Numbers not expressible as a ratio of two integers. Examples: $\sqrt{2},\phi,e,\pi,\zeta(3)$. Some of them are algebraic ($\sqrt{2},\phi$) and some transcendental ($e,\pi$).

learn more… | top users | synonyms

1
vote
1answer
99 views

When was the significance of $i$ first noticed?

Complex analysis is an entire field of mathematics that focuses on the use of the complex constant $i$. When was the significance of $i$, an imaginary number, first noticed? If I did not know some ...
8
votes
5answers
382 views

Arithmetic of irrationals and the Vedanta behind it..

I am really curious about the Vedanta behind the arithmetic operations on irrational numbers. It still got aggrevated after the productive discussions with my friend. So I decided to ask it here. ...
4
votes
0answers
112 views

Finding a closed expression for a calculated value.

Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value ($a$) I have found might be a quadratic ...
2
votes
1answer
120 views

Is there a rational univariat polynomial of degree 3 with 3 irrational roots?

The title pretty much asks my question: Does $f\in\mathbb{Q}[x]$ such that $$ f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3),$$ where $\alpha_1, \alpha_2, \alpha_3\in\mathbb{R}\setminus\mathbb{Q}\ $ ...
8
votes
2answers
1k views

Has Euler's Constant $\gamma$ been proven to be irrational?

I found a paper by Kaida Shi called "A Proof: Euler’s Constant γ is an Irrational Number" which claims to have proven the irrationality of $\gamma$. I know people have been trying to prove that ...
23
votes
5answers
3k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
4
votes
2answers
498 views

Use of the Reciprocal Fibonacci constant?

The Reciprocal Fibonacci constant ($\psi$) is defined as $$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$ where $F_{k}$ is the $k^{th}$ Fibonacci number. The irrationality of $\psi$ has been proven. ...
5
votes
1answer
225 views

Rotation $x \to x+a \pmod 1$ of the circle is Ergodic if and only if $a$ is irrational

I have a book, Ergodic problems of classical mechanics by Arnold/Avez, and in it they prove that rotation $Tx = x+a \pmod 1$ of the circle $M=\{x \pmod 1\}$ is Ergodic if and only if a is irrational. ...
16
votes
2answers
436 views

Which results depend on the irrationality of $\pi$?

Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it ...
5
votes
3answers
256 views

certain proofs of the irrationality of $\sqrt{2}$

I had the impression that there might be proofs of the irrationality of $\sqrt{2}$ that showed that $$ \left|\frac a b - \sqrt{2} \right| \ge (\text{something possibly depending on $a$ or $b$}) >0 ...
6
votes
3answers
2k views

For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$

I am not able to prove that this set is dense in $\mathbb{R}$. Will be pleased if you help in a easiest way, $\{a+b\alpha: a,b\in \mathbb{Z}\}$ where $\alpha\in\mathbb{Q}^c$ is a fixed irrational.
2
votes
4answers
544 views

Computing decimal digits of irrational numbers

How to compute the decimal digits of irrational number(non-transcendental) with an arbitrary precision? eg. Expansion of $\sqrt{ 2}$ with a precision of 500.
3
votes
0answers
84 views

Linear independence of reciprocals of logarithms

I would like to ask whether there is a proof of the following statement: Let $p$, $q$ be primes and $n$ positive integer coprime with $pq$. Then $\frac1{\log p}$, $\frac1{\log q}$ and $\frac1{\log n}$ ...
2
votes
3answers
388 views

Constructing the proof that $\sup \{x:x\in\mathbb Q \wedge x<\sqrt 2 \}$ doesn't exist.

I know there is a very well known proof that for any rational such that $$x=\frac p q < \sqrt 2$$ there exists another rational $y=\dfrac mn$ such that $$x=\frac p q < \frac m n <\sqrt 2$$ ...
1
vote
1answer
201 views

Is it transcendental? Also normal?

The number we are considering is as follows: $0.a_1 a_2 a_3 \cdots $, where $a_{2n-1}=(n)_{(2)}, a_{2n}=(n)_{(3)}.$ So, the number is $$0.(1)(1)(10)(2)(11)(10)(100)(11)(101)(12)\cdots.$$ Is the ...
3
votes
2answers
664 views

“GCD” of any two real numbers

This isn't really a GCD question, because GCD is only defined for integers. I'm interested in the the existence of a common divisor of any two non-zero real numbers. In other words can you prove or ...
5
votes
1answer
92 views

powers of $\frac{1+\sqrt a}2$

For any a which is not a perfect square, let $x=\frac{1+\sqrt a}2$. $x^n$ can be written uniquely as $b_nx+c_n$, where b and c are rational. Apart from $a=0, a=1, a= 1 \pm 2^m$ for $m>2$, are ...
-1
votes
1answer
236 views

Is a transcendental number necessarily irrational?

Being transcendental implies necessarily being irrational?
2
votes
2answers
637 views

Rational numbers- sticks and stones

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. For all rational numbers, we will have a stick of variable length ...
4
votes
4answers
333 views

Is there an algorithm that can tell whether the power of two rational numbers is rational?

It has been known since Pythagoras that 2^(1/2) is irrational. It is also obvious that 4^(1/2) is rational. There is also a fun proof that even the power of two irrational numbers can be rational. ...
2
votes
2answers
224 views

Is there a rational way to conceptualize an irrational number?

