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 ($e,\pi$).
3
votes
3answers
96 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
84 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
187 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
64 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
39 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?
25
votes
2answers
426 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
69 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
82 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
56 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 ...
4
votes
1answer
110 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 ...
-1
votes
3answers
152 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 $q_i,p\in \mathbb{Q}$ ...
-1
votes
2answers
90 views
$\mathbb{R} \setminus \mathbb{Q}$:'a stamping tool' [closed]
What does it mean for the polynomial
$$ a_1x_1+\cdots+a_nx_n=b $$
to have solutions in $\mathbb{R} \setminus \mathbb{Q}$, where $a_i,b\in \mathbb{Q}$?
0
votes
3answers
76 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
79 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
57 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 ...
20
votes
1answer
334 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
63 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?
3
votes
1answer
77 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 ...
11
votes
0answers
163 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
67 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
38 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
47 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 $x$ ...
8
votes
2answers
82 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
51 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
59 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 ...
1
vote
1answer
96 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 ...
8
votes
1answer
212 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 ...
13
votes
1answer
155 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
53 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 ...
8
votes
4answers
215 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
117 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
66 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
147 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
3answers
100 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
114 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
77 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 ...
4
votes
1answer
69 views
Do irrational number contain infinate/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
41 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 ...
0
votes
2answers
52 views
Supremum of set not in the set?
Please help me understand the question with solution.
Consider $S = \{ x \in \mathbb Q : -1 < x < \sqrt 2\}$. Show that $\sup S = \sqrt 2$. $\sup S \in S$ in this case?
1
vote
1answer
38 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
149 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
76 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
51 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
121 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
152 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 $p$.
14
votes
3answers
280 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
65 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
84 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
67 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)?
15
votes
10answers
1k 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.
$$\Big(\frac{m}{n}\Big)^2 = 12$$
...
