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.

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5
votes
1answer
151 views

If $q>1$ is not an integer, can $q^n$ be made arbitrarily close to integers?

This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to ...
11
votes
4answers
6k views

Prove that $\sqrt 5$ is irrational

I have to prove that $\sqrt 5$ is irrational. Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common ...
0
votes
1answer
83 views

Transcendental proofs vs. Irrational proofs

Why are proofs of the transcendence of certain numbers usually harder than irrationality proofs of those same numbers (for example, Lindemann's proof of the transcendence of pi vs. Niven's proof of ...
2
votes
3answers
819 views

Positive integer multiples of an irrational mod 1 are dense [duplicate]

I'm not sure how to solve this one. Thank you! $2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha ...
22
votes
2answers
2k views

Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
17
votes
7answers
7k views

What rational numbers have rational square roots?

All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$). My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer ...
1
vote
1answer
54 views

How do you show this isomorphism?

How do you show that $\mathbb{R} \backslash \mathbb{Q} \cong \mathbb{N}^{\mathbb{N}}$? What is a good starting point in showing this?
17
votes
3answers
4k views

irrationality of $\sqrt{2}^{\sqrt{2}}$.

The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved ...
0
votes
0answers
57 views

Any irrational number can be raised to a power so that the result is an integer number [duplicate]

Does it hold in general, that for every irrational number there exists a power to which when raised, the result will be an integer? Does there exist a counterexample, of which it can be showed that no ...
7
votes
4answers
1k views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
3
votes
4answers
235 views

What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$?

In this proof extracted from the Wikipedia A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it could be expressed as ...
0
votes
1answer
42 views

How to solve the following problems with exponent?

If $9^{x+2}= 240+9^x$ then x= ? $10^x = 64$ what is the value of $10^{(x/2)+1} = ?$ $x/x^{1.5} = 8*x^{-1}$ and x > 0 , then x = ? $x^{-2} = 64$, then $x^{1/3} + x^0$ = ? $4^x - 4^{x-1} = 24 $ then ...
0
votes
2answers
2k views

Numbers that cannot be expressed as fractions

What are Numbers that cannot be expressed as Fractions called?
2
votes
1answer
63 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 $(a-b)k - bf$?
4
votes
0answers
146 views

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

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 ...
6
votes
3answers
693 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.
16
votes
2answers
462 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
476 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
133 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
258 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
88 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
541 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
580 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 ...
15
votes
4answers
7k 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
42 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 $x^3-x-1=0$?
-9
votes
5answers
1k views

How does the number $ e $ relate to the limit of $(1+1/n)^n$ as $n \to \infty$? [closed]

The number $e$ is sometimes defined as $\lim_{n\to\infty}\left(\frac{n+1}{n}\right)^n$. Does this mean we can say that $e = \left(\frac{\infty + 1}{\infty}\right)^{\infty}$? Why or why not? And what ...
1
vote
5answers
173 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
votes
3answers
423 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.
3
votes
1answer
488 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
2answers
787 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 ...
2
votes
2answers
1k 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?
6
votes
2answers
675 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
297 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
75 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?
41
votes
2answers
2k 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
160 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?
2
votes
3answers
399 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
622 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 ...
10
votes
7answers
577 views

When is an Integer a Rational Number, and are All Ratios Rational, Even $\frac{\sqrt{7}}{2}$?

$$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$ In other words, a rational number is a number that can be written as one integer over another. ...
1
vote
3answers
2k 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..
4
votes
2answers
134 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
92 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
718 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
186 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
197 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
589 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
149 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
62 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
79 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 $x$ ...