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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0
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2answers
755 views

Numbers that cannot be expressed as fractions

What are Numbers that cannot be expressed as Fractions called?
2
votes
1answer
57 views

Irrational sum to integers?

Is it possible for $(a-b)k + bf$ to be an integer if $k,f$ are irrational numbers and $a,b$ are integers? What about $(a-b)k - bf$?
2
votes
0answers
117 views

Can every string of numbers be found in the number pi (cfr. infinite monkey theorem)?

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of ...
5
votes
3answers
660 views

Is $e^{i+\pi}$ irrational or not?

Since we know that the value of $e$, $i$, and $\pi$ are irrational reals, how about $$e^{i+\pi}\;?$$ Is it still irrational (that is, not a Gaussian rational)? The problem make me curious until now.
87
votes
1answer
20k views

Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can ...
14
votes
2answers
394 views

What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime?

What is the value of $\sum\limits_{i=1}^\infty\dfrac{1}{p_{p_i}}$ where $p_n$ is the nth prime (and so $p_{p_n}$ is the $k$th prime, where $k$ is the $n$th prime) ? Thus ...
12
votes
7answers
418 views

Numbers with no finite representation on paper

It occurred to me that there must be a lot of numbers without any form of finite representation on paper. Is there a name for these numbers? For example... Integers and rationals have a very simple ...
3
votes
3answers
112 views

More three-term arithmetic progression questions

This question was inspired by a recent question about whether $\frac1{2}$, $\frac1{3}$ and $\frac1{15}$ can be (possibly non-consecutive) terms in an arithmetic progression. My question(s): Which of ...
2
votes
1answer
233 views

Geometric proof that if n is a non-perfect square, then √n is irrational.

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere. ...
2
votes
2answers
80 views

Could you prove convergence?

Two questions: Prove $a_n$ is bounded by 2 if $a_1=0, a_2=\sqrt2,\ldots,a_{n+1}=\sqrt{2+a_n}$ Prove (I've already proven $s_n$ is monotonically increasing and bounded above by 3) $\lim_{n \to ...
14
votes
3answers
376 views

Is there a simple proof that $e^2$ is irrational using a positional numeral system?

My favorite proof that $e$ is irrational goes something like this. Observe that we can write any real number $r$ as $$ a \,+\, \frac{b_2}{2} \,+\, \frac{b_3}{3!} \,+\, \frac{b_4}{4!} \,+\, ...
0
votes
1answer
213 views

Check my answer: $\Bbb Q$ is neither open nor closed

I have a so easy question. I have done it's answer by myself. I want you to only check my answer please. Does there exist any mistake or the missing? The set of irrational numbers - $\Bbb Q$ is ...
10
votes
4answers
3k views

Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
1
vote
1answer
38 views

Does $x^2>x^3+1 \implies x < -{1\over P}$?

How could one prove that: $$x^2>x^3+1 \implies x < -{1\over P}$$ where $P$ denotes the plastic constant, the unique real root of $x^3-x-1=0$?
-10
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5answers
798 views

What is $ e $? How does $ e $ relate to its limit as $n \to \infty$? [closed]

Why does $\left(\frac{\infty + 1}{\infty}\right)^{\infty} = e$? Does this account for the disparity between the countable and uncountable $\infty$? Why?
1
vote
5answers
140 views

Find two rationals, one greater and one smaller than a given irrational number.

Given an irrational number 0 < i < 1. Find two rational numbers a and b such that 0 < a < i < b < 1.
9
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3answers
409 views

Is there a (real) number which gives a rational number both when multiplied by $\pi$ and when multiplied by $e$?

Besides $0$ of course. What about addition and exponentiation? I would think there's no such number, but I'm not sure if I could prove it.
1
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1answer
276 views

Non-integer bases and irrationality

I read somewhere: When it comes to properties like prime, irrational, rational, divisible by 2, etc., nothing changes when you change base. But I'm not sure about the rational/irrational one. ...
4
votes
3answers
455 views

Are all integer fractions rational?

Any repeating decimal can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers. But is the reverse true. Will any fraction $\frac{a}{b}$ where $a$ and $b$ are integers produce a ...
1
vote
2answers
405 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
519 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
141 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
62 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?
35
votes
2answers
956 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
128 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
232 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
464 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 ...
5
votes
1answer
334 views

Linear equations; real solution; rational solution?

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have ...
0
votes
3answers
179 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
985 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
105 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
86 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
649 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
134 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
135 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
531 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
119 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
242 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
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1answer
235 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
98 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
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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
330 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
368 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
186 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 ...
10
votes
4answers
315 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
726 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
578 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 ...