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

2
votes
2answers
618 views

How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?

How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?
5
votes
2answers
574 views

How to prove to be an irrational number? Like $\sqrt{2}$ $\sqrt{3}$ or $\sum\limits_{k=1}^{\infty} \frac{1}{n^2}=\pi^2/6$

As we know $\sqrt{2},\sqrt{3}$ are irrational numbers. And I see some proofs on the net. So I doubt that how $e,\pi$ or already known irrational numbers are proved to be irrational. In fact, I got ...
1
vote
2answers
171 views

How do I evaluate the following expression?

How to evaluate the following expression: $\displaystyle \frac{1}{\sqrt{2}+1}+ \frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}} +\cdots +\frac{1}{\sqrt{9}+\sqrt{8}}$
2
votes
4answers
72 views

What would be the value of $a$ and $b$ in following rational expression?

If $(5 + 2\sqrt{3})/(7 + \sqrt{3}) = (a - \sqrt{3b})$, How do I find the value of $a$ and $b$ where $a$ and $b$ are rational numbers?
36
votes
2answers
1k views

Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
1
vote
1answer
137 views

Area of a circle is $A = \pi r^2$. Is it possible that both $A$ and $r$ are perfect integers.

Can you produce an example where both the area of a circle and it's radius are integers?
1
vote
3answers
290 views

Direct proof for the irrationality of $\sqrt 2$. [duplicate]

Prove that $\sqrt 2$ is irrational using direct proof. I have seen TONS indirect proofs (e.g. proof by contradiction) for it, and people say that it's difficult to proof this directly. So is this ...
2
votes
2answers
530 views

Proof f(x) is continuous given $x$ rational and irrational.

How can I resolve the task below: Given $f(x)= \begin{cases} x, &x\in \mathbb{Q}\text{ }\\ 1-x, &x\notin \mathbb{Q}\text{ (irrational)} \end{cases}$, $0 \leq x \leq 1$. Show $f(x)$ is ...
0
votes
3answers
185 views

I'm just curious, what exactly is $\mathbb{R}\setminus\mathbb{Q}$? [duplicate]

What exactly is $\mathbb{R}\setminus\mathbb{Q}$? How many different kinds of things live in this place? For $n>1$ how does $$ q_1x_1+\cdots+q_nx_n=p $$ have a solution for $q_i,p\in \mathbb{Q}$ ...
0
votes
3answers
1k views

plot any irrational number on number line.

I have a basic question that can we plot any irrational number on number line?As I can plot all integers and rational number but how to plot any irrational number on it like $\sqrt2$,$\sqrt3$ etc..
3
votes
2answers
106 views

The sum of the series $\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$ is an irrational number

Let $\{\epsilon_n\}$ be a sequence where $\epsilon_n$ is either $ 1$ or $-1$. How could I Show that the sum of the series $$\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$$ is an irrational number.
1
vote
1answer
88 views

Can you produce a number like 1.01010101… by just addition and subtraction?

I'm working on a program in C# where a Decimal variable can hold negative and positive values including 0 and those values can only change by addition and subtraction. I have a conditional where if ...
31
votes
1answer
665 views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
3
votes
2answers
151 views

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?
2
votes
1answer
146 views

Can we take an $i$th root?

I misread this question and began thinking about the value $e^\pi$. This lead me to the Wikipedia on Gelfond's Constant, which suggests deriving a numerical value for $e^{\pi}$ by using Euler's ...
20
votes
1answer
550 views

Any proof to $\pi^{e}$'s irrationality?

I've searched for this for a while but get nothing... There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
2
votes
1answer
132 views

Infinite irrational number sequences?

Is an irrational number, such as $\pi$ or $\sqrt2$, guaranteed to contain every possible digit sequence somewhere within it? Is there no proof for this? Is there any clue as to whether this is so? It ...
2
votes
1answer
59 views

How uniform is the distribution of $n+sm$ for an irrational $s$?

It's not difficult to prove that for $s\in\mathbb{R}\setminus\mathbb{Q}$ the set $S=\{n+ms\;|\;n\in\mathbb{Z},\;m\in\mathbb{N}\}$ is dense in $\mathbb{R}$. When trying to solve this question, I come ...
1
vote
1answer
73 views

Looking for name of theorem: “rational $\Leftrightarrow$ fractional part terminates or repeats”

I am looking for the name of the theorem that says that a number $x$ is rational if and only if its fractional part terminates or repeats (where "fractional part" refers to the representation of $x$ ...
12
votes
2answers
271 views

Irrational numbers, decimal representation

Can this even be proved? (Or disproved?) Any irrational number without a 0 (zero) in its decimal representation is transcendental. Not sure where to start on this one...
0
votes
1answer
276 views

A calculator's solution to irrational exponent

An irrational number cannot be represented by $\frac{p}{q}$ where $p$ and $q$ are integers. And when we encounter exponents with decimal points, it is a possible way and a rather simple one to turn ...
5
votes
1answer
103 views

For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?

Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$? The original question concerned $a=e$, the usual ...
2
votes
1answer
131 views

Interesting question about irrational numbers

Find all solutions in un-ordered integers $(a,b)$ to $7-a-b=2\sqrt{10}-2\sqrt{ab}$. It would appear that the only solution to this is $a=2, b=5$. But how to prove this rigorously? Do irrational ...
11
votes
1answer
336 views

Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?

