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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6
votes
0answers
146 views

Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
7
votes
4answers
625 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
-1
votes
4answers
1k views

Prove that $\log_2 3$ is irrational [closed]

Prove that $\log_2 3$ is irrational Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?
3
votes
1answer
141 views

$f$ differentiable, $f(x)$ rational if $x$ rational; $f(x)$ irrational if $x$ irrational. Is $f$ a linear function?

Let $f$ be an everywhere differentiable function whose domain consists of all real numbers. Assume that $f(x)$ is rational for rational $x$ and irrational for irrational $x$. Can we conclude that $f$ ...
3
votes
1answer
160 views

How is this a proof of the irrationality of $\sqrt2$

Proof. Suppose for the sake of contradiction that $\sqrt2$ is rational, and choose the least integer, $q \gt 0$, such that $(\sqrt2 − 1)q$ is a non negative integer. Let $q':=(\sqrt2 − 1)q$. Clearly ...
0
votes
1answer
93 views

Will negative bases with irrational exponents get a real or imaginary number?

Here are a few examples: $$(-1)^{\sqrt{2}},(-2)^{\pi},(-3)^{e}$$ From what I've learned, negative bases must have denominators of the exponent odd. Normally if we do $(-2)^{0.258}$ it would be the ...
1
vote
2answers
46 views

A question about limit and sequence

Let $\{p_n\},\{q_n\}$ be sequences of positive integers such that $$ \lim_{n \to \infty}\frac{p_n}{q_n}=\sqrt{2}. $$ Show that $$\lim_{n\to\infty}\dfrac{1}{q_n}=0.$$ I have no clue in proving this. ...
6
votes
5answers
1k views

Teaching irrational numbers?

I'm interested in teaching the irrational numbers to high-school students, and I need your ideas on how to do this in an 'optimal' and innovative way. And my question is: What should the teacher know ...
6
votes
1answer
255 views

Predicting digits in $\pi$

Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational? Basic assumption is that the person doesn't know a ...
1
vote
3answers
98 views

$x^2-p=0$, with $p$ prime, have irrational roots.

Unaware that $\sqrt{p}$ is irrational, prove that as $x^2-p=0$ have irrational root for $p$ prime? How would you use the criterion of Eisenstein?
3
votes
2answers
164 views

Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$

What is a polynomial $P(x)\in \mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$? I write $x=\sqrt[3]{2}+\sqrt[3]{3}$, so $(x-\sqrt[3]{2})^3=3$, but the expansion of the left side contains two cube ...
0
votes
2answers
298 views

Proof e is irrational — Choice of Sign of m/n = e and Uncanny Step

Assume $e$ is rational. Then, there exist coprime integers $m$ and $n$, and we can choose $n$ to be positive, such that: $\displaystyle \frac m n = e = \sum_{i \mathop = 0}^\infty \frac 1 {i!}$ from ...
22
votes
2answers
312 views

Is there an elementary proof that $\sum_{n=1}^\infty {1\over n^s\{n\pi\}}<\infty$ for some $s>0$?

Edit: David Speyer's answer made me realize a couple of things and I would like to clarify. Sorry if the length of this is getting out of hand. First, it is now clear that no estimate can be obtained ...
0
votes
5answers
608 views

Direct proof: $\sqrt{13}$ is irrational

Show that $\sqrt{13}$ is an irrational number. How to direct proof that number is irrational number. So what is the first step.....
2
votes
3answers
1k views

Proof that there are infinitely many irrational numbers!

I want to prove that there are infinitely many irrational numbers! How can I do that? I don't know where or how to start so any hint is appreciated. Thanks! :)
3
votes
1answer
96 views

Irrational number?

Is the solution of the equation $$x + \arctan(x) = \pi$$ irrational ? The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is ...
12
votes
1answer
397 views

Prove that this number is irrational

The number $a=0.12457...$ is defined as follows: The digit on the $n$-th place after the dot is the first digit left to the dot of the number $n\sqrt2$. For example, for $n=1$ we have ...
8
votes
1answer
119 views

Number made from ending digits of primes

Consider the number 0.23571379391713739171393971379371799173739113791379391173917133713717793 ... The number is formed by the ending digits of the prime numbers. Is it known whether this number is ...
13
votes
4answers
250 views

$\sqrt{2}\notin\mathbb{Q}$ but …

Ok, it's easy to prove that prime roots are irrational (i.e. $ \sqrt{p} \not\in \mathbb{Q}, \text{ if } p \in \mathbb{P} $) Consider $ \sqrt{2} $. We can quickly prove that $ \sqrt{2} \not\in ...
11
votes
1answer
243 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $$S(\alpha)=\big\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \big\}.$$ Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers. ...
-4
votes
2answers
449 views

Do irrational numbers really exist? [closed]

Isn't it possible that an irrational number is in reality the quotient of two infinitely long integers that even if there were repeating sections in it, it would take infinite digits to find the first ...
0
votes
2answers
265 views

Draw an irrational number on the number line (without pythagoras sentence)

Let's say im a guy for ancient greece and I only have a string and a pencil. And I want to draw a line, the width of the line is the square root of 6. And I only know how to draw a line in the width ...
1
vote
2answers
236 views

How to represent the square root of 90 as a fraction?

How to represent the square root of 90 as a fraction? I can usually do this with a calculator but it's not wanting to play nice. I need it to find the distance between two points but, ...
3
votes
3answers
949 views

Sum of all real number for any interval.

We know that sum of natural numbers over any interval always exists. For example sum from 0 to 10 of all natural numbers is $$S=\sum_{n=0}^{10}{n}=\frac{0+10}{2}\times{10}=55$$ But what about real ...
3
votes
2answers
66 views

Does $E^2 \; ( E \approx 1.2640847\ldots)$ equal $D \approx 1.5979102\ldots$?