This is a request for help, not an attempt to challenge anything. Since $\pi$ is irrational, this tells me that it's impossible to express the distance around a circle in terms of the distance ...
11
votes
3answers
582 views

Irrationality of “primes coded in binary”

For fun, I have been considering the number $$ \ell := \sum_{p} \frac{1}{2^p} $$ It is clear that the sum converges and hence $\ell$ is finite. $\ell$ also has the binary expansion $$ \ell = ...
1
vote
0answers
116 views

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let's say $p$ is even. I'm interested in the asymptotic behavior of this for $n$ ...
3
votes
2answers
92 views

How do continuity, distance and irrationals arise from discreteness?

Consider a square as rendered on a computer screen: its width and height are $N$ pixels each, and its area is $N^2$ pixels. Its diagonal, when measured in pixels, is also $N$ pixels long. If you ...
59
votes
4answers
7k views

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
3
votes
5answers
3k views

How can one prove that the cube root of 9 is irrational?

Of course, if you plug the cube root of 9 into a calculator, you get an endless stream of digits. However, how does one prove this on paper?
20
votes
4answers
1k views

Uncountable set of irrational numbers closed under addition and multiplication?

Is such a thing even possible? There's not much to say really. Obviously if there was a set it would be full of transcendental numbers. This led me to think of a function generating transcendental ...
37
votes
6answers
3k views

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and ...
1
vote
3answers
509 views

Dedekind's method for irrational number

I am now reading the definition of irrational number, which we can describe by the following terms: suppose that we have divided all rational numbers into two classes, a lower class and an upper ...
2
votes
3answers
711 views

What is the ratio of rational to irrational real numbers?

There exists an infinite amount of rational and irrational numbers. But is there more irrational numbers than rational? And if so can a ratio of one to the other be calculated?
6
votes
1answer
3k views

Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...
23
votes
9answers
5k views

Prove $2^{1/3}$ is irrational.

Please correct any mistakes in this proof and, if you're feeling inclined, please provide a better one where "better" is defined by whatever criteria you prefer. Assume $2^{1/2}$ is irrational. ...
6
votes
0answers
324 views

Is ${^5\pi}$ an integer? [duplicate]

Possible Duplicate: How to show $e^{e^{e^{79}}}$ is not an integer Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it? Here ${^5\pi}$ means the result of tetration ...
23
votes
2answers
830 views

Is there any real number except 1 which is equal to its own irrationality measure?

Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
18
votes
3answers
962 views

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational. I can prove that $x$ is irrational by showing that it's a root of a polynomial with integer coefficients ...
0
votes
1answer
249 views

When are $\theta$ and $\sin\theta^\circ$ both rational? [duplicate]

Possible Duplicate: Sine values being rational I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it ...
7
votes
4answers
2k views

When is $\sin(x)$ rational?

Obviously, there are some points (like $\pi,30$) but I am unsure if there are more. How can it be proved that there are no more points, or what those points will be? EDIT: I largely meant to ask ...
10
votes
1answer
180 views

Is there a dense subset of $\mathbb{R}^2$ with all distances being incommensurable?

Is there a set $S$ of points on the real plane $\mathbb{R}^2$ such that: there is a point belonging to $S$ in any neighborhood of every point of $\mathbb{R}^2$ (so, $S$ is dense) and ratio of any ...
11
votes
1answer
136 views

Irrationality of Two Series

Show that if the integers $1<b_1<b_2<\cdots$ increase so rapidly that$$\frac{1}{b_{k+1}}+\frac{1}{b_{k+2}}+\cdots<\frac{1}{b_{k}-1}-\frac{1}{b_{k}},\quad k\geq 1,$$ then the number ...
1
vote
1answer
164 views

Does anybody know the formula for a quasicrystal structure?

I am an architecture student researching into quasicrystals with the hope of applying it to form a complex truss system. I was wondering if anyone new of a formula for the structure? thanks in ...
5
votes
4answers
315 views

Existence of irrationals in arbitrary intervals

I was studying for my analysis mid-term paper and was going over the properties of real numbers. I was wondering how to prove the following statement: (Not a textbook problem, it just popped into my ...
14
votes
2answers
3k views

ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles ...
1
vote
3answers
289 views

How to prove $e$ isn't a $\frac {a}{b}$. Not irrationality with other ways or about transcendental, only about fractions

I would like a proof that $e$ isn't a fraction $\frac{a}{b}$, for $a,b \in Z$ and $mdc(a,b)=1$. Just a observation =) I'd like a proof with fractions, not about $e$ irrationality or if $e$ is ...
1
vote
0answers
281 views

What is the exact definition of a rational power?

I was taught in school that $$x^{a/b} = \sqrt[b]{x^a}$$ however, wolfram says this is not always true: $\sqrt[3]{x^2} \ne x^{2/3}$ ...
21
votes
3answers
2k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
0
votes
1answer
116 views

non-perfect square of number [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I would like to know the better proof for the following one. question: non perfect square of any integer is an ...
0
votes
1answer
232 views

How to prove $\sqrt3$ is irrational? [duplicate]

How to prove $\sqrt3$ is irrational using Fermat's infinite descent method? Like says in Carl Benjamim Boyer's book. Isnt the same prove to $\sqrt2$, in Boyer's book says something like this. ...
11
votes
3answers
436 views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...
4
votes
0answers
151 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
4
votes
1answer
394 views

rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...