(somewhat related to my earlier question) Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$). The ...
19
votes
1answer
381 views

Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
0
votes
3answers
218 views

Show that there is no rational number $r=m/n$ such that $r^3=3$ [duplicate]

How do I solve this by prime factorization? I came across a similar problem on MSE just recently, but I can't find it and I thoroughly searched for it. If anyone can find it, please post it in the ...
11
votes
4answers
329 views

Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number

How can one prove that the series $\displaystyle \sum\limits_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number? There's no need to use Taylor expansion, integrals or any ...
3
votes
4answers
928 views

Non-existence of irrational numbers?

I realize the title of my question will probably cause the raising of some eyebrows, so let me explain. Not sure whether to file this under "math" or "philosophy". This also might be able to be ...
4
votes
4answers
669 views

Can you raise a Matrix to a non integer number? [duplicate]

So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
6
votes
3answers
2k views

Prove that the Tangent of 75 degrees equals 2 plus the square-root of 3

My (very simple) question to a friend was how do I prove the following using basic trig principles: $\tan75^\circ = 2 + \sqrt{3}$ He gave this proof (via a text message!) $1. \tan75^\circ$ $2. = ...
4
votes
4answers
543 views

On comparing fractions , fraction with smaller difference between numerator and denominator is greater than the other

A text book proposed that "when comparing fractions ,if the compared fractions's are such that numerator is smaller than denominator ,then fraction with more difference(absolute) between numerator ...
4
votes
2answers
281 views

Mystery about irrational numbers

I'm new here as you can see. There is a mystery about $\pi$ that I heard before and want to check if its true. They told me that if I convert the digits of $\pi$ in letters eventually I could read ...
2
votes
2answers
170 views

The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.

Problem Is there an irrational $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ such that the set $S= \{\,\{2^N\alpha\} :N\,\in\mathbb{N}\}$ is not dense in $[0,1]$. Here $\{x\}=x-\lfloor x\rfloor$ is the ...
6
votes
1answer
291 views

Do irrational number contain infinite/every patterns of sequences?

I guess the question is "does an 'infinite' number of patterns imply 'every' number of patterns?" For instance, if you could quickly calculate the decimal sequence of π, could you not (in ...
1
vote
1answer
55 views

rational fractions and the negative sign

Say we have the expression $$\frac{a}{b}=\frac{a+3}{b-8}$$ When we cross-multiply the terms we end up with $$a(b-8)=b(a+3)$$ If we try $a=-3$ and $b=8$ in the previous expression we get ...
3
votes
1answer
136 views

Extending the rationals using exponentiation

The set of integers can be constructed as an equivalence relation over the natural numbers using the the binary operation of addition, and a similar process yields the rationals from integers and ...
2
votes
2answers
383 views

Is there a proof that $\mathbb{R}$ is connected?

Is there a proof that the set $\mathbb{R}$ of all real numbers is connected? I've been assuming that $\mathbb{Q}$ is discrete, with a (very small) gap existing between any two elements ...
0
votes
2answers
154 views

Is the Copeland–Erdős constant a random number? How is it normal?

The Champernowne constant is not random. Is the Copeland–Erdős constant random? Also if Copeland–Erdős number is normal, then shouldnt the number of $5$s and even digits be low because they cannot ...
1
vote
1answer
162 views

Type of periodicity in champernowne constant.

Digits of Champernowne constant are aperiodic, else it will be rational. Fine! But it is not random because I can write a program which will give me the position of every digit. E.g. I can calculate ...
1
vote
2answers
1k views

Formula to reverse digits

Is there a formula that can be used to reverse the digits in a number, given a certain base b? E.G., $$F_{10}(32) = .23$$ $$F_{10}(123.456) = 654.321$$ If not, how can you write this out to show ...
0
votes
3answers
404 views

When a prime number p divides $ab$ then we have either p divides a or p divides b.Prove that $\sqrt {p} $ is not rational for any prime number p.

When a prime number $p$ divides $ ab $ then we have either $p$ divides $a$ or $p$ divides $b$. Prove that $ \sqrt p $ is not rational for any prime number $p$.
19
votes
3answers
532 views

Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational

Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums ...
0
votes
1answer
449 views

Surds - Finding square roots.

To find square root of surd like this : $a+\sqrt{b}+\sqrt{c}+\sqrt{d} $ We put it equal to $\sqrt{x}+\sqrt{y}+\sqrt{z}$ To find the square root of : $21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15} $ can we put ...
1
vote
1answer
178 views

Proof by contradiction that irrational numbers conform to $f(x)=x^2$

I am having some difficulty with this proof for my Real Variables class. I know that $f(x)$ is a continuous function defined on $R^1$, and $f(x)=x^2$ for any rational $x$. I also have the definition ...
2
votes
2answers
109 views

Computationally complex irrational numbers

Are there irrational numbers for which we know that computing its nth digit would take (at least) linear/polynomial/exponential/superexponential time (wrt to length of n and with "big enough" n)?
16
votes
10answers
2k views

Critiques on proof showing $\sqrt{12}$ is irrational.

My only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ ...
17
votes
2answers
809 views

Does $\sin(x)=y$ have a solution in $\mathbb{Q}$ beside $x=y=0$

Is there a way to show, that the only solution of $$\sin(x)=y$$ is $x=y=0$ with $x,y\in \mathbb{Q}$. I am seaching a way to prove it with the things you learn in linear algebra and analysis 1+2 ...
2
votes
3answers
450 views

Can an irrational number have a finite number of a certain digit?

This question came up because I was wondering the following: If the digits of PI are placed in ascending order, what is the <insert-large-finite-number-here>th digit? I believe that the answer ...
13
votes
2answers
1k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
4
votes
2answers
212 views

Exponentiation when the exponent is irrational

I am just curious about what inference we can draw when we calculate something like $$\text{base}^\text{exponent}$$ where base = rational or irrational number and exponent = irrational number