Does $E^2=D$? Where $E$ is a constant used in the closed form of the Sylvester Sequence (see: Closed form formula and asymptotics) and $D$ is a constant for the closed formula of the sequence A007018 ...
18
votes
0answers
397 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Question : For every even $k\ge 4$, is the following $(\star)$ true? $$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb ...
4
votes
2answers
48 views

probability that a number defined by a random process is irrational

What if we write $0$. and then throw a coin and depending on the result continue the number with 1 or $0$ and continue this process indefinitely. It is clear that the result of this procedure is a ...
0
votes
1answer
58 views

recipe for infinitely many irrational numbers - or is it?

What if we write 0. and then throw a coin and depending on the result continue the number with 1 or 0 and continue this process indefinitely. It seems like a recipe for producing irrational numbers. ...
5
votes
1answer
115 views

Determine if the number $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ is rational

$ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ I have tried to raising it to the square, but I can't obtain the result. $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - ...
9
votes
2answers
177 views

$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
1
vote
0answers
38 views

Proving irrationality i p,s and k are primes number?

Can you prove that $\frac{k^{\frac13}-p^{\frac13}}{s^{\frac13}-p^{\frac23}}$ is irrational if p, k and s are different prime numbers. I am certain it is but i dont know how to prove it.
3
votes
2answers
77 views

Existence of five real numbers satisfying a given condition.

Let $a_1,\dots,a_5$ be five distinct non-zero real numbers. Suppose that for $i\neq j$ either $a_i+a_j$ or $a_ia_j$ or both are rational numbers, does it implies that $a_i^2$ are rational numbers for ...
2
votes
2answers
82 views

Does $\pi \ | \ 2 \pi$

Does $\pi$ divide $2 \pi?$ Clearly $\frac{2 \pi}{\pi}=2$ and 2 is an integer, so it would seem to make sense to say that $\pi \ | \ 2 \pi$. Does it make sense to write, for example, $$\pi \ | \ x ...
0
votes
1answer
212 views

Logic: Prove Log(base 9) 15 is irrational

Im having trouble with the following proof... Ill post what I have completed so far.. Prove $\log_915$ is irrational. Ill attempt by contradiction assuming $\log_915$ is rational. So, $\log_915 = ...
0
votes
1answer
113 views

The shape of a graph of a function with $n$th-roots?

Not just these type of functions: $$\sqrt[3]{x}=x^{1/3} \;\;\;\text{and} \;\;\; \sqrt[8]{x}=x^{1/8}$$ But also more complicated expressions, like expressions that have $n$th roots inside of ...
2
votes
1answer
250 views

Can you get any irrational number using square roots?

Given an irrational number, is it possible to represent it using only rational numbers and square roots(or any root, if that makes a difference)? That is, can you define the irrational numbers in ...
3
votes
1answer
78 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
2
votes
2answers
100 views

Can the exponentiation of an integer by a rational be a non-integer rational?

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. My question is the following: what is the nature of $n^r$? My first guess is that $n^r$ is an ...
1
vote
1answer
157 views

Subsets of real line without accumulation points; also, accumulation points of irrationals

I had to answer three questions about accumulation points. I think my work is correct, but I'd appreciate if someone would look over them for me. (I'm not sure if I read question 2 correctly.) In my ...
0
votes
0answers
1k views

The Conjugate Roots Theorem for Irrational Roots

The Conjugate Roots Theorem for Irrational Roots states that for a polynomial $f(x)$ with integer coefficients, if a root of the equation $f(x) = 0$ is expressed as $a+\alpha$, where $a\in\mathbb{Q}$ ...
1
vote
0answers
425 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
-1
votes
1answer
167 views

ZFC and irrational numbers [duplicate]

I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed? If not, are there other axiomatic set theories able to express ...
0
votes
3answers
96 views

The irrationality of the square root of 2 [duplicate]

Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?
-2
votes
1answer
338 views

how $\pi$ is irrational if it is a ratio [duplicate]

How can $\pi$ be an irrational number if it is a ratio of the circumference over the diameter? Thanks!
0
votes
3answers
233 views

Can there exist a function with discontinuity at Cantor's Set union Z?

I know there can't exist function with discontinuities only at irrational points,since cantor set is also uncountable like irrational numbers,I thought that the answer is no. Also if yes can you give ...
0
votes
3answers
294 views

Proving that $\sqrt{3} +\sqrt{7}$ is rational/irrational [duplicate]

I took $\sqrt{3}+\sqrt{7}$ and squared it. This resulted in a new value of $10+2\sqrt{21}$. Now, we can say that $10$ is rational because we can divide it with $1$ and as for $2\sqrt{21}$, we divide ...
2
votes
7answers
943 views

Why is a repeating decimal a rational number?

$$\frac{1}{3}=.33\bar{3}$$ is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ...
2
votes
1answer
326 views

When to rationalize numerator and/or denominator? [duplicate]

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
3
votes
0answers
187 views

Minkowski's question mark function iterations

The Minkowski's question mark function (we use the sign $?$ to note this function) was designed in 1904 by Minkowski. It can be defined as an increasing bijection between $\mathbb Q$ and the set of ...
2
votes
1answer
73 views

If $(F_n)$ is increasing and $\lim_{n\to\infty}\frac{F_1\dotsb F_n}{F_{n+1}}=0$ then $\sum\limits_{n=1}^\infty\frac1{F_n}$ is irrational [closed]

Let $F_n$ be integers, and $F_1<F_2<\cdots<F_n<\cdots$. Suppose that $$\lim_{n\to\infty}\frac{F_1F_2\cdots F_{n-1}}{F_n}=0.$$ Prove then $$\sum_{n=1}^\infty \frac{1}{F_n}$$ is convergence, ...