Tagged Questions

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 ($e,\pi$).

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2
votes
3answers
711 views

(How to/Can I) show irrational numbers?

This might sounds stupid, but I really don't know can I show Irrational numbers in proves? And if so, how to show it? For example, when I want to show Rational numbers, I do this: $\frac{m}{n} $ , ...
3
votes
3answers
215 views

Closed form representation of an irrational number

Can an arbitrary non-terminating and non-repeating decimal be represented in any other way? For example if I construct such a number like 0.1 01 001 0001 ... (which is irrational by definition), can ...
3
votes
3answers
2k views

Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational. [duplicate]

Possible Duplicate: $\sqrt a$ is either an integer or an irrational number. I have this homework problem that I can't seem to be able to figure out:Prove: If $n\in\mathbb{N}$ is not the ...
0
votes
1answer
8k views

Online tool to check if number is rational or irrational?

I am new to this forum. I've been programing for some time, and now starting my engineering degree. I am trying to find an online utility that will help me grasp the concept of irrational numbers ...
1
vote
1answer
189 views

Arbitrary Sequence of Digits in Irrational Number

What are numbers in which we can find arbitrary sequence of digits (in a certain base-$n$ expansion)? I know that $0.123456789101112131415\cdots$ does (and its analogues in other bases), but does this ...
17
votes
2answers
1k views

Why is $\pi$ irrational if it is represented as $c/d$?

$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction. Then why is $\pi$ an irrational number?
2
votes
1answer
71 views

Approximate rational dependence

After seeing question Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer? I wonder if there is an algorithm that can find approximate rational dependence?! I pick any irrational numbers ...
5
votes
1answer
1k views

$\log_7 n$ is either an integer or an irrational number

Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number. I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does ...
2
votes
3answers
2k views

Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational [duplicate]

Possible Duplicate: $\sqrt a$ is either an integer or an irrational number. I'm a total beginner and any help with this proof would be much appreciated. Not even sure where to begin. ...
6
votes
1answer
624 views

Deciding whether $2^{\sqrt2}$ is irrational/transcendental

Is $2^\sqrt{2}$ irrational? Is it transcendental?
8
votes
1answer
239 views

Irrationality of a series

Here is a series in which $m \geq 2 $. I want to ask how to prove the below series is irrational: $$\sum _{n=1}^{\infty} \frac{1}{m^{n^2}}$$
3
votes
0answers
92 views

how many numbers of irrationality measure $x$

Does there exist $x>2$ such that uncountably many reals have irrationality measure x? Must there exist at least one number of irrationality measure $x$? related question on sets of constant ...
20
votes
3answers
1k views

What is the simplest way to prove that the logarithm of any prime is irrational?

What is the simplest way to prove that the logarithm of any prime is irrational? I can get very close with a simple argument: if $p \ne q$ and $\frac{\log{p}}{\log{q}} = \frac{a}{b}$, then because ...
4
votes
1answer
174 views

Is it assumable that $2^{1/12}$ is irrational because $2^{1/2}$ is?

I need to prove that $2^{1/12}$ is irrational but I need to connect this to $2^{1/2}$ being irrational. I know how to prove that $2^{1/2}$ is irrational, but can I assume that $2^{1/12}$ is irrational ...
5
votes
3answers
610 views

Is this proof that $\sqrt 2$ is irrational correct?

Suppose $\sqrt 2$ were rational. Then we would have integers $a$ and $b$ with $\sqrt 2 = \frac ab$ and $a$ and $b$ relatively prime. Since $\gcd(a,b)=1$, we have $\gcd(a^2, b^2)=1$, and the fraction ...
2
votes
4answers
1k views

Why doesn't the indirect proof of irrational roots apply to rational roots?

When trying to prove that a particular root (say $\sqrt{2}$ or $\sqrt{10}$) cannot be rational, I always see a particular indirect proof that goes something like this: Suppose $\sqrt{x}$ were ...
30
votes
1answer
3k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
21
votes
3answers
14k views

The sum of irrationals is irrational?

If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational? Thanks for your help
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
383 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
113 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
501 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
232 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
440 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
258 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
549 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
88 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
392 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
203 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
670 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
237 views

Is a transcendental number necessarily irrational?

Being transcendental implies necessarily being irrational?
2
votes
2answers
638 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
335 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
583 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
2k 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
531 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
718 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. ...