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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3
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2answers
101 views

Understanding proof that $\pi$ is irrational

Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part: Since $n!f(x)$ has integral coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and ...
0
votes
1answer
25 views

Farey sequence problem with irrational numbers

If an irrational number $\theta$ lies between two consecutive terms $a/b$ and $c/d$ of the Farey sequence of order n, prove that at least one of the following holds: $|\theta- a/b| < 1/2b^2$ or ...
3
votes
3answers
141 views

Check that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational

How to prove that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational. I will appreciate any proof, but I had such exercise during lecture in field theory. Thanks.
3
votes
2answers
81 views

How to prove that: $\sqrt{25!+3} \in \mathbb{R}\setminus\mathbb{Q}$

How can I prove that: $$\sqrt{25!+3} \in \mathbb{R}\setminus\mathbb{Q}?$$ Thanks!
3
votes
5answers
219 views

Prove the irrationality of $0.235711131719…$

How can I prove that the number formed by concatenating the primes in order i.e. $0.235711131719...$ is irrational. I know I have to demonstrate that it has no period, but I'll be so thankful if ...
3
votes
1answer
47 views

Cantor's proof on uncountability of irrationals

I have a question regarding Cantor's proof that the set of irrational numbers is uncountable. As far as I know, to prove this, Cantor proves that there exists a mapping between irrationals and ...
5
votes
2answers
441 views

Is the Nested Radical Constant rational or irrational?

Given the sequence $A_n=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$: Are there any known rational elements in $A_n$, or has it been proved that all are irrational? Is there any proof for ...
2
votes
1answer
67 views

Irrational fractional parts - patterns

Some fractional-part list plots are: $\text{listplot of }|[\pi x]-\pi x|\text{, for }x \in \mathbb{Z} \text{ and } \text{listplot of }|[ex]-ex|\text{, for }x \in \mathbb{Z}$ $\text{listplot of ...
2
votes
0answers
45 views

How to represent an algebraic number in algorithm

I'm studying the problem of computing $n$th digit of given (irrational) algebraic number. Is there some canonical way to represent an algebraic number as a string I could use in computing? Basically ...
2
votes
0answers
72 views

Question about $e^{e^{e^e}}$

Is there a proof that the power tower of length $4$ of $e$ is irrational? Is it known whether or not $$e^{e^{e^e}}$$ is transcendental?
8
votes
2answers
235 views

Is $\ \Large\pi^e$ rational?

Is the number $\ \Large\pi^e$ rational?
6
votes
3answers
142 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
2
votes
2answers
78 views

Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$

Question : Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$. My attempt: Definition 2.7.8 states that a number $s$ is an algebraic number when there exists some $p \in\Bbb Z[x]$ such that ...
3
votes
3answers
145 views

How can you proof that the sum of three roots is irrational?

I would like to know how to prove that $\sqrt{2} + \sqrt{5} + \sqrt{7}$ is an irrational number. I know how to do the proof for a sum of two roots. Can I just define $\sqrt{2} + \sqrt{5} :=c$ and then ...
3
votes
5answers
124 views

$\sqrt[3]{2}$ is not the root of a quadratic with rational coefficients?

How can one show that $\sqrt[3]{2}$ is not a root of a quadratic with rational coefficients? It is clear that if $\sqrt[3]{2}$ is the root of such a quadratic, then it is also the root of a quadratic ...
1
vote
1answer
111 views

2.71828. And then another 1828.

This may qualify as the silliest math.SE question ever, but am I really the first person ever to worry about this? The decimal expansion of $e$ has a 2. And then a 7. And then a 1828. And...well, then ...
2
votes
2answers
67 views

Values $nx - [nx]$ are distinct for an irrational number

When reading the proof for Dirichlet's Approximation Theorem, I came across the following statement: If $x$ is irrational, then $nx - [nx]$ are distinct for all $n \in \mathbb{Z}$. I don't ...
0
votes
3answers
128 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
0
votes
1answer
35 views

If $\frac1\alpha+\frac1\beta=1$, irrational, then $\{\lfloor n\alpha\rfloor:n\in\Bbb N\}\uplus\{\lfloor n\beta\rfloor:n\in\Bbb N\}=\Bbb N$

Let $\alpha,\beta\in\Bbb R\setminus\Bbb Q$ such that $\frac1\alpha+\frac1\beta=1$, and define $S(x)=\{\lfloor nx\rfloor:n\in\Bbb N\}$. (Note that my convention takes $0\notin\Bbb N$.) The claim is ...
7
votes
3answers
113 views

How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$

A few month ago I had to prove $\lambda(\mathbb{Q}) = 0$ (where $\lambda$ is the one-dimensional Lebesgue measure). The idea: Let $\varepsilon \gt 0, r_n := \frac{\varepsilon}{2^n}$ and $\mathbb{Q} = ...
1
vote
1answer
49 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
1
vote
1answer
95 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
4
votes
1answer
72 views

Do rational and irrational numbers flip-flop?

