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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2
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1answer
41 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
27 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 ...
10
votes
1answer
146 views

Does $\lfloor(4+\sqrt{11})^{n}\rfloor \pmod {100}$ repeat every $20$ cycles of $n$?

I recently came across a post on SO, asking to calculate the least two decimal digits of the integer part of $(4+\sqrt{11})^{n}$, for any integer $n \geq 2$ (see here). The author presented a Java ...
0
votes
1answer
22 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
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3answers
92 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
votes
0answers
22 views

Irrational number problem

Prove that if $x$ is irrational, then there exists $p_n,q_n \in \mathbb{Z}$ such that $\bigg|x -\dfrac{p_n}{q_n}\bigg| < \dfrac{1}{q_n^2}$
1
vote
1answer
65 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. ...
1
vote
1answer
33 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 ...
27
votes
8answers
3k views

How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
3
votes
1answer
43 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 ...
0
votes
1answer
28 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 ...
3
votes
3answers
228 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
2
votes
1answer
15 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 ...
3
votes
1answer
59 views

Question regarding the golden/silver ratio

$\Phi$, 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 ...
6
votes
0answers
55 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 $\mathcal{C}$ ...
1
vote
5answers
125 views

Find two rationals, one greater and one smaller than a given irrational number.

Given an irrational number 0 < i < 1. Find two rational numbers a and b such that 0 < a < i < b < 1.
14
votes
3answers
288 views

Is there a simple proof that $e^2$ is irrational using a positional numeral system?

My favorite proof that $e$ is irrational goes something like this. Observe that we can write any real number $r$ as $$ a \,+\, \frac{b_2}{2} \,+\, \frac{b_3}{3!} \,+\, \frac{b_4}{4!} \,+\, ...
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
50 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
0answers
93 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
11
votes
1answer
350 views

Is the difference of the natural logarithms of two integers always irrational or 0?

If I have two integers $a,b > 1$. Is $\ln(a) - \ln(b)$ always either irrational or $0$. I know both $\ln(a)$ and $\ln(b)$ are irrational.
0
votes
2answers
73 views

Let a, b, c, d be rational numbers…

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
107 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
58 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
48 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 $x \in \mathbb{Q}$ ...
6
votes
0answers
113 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. $\pi^a = e^b$ ?
0
votes
2answers
330 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?
2
votes
1answer
51 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 ...
9
votes
7answers
238 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 ...
0
votes
4answers
86 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
1answer
155 views

Does anybody know the formula for a quasicrystal structure?

I am an architecture student researching into quasicrystals with the hope of applying it to form a complex truss system. I was wondering if anyone new of a formula for the structure? thanks in ...
1
vote
0answers
48 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 ...
32
votes
9answers
6k views

$\sqrt a$ is either an integer or an irrational number.

I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown ...
1
vote
4answers
47 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
36 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 ...
7
votes
2answers
264 views

Proving that $e$ is irrational

Show that $e$ is irrational. Recall $\mathrm{e} = \exp(1)$ so assume $\mathrm{e}$ is rational , then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b}\quad \text{for some}\,\,\, a,b \in ...
1
vote
0answers
56 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 ...
6
votes
4answers
324 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 ...
5
votes
1answer
117 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
48 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
55 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
311 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?
1
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3answers
82 views
2
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2answers
386 views

Pi might contain all finite sets, can it also contain infinite sets?

In a previous, and quite popular, question it was discussed about whether or not $\pi$ contains all finite number combinations. Let us assume for a moment that $\pi$ does in fact contain all finite ...
0
votes
1answer
59 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
2
votes
2answers
112 views

Prove that if $a$ is irrational then $\sqrt a$ is irrational

Just hints but solution thx. Any hints for me? I simply suppose that $a = \dfrac mn$ then $\sqrt a = \sqrt{\dfrac mn}$ But this does not make sense ..
7
votes
3answers
427 views

Sequences of Rationals and Irrationals

Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational? Let $(y_n)$ be a sequences that converges to the rational number ...
5
votes
4answers
103 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
0
votes
1answer
26 views

Filling up space with irrational fractional parts [duplicate]

While trying to generalise a mechanics exercise with a friend, we came up with this question, in an attempt to understand wether sine curves with irrational period defined inside an annulus will end ...
3
votes
1answer
55 views

Enough Dedekind cuts to define all irrationals?

Assuming that there are uncountably infinitely many irrationals between any two consecutive rationals, how can the Dedekind cuts (defined on the countably infinite rationals) define all the ...