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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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
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3answers
568 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
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2answers
104 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.
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1answer
83 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 ...
30
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1answer
605 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
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2answers
120 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
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1answer
112 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
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1answer
502 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
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1answer
114 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
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1answer
57 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 ...
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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 $x$ ...
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2answers
178 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
206 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
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1answer
93 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
130 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
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1answer
321 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 ...
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1answer
332 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?
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3answers
180 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 ...
9
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4answers
285 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
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4answers
598 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
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4answers
495 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
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3answers
1k 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. = ...
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4answers
431 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
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2answers
220 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
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2answers
149 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 ...
5
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1answer
159 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 ...
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1answer
53 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
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2answers
326 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?
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1answer
108 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
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2answers
267 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 ...
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2answers
124 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 ...
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1answer
143 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 ...
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2answers
894 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
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3answers
341 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 $p$.
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3answers
442 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
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1answer
372 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
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1answer
143 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
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2answers
103 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)?
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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
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2answers
797 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
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3answers
351 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 ...
11
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2answers
844 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
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2answers
185 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
2
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0answers
109 views

Irrationality of $\pi$ from the spigot algorithm?

The spigot algorithm for BPP formula gives hexadecimal digits of $\pi$ one at a time. Is it possible to prove directly that this algorithm cannot be computed with bounded-memory? (From R J Lipton It ...
3
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1answer
194 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
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5answers
568 views

Algorithms for “solving” $\sqrt{2}$

The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
4
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2answers
3k views

Is a non-repeating and non-terminating decimal always an irrational?

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ...
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5answers
243 views

Will we get all real numbers if we add all limits?

Consider a set of all rational numbers from 0 to 1 inclusive. If we add to this set all limits of all convergent sequences of these numbers, will we obtain a set of all real numbers from 0 to 1?
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236 views

Difference between irrational numbers with and without a pattern.

I'm not sure how to talk about what I want to talk about, so I'll give some examples. The number $\pi$ is irrational and has no repeating pattern, but is computable by an easy rule; divide the ...
3
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0answers
141 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...