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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3answers
69 views

Prove that there is no rational number solution for an equation.

Prove that there is no rational number solution to the equation $x^2-3x+1=0$. (Note, we do not assume that we know all the solutions of $x^2-3x+1=0$ are given by quadratic formula)
6
votes
5answers
195 views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
2
votes
1answer
47 views

What Will Happen Without Decimal Expansion?

After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers ...
2
votes
3answers
33 views

A good site documenting approximations of irrationals

I'm thinking of Sloane here but I believe that only takes sequences/series into account. Basically I've derived an interesting, appealing formula for e and want to know if it's already been ...
8
votes
0answers
150 views

Irrationality of $\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
5
votes
3answers
631 views

Is a cube root of a prime number rational?

The question is: if $P$ is prime is $P^{1/3}$ rational? I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube ...
1
vote
1answer
53 views

How do you prove $\sqrt{n}$ is an integer or it is irrational? [duplicate]

I have tried this problem five times but I keep getting stuck. I keep following the proof for $\sqrt{2}$. I know that I have to prove that the set is nonempty. Which I do by induction. $2^1 > 1$ ...
7
votes
2answers
151 views

Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?

I'm sure this gets asked all the time but I swear I googled with no useful result. What I'm looking for is a reasonably intuitive answer. Those two constants have some pretty interesting properties. ...
0
votes
1answer
39 views

representation of rational field

I want to know how is represented general form of rational field, for example definition of ${\mathbb Q}(\sqrt{2})$ is represented as $p+q \sqrt{2}$, where $p$ and $q$ are rational numbers, for ...
5
votes
3answers
306 views

Prove that 2.101001000100001… is an irrational number.

My try: This number is non-terminating and non-repeating, so this is an irrational number. But how do I prove it more formally in a more mathematically rigorous way?
2
votes
2answers
122 views

Proof of irrationality of $\dfrac{\sqrt{8}}{\sqrt{7}}$

We have to prove that $\dfrac{\sqrt{8}}{\sqrt{7}}$ is irrational(try not to use the Rational Root Theorem) At first,we prove that the expression is not an integer. ...
0
votes
5answers
209 views

What is the name of $0.\overline{0}1$

Short question: What is the name of the number closest but not equal to zero? Long question: Some programmers were discussing about the smallest number close to zero, which is ...
0
votes
1answer
64 views

Calculation of irrational numbers

I am looking for the fastest parallelized algorithm to calculate 1 billion decimal places of ANY irrational number, be it e, pi or any other. May main aim is to obtain 1 billion of non repeating ...
6
votes
0answers
116 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?
5
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4answers
286 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
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4answers
113 views

Help me to Prove that log2 3 is irrational. [closed]

seemingly simple homework assignment, help? Was never the best with logarithms, how would I go about proving? Sorry the question read IRrational!
3
votes
1answer
77 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
144 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
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1answer
57 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 ...
6
votes
4answers
337 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 in an 'optimal' and innovative way. And my question is: What should the teacher know ...
6
votes
1answer
130 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
85 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
91 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 ...
1
vote
0answers
49 views

The rational numbers can't be the set of points on which a real function is continous

Let $A$ be the set of all points on which $f:\mathbb{R}\rightarrow \mathbb{R}$ is continous. How can I prove that $A$ can never be $\mathbb{Q}$? edit: also, is it possible to generalize it for any ...
4
votes
3answers
193 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational zeros theorem, we can conclude that $ \pm 1$ and $ \pm 23 $ are the only possible rational solutions*. ...
12
votes
0answers
175 views

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

For $x\in\mathbb R$, let $\{x\} = x-\lfloor x\rfloor$ be the fractional part of $x$. I'm searching for an elementary proof that there exists a positive real number $s$ such that $$ \sum_{n = 1}^\infty ...
0
votes
0answers
52 views

Solutions of $3a^2-2b^2=1$ [duplicate]

The question is to find the integral solutions of $$3a^2-2b^2=1$$ In the solution it is given that: The solutions of $3a^2-2b^2=1$ can be obtained from ...
1
vote
5answers
283 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
112 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
74 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
330 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
72 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 ...
12
votes
4answers
165 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 ...
9
votes
1answer
177 views

Integer parts of multiples of irrationals

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

Do irrational numbers really exist?

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
84 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 ...
-4
votes
1answer
63 views

Proving rational numbers [closed]

Definition 33: Define a relation $\sim$ on $\mathbb Z \times (\mathbb Z \setminus \{0\})$ by setting $(a,b) \sim (c,d)$ if $ad - bc = 0$. Proposition: The relation $\sim$ defined above is an ...
3
votes
3answers
358 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
58 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 ...
4
votes
2answers
31 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
37 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. ...
9
votes
2answers
123 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
35 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
70 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
76 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
100 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
71 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
130 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
33 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
89 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 ...