I have found out that between every 2 rational numbers there is an irrational number, and between every 2 irrational numbers, there is a rational number. Does this mean that the rational and ...
2
votes
1answer
130 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
0
votes
1answer
47 views

rational numbers

It is by the rule that a rational number can be expressed in P/Q form and an irrational number cannot be expressed in such form. It is even said that (in order to promote the P/Q form for expressing ...
5
votes
3answers
306 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi = \int_{0}^{1}\frac{x^{4}\left(1 - x\right)^{4}}{1 + x^{2}}\,{\rm d}x $$
2
votes
1answer
32 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
4
votes
2answers
117 views

Generalizations of the golden and silver ratios, and their significance

$\Phi$, or the golden ratio, is basically $\frac{a+b}{a}=\frac{a}{b}$. The silver ratio corresponds to a similar idea of: $\frac{2a+b}{a}=\frac{a}{b}$. I've read on Wikipedia that both of these ratios ...
0
votes
0answers
79 views

irrational, proving or finding counterexapmle [duplicate]

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational. Can anyone give me a hint just about how to start cause i ...
6
votes
0answers
77 views

Does the Cantor set contain any irrational algebraic numbers?

I've been trying to characterise the irrationals in the Cantor set $\mathcal{C}$ and this is proving to be surprisingly difficult. In particular I am trying to investigate whether $\mathcal{C}$ ...
0
votes
5answers
163 views

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational.

The following is a Homework Question that I've been working on and I would like some feedback on my answer: Prove or find a counterexample: For all real numbers $x$ and $y$ it holds that $x + y$ is ...
19
votes
4answers
2k views

Can every irrational number be written in terms of finitely many rational numbers?

Consider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...
4
votes
1answer
66 views

Are there numbers that if proven rational (or irrational) will have important consequences to mathematics?

We see all the time conjectures and proofs that specific (real) numbers are (more often than not) irrational. I'm wondering that apart from the mathematical curiosity motivating such proof attempts, ...
2
votes
2answers
159 views

Let a, b, c, d be rational numbers… [closed]

Let $a, b, c, d$ be rational numbers, where $\sqrt{b}$ and $\sqrt{d}$ exist and are irrational. If $a + \sqrt{b} = c + \sqrt{d}$, prove that $a=c$ and $b=d$.
6
votes
1answer
141 views

Is $\log 2\pi$ rational?

Is it known whether $\log 2\pi$ is rational (where the base of the logarithm is $e$)? Or algebraic?
1
vote
1answer
68 views

Confusing rational numbers

Question: If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the value of $x$ to ...
2
votes
1answer
50 views

Logic verification: $x^3$ is irrational, then $x$ is also irrational

Prove, by contraposition, if $x^3$ is irrational, then $x$ is also irrational. Just a verification do I need to show that given $x$ is rational $x^3$ is also rational? Suppose $x \in \mathbb{Q}$ ...
5
votes
0answers
133 views

Are there integers $a, b$ s.th. $\pi^a = e^b$?

Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?
2
votes
1answer
86 views

A general method to efficiently calculate the floor of an element of $\mathbb{Q}[\sqrt{2}]$

I had to decide whether to post this question here on the Mathematics Stack Exchange or on Stack Overflow, but I decided that the question was essentially a mathematical one despite being inspired by ...
10
votes
7answers
296 views

Proving that $\sqrt{2}+\sqrt{3}$ is irrational [duplicate]

This is from Spivak. Prove that $\sqrt{2}+\sqrt{3}$ is irrational. So far, I have this: If $\sqrt{2}+\sqrt{3}$ is rational, then it can be written as $\frac{p}{q}$ with integral $p, q$ and in ...
1
vote
0answers
53 views

Products of irrational numbers field of mathematics?

Recently a friend posed the question "can the product of two irrational numbers be rational?" We the trivial answers like for example $\sqrt{2}\sqrt{8} = 4$. I have become somewhat obsessed with the ...
0
votes
4answers
95 views

why the occurrence of 4,5,6 and 9 in pi differs?

i´m playing around with pi, i have this document with the first 5million decimal numbers after comma. http://www.aip.de/~wasi/PI/Pibel/pibel_5mio.pdf and i build a script that i put in for example ...
1
vote
4answers
65 views

Cancelling out square roots gives 2?

Question: If $$N = \frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}$$Find N (This is a subset of a larger question) My approach: After rationalizing the denominator, by ...
0
votes
0answers
41 views

Combination of integers and irrationals numbers

Show that if $j$ is a positive irrational number, then for all $\varepsilon>0$ there exist integers $h$ and $k$ such that $0<h+kj<\varepsilon$. I need this result in a problem, but I don't ...
1
vote
0answers
60 views

Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
13
votes
3answers
543 views

Proving that e is irrational

Prove that that e is irrational. Recall that e $=\sum_{n=0}^\infty\frac{1}{n!}$, and assume $\mathrm{e}$ is rational. Then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b}\quad \text{for some ...
5
votes
1answer
125 views

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Prove that for a strictly increasing natural sequence $(n_k) $ satisfying $\lim_{n \to \infty} n_k^{1/2^k}=\infty$, $\sum_{k=1}^{\infty} 1/n_k$ is irrational. This is another problem "problems in ...
1
vote
1answer
55 views

Proving that a series converges to an irrational number.

I am taking honors Calculus II and have been doing reasonably well in the course until the current problem set which is due tomorrow. One exercise that is really giving me trouble is this: Prove ...
2
votes
2answers
79 views

Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
1
vote
3answers
415 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?