\Bbb R→\Bbb R$ where $f(x)\in\Bbb Q$ $\Leftarrow\Rightarrow$ $f(x+1)\notin\Bbb Q$ [closed]

Prove that there exists at least one continuous function with the specified property, or proof that no such function can exist. The property is equivalent so if $f(x)\in\Bbb Q$ then $f(x+1)\notin\Bbb ...

1
2 3 4 5
40