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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$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

$m_1 , m_2, \cdots m_n$ are natural number that at least one of them is'nt perfect square. how to prove that $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational?
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2answers
440 views

Irrationality of sum of two logarithms

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
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12answers
1k views
+50

Irrational numbers in reality

I have a square stone slab 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another is given by $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
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1answer
19 views

Rational vs irrational

What is the difference between a rational and an irrational number? I'm am interested to understand what the difference. Is one like a decimal and one not? Thanks a ton in advance.
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2answers
137 views

Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
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1answer
25 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
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2answers
55 views

How to find irrational approximates

Say I have a rational number, $n$, that approximates an irrational number of the form: $$n \approx {a+\sqrt b \over c}$$ in terms of being irrational. What is a good way of finding the unknown ...
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0answers
45 views

Proving that an infinite sum of irrationals is irrational

First of all, I know this question may be closed because it is off topic, but I do have a valid question. Problem: Is is possible to prove that an infinite sum of distinct and different irrational ...
36
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2answers
4k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
12
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0answers
96 views

$\log_2 13$ is irrational.

$\log_2 13$ is irrational. Is it true? $x=\log_2 13$ $\implies 2^x=13$ So, it will be an irrational number, if not,$$x=\frac p q$$ and $$2^{\frac p q}=13$$ $$\implies 2^p=13^{q}$$ Since, $13$ is ...
6
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1answer
270 views

Irrationality of reciprocal Fibonacci constant

I read that it was proved that reciprocal Fibonacci constant $$\sum_{n} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \frac{1}{21} ...
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4answers
283 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
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8answers
5k views

Proof that dividing irrational number by an irrational number can result in an integer?

How can I prove that dividing an irrational number by an irrational number (besides himself) can result in an integer?
24
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2answers
420 views

Can both $x$ and $\sin(x)$ be rational at the same time?

Except, of course, trivial $x=0$ case ($\sin0=0$); $x$ is measured in radians. The question turned out to be more complicated than it seemed to me at the first sight. All I came up with, that posed ...
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4answers
152 views

Is this attempt of showing the square root of two is irrational valid?

Any odd integer squared is always odd, and likewise, any even integer squared is always even. Therefore, the square root of an odd number must be odd and the square root of an even number must be ...
6
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3answers
3k views

Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...
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2answers
272 views

$e^{\pi\sqrt N}$ is very close to an integer for some smallish $N$s. What about $\pi^{e\sqrt N}$?

Heegner numbers (1, 2, 3, 7, 11, 19, 43, 67, 163 - let's use symbol $H_n$) are know for peculiar property that $e^{\pi\sqrt{H_n}}$ are almost integers: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...
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2answers
78 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
2
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4answers
114 views

Rationality of $e + \pi$

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often. Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = ...
6
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7answers
2k views

How to show that the product of two irrational numbers may be irrational?

Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot ...
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2answers
76 views

What do we mean when we say an irrational number can't be expressed as a fraction?

An irrational number is one such that it cannot be expressed by a fraction, but consider the definition of the Golden Ratio. Two line segments, call one a and the other b, are said to be of the ...
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2answers
259 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 ...
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1answer
47 views

power e irrational number [closed]

I have a question and it will be appreciated that you tell me some more details. Here is the question. For an arbitrary natural integer n, prove for any $n$, $e^n$ is irrational.
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1answer
52 views

Is there a systematic method of knowing if $n = \sqrt a $ is or is not a rational number?

Consider $$n = \sqrt a $$ where $a$ is any integer. Is there a rigorous, systematic method of figuring out if $n$ will or will not be a rational number?
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1answer
41 views

Can we deduce that $⌊r^{n}α⌋≃r^{n}α$ when $r→∞$?

Let $α∈(0,1)$ be an irrational number and let $n≥1$ be a fixed positive integer. For any $r>4$ we define the positive integer $$k=⌊r^{n}α⌋$$ where $⌊.⌋$ denotes the floor function. My question is: ...
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4answers
926 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
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2answers
2k views

Proving/Disproving Product of two irrational number is irrational

I saw this question where I had to prove/disprove that: Ques. Product of two irrational number is irrational. I tried 'Proof by Contraposition'. Product of two irrational number is irrational. p ...
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3answers
1k views

(A twist in a classical question) Sum and product of two irrational numbers is rational?

So I know that it is possible for the sums and products of irrational numbers to be rational. But, the only instances I know of that happening is when a certain combination of additive or ...
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2answers
63 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a faction, i.e. $x \in \mathbb{Q} - \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
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378 views

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

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 ...
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0answers
32 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
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0answers
170 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 ...
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3answers
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How to evaluate a sum which contains limit variables?

For example: $$\lim_{n\to\infty}\sum_{i=1}^n\frac{n-1}n\frac{1+i(n-1)}n $$ And would the result necessarily be rational, because each term appears to be the multiplication of two rational fractions? ...
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1answer
70 views

Rational summation of irrational numbers

Is the sum of all irrational numbers between any two integer constants rational? I think it should be, because every irrational number should have another irrational with which it would sum to a ...
3
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1answer
238 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 ...
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4answers
496 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 ...
3
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3answers
455 views

Irrationals forming rationals

Can we obtain every rational number from the multiplication of two irrational numbers? If not, which ones can we not obtain?
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2answers
85 views

If $a$ is irrational, does there exist a natural number $n$ such that $na$ is rational?

For some irrational $a$, does there exist an $na$ which is contained within the rational numbers for some natural $n$?
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2answers
67 views

Rational and irrational numbers

Consider $x$ a rational number. Let $\epsilon \geq 0$ be the minimal value such that $x + \epsilon$ is irrational, and let also $\gamma > 0$ be the minimal value such that $x+\gamma$ is rational. ...
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3answers
449 views

Transcendental a infinitely close to rationals?

Apologies that this question is rather vague, but I do not know how to state it more precisely. Is, say pi, infinitely "close" to some rational number? More importantly, are all transcendental numbers ...
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0answers
42 views

Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
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1answer
28 views

Does ${\frac{k}{2\left(1-H\right)}} + \frac{1}{H}\in Z$ when $H$ is irrational and $k \in Z^{+}$?

While working on something, I have stumbled across the following expression $$\frac{\Gamma \left({\frac{k}{2\left(1-H\right)}} + \frac{1}{2H}\right)}{\Gamma \left(\frac{1}{2H}\right)}$$ where $0 < ...
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5answers
991 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 ...
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1answer
56 views

Find $(1+i)^i$ in simpler terms, without imaginary exponents. [duplicate]

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent. since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
5
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2answers
294 views

Simple proof that $\pi$ is irrational - using prime factors of denominator

Simple proof that $\pi$ is irrational Consider the Gregory - Leibniz series for $\pi/4$: $$\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 + \cdots $$ Let $A_n/B_n$ be the irreducible fraction given by ...
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3answers
124 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
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4answers
272 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
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0answers
54 views

Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ ...
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31 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
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2answers
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Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...