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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4answers
793 views

The n-th root of a prime number is irrational

If $p$ is a prime number, how can I prove by contradiction that this equation $x^{n}=p$ doesn't admit solutions in $\mathbb {Q}$ where $n\ge2$
4
votes
6answers
1k views

Prove $\sqrt{2} + \sqrt{5}$ is irrational [duplicate]

How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational? I tried to prove it by contradiction and got this equation: $a^2/b^2 = \sqrt{40}$.
0
votes
2answers
44 views

algebraically determining if a number is irrational or not

Is it possible to use an algebraic formula, equation, concept, or principle to determine with perfect accuracy (or high precision, if not perfect) whether or not a number is rational? An example ...
4
votes
1answer
172 views

Proof that ${\pi}$ can(not) be expressed as a root or as a root in combination with a fraction

I was doing some math for a programming project of myself and ran into decimal numbers and how to define them without losing precision while calculating an expression, so I tried writing them down as ...
8
votes
5answers
239 views

Different ways to prove $\sqrt p$ irrational for $p$ prime.

I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.
5
votes
6answers
3k views

what's the difference between a rational number and an irrational number?

I tried to understand the difference between rational numbers and irrational numbers. I understand what is a rational number (a number that can be expressed as the ratio of two numbers p/q). what ...
2
votes
9answers
996 views

Why does $(3\sqrt3)^2 = 27$?

How does $(3\sqrt3)^2 = 27$? I've tried to solve this using binomial expansion and using the FOIL method from which I obtain $9 + 3\sqrt3 +3\sqrt3 + 3$. it has been a while since I've done this kind ...
1
vote
1answer
69 views

Question on a subset $S$ of $[0,1]\times[0,1]$ where for each $(x,y)\in S$ at least one of $x$ and $y$ is irrational

If $S$ is a subset of $[0,1]\times[0,1]$ such that one point of the ordered pair is rational and the other is irrational or both are irrationals. Then which of the following is true? a) $S$ is closed ...
3
votes
3answers
85 views

Number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$?

The number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$ is (A)0 (B)2 (C)3 (d)4 Actually im a 10 class student i don't know any of it,but my elder brother(IIT Coaching) cannot ...
4
votes
1answer
98 views

$f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ nonconstant, continuous, with period $1, \sqrt{2}$, respectively, then $f_1 + f_2$ is not periodic

I've been working on this problem for several hours, but I keep getting stuck. Suppose $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ periodic with period $1, \sqrt{2}$, respectively, and that each of ...
1
vote
1answer
56 views

Prove $k!(e-s_k)$ is irrational.

Given that $\frac{p}{q} = e = 1 + \frac{1}{1!} + \frac{1}{2!} + ... + \frac{1}{k!} + \frac{e^{z}}{(k+1)!}$ for some $z$ in $[0,1]$ (using Taylor's theorem), and that $s_k = 1 + \frac{1}{1!} + ...
0
votes
2answers
2k views

Can we ever get an irrational number by dividing two rational numbers?

If we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and ...
2
votes
1answer
62 views

If $a$ and $b$ are rational, then $a + b{\sqrt{2}} \ne {\sqrt{3}} $

If $a,b\in\mathbb{Q}$ then demonstrate:$$a + b{\sqrt{2}} \ne {\sqrt{3}} $$ I raised and squared the equation but it didn't work.
2
votes
0answers
45 views

Apery's constant

I read that it is unknown if $\zeta (3)$ is algebraic but it is known to be irrational. Has anyone proved anything of the form $\zeta (3)$ is not a root of a polynomial of degree $12345$ with integer ...
2
votes
0answers
43 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
2
votes
3answers
539 views

Set of irrationals between two reals is uncountable

I know that between any two reals, there is an irrational number. See: Proving that there exists an irrational number in between any given real numbers Now let a, b $\in$ $R$ such that a < b. And ...
1
vote
2answers
804 views

Check if this proof about real numbers with an irrational product is correct.

Can anyone confirm if my proof is correct, please? Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” Proof:- Assume that both ...
1
vote
3answers
800 views

How to write this in mathematical notation?

I have the following claim: “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” I'm supposed to write this in mathematical notation. It's ...
-1
votes
1answer
263 views

Proof of $\pi$+$e$ irrational

The wikipedia tells that it is not known that $\pi+e$ is irrational? Immediately after reading this my mind came with this proof- Let $x =\sqrt{\pi^2}+\sqrt{e^2}$ be rational, then $ \quad ...
5
votes
2answers
168 views

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
4
votes
2answers
364 views

If $x$ and $y$ are rational then is $x^y$ also rational?

I can think of the counter example $x = 2$ and $y = 1/2$ but how would a proof to disprove this look like?
-1
votes
1answer
214 views

Is this a rational or irrational number?

It is given that $$z=\sqrt\frac{\sqrt{3x+1}}{\sqrt{3x-1}}$$ How does one find whether $z$ is a rational or irrational number?
1
vote
1answer
318 views

a square root of an irrational number

I wonder if a square root of an irrational number is always irrational? I would tend to think that yes, but I can´t think of any justification. Also there are cases which are rather hard to decide ...
5
votes
3answers
1k views

sum of irrational numbers - are there nontrivial examples?

I know that the sum of irrational numbers does not have to be irrational. For example $\sqrt2+\left(-\sqrt2\right)$ is equal to $0$. But what I am wondering is there any example where the sum of two ...
0
votes
1answer
235 views

Spiral of Theodorus - Discussion

The fact that $\sqrt2$ is not rational goes back to Theodorus of Cyrene from the school of Pythagoras, and is discussed in Plato's dialog "Theaetetus". Of course, $\sqrt n$ is not rational for any ...
1
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1answer
165 views

integral factors of an irrational number

If the radicand of a square root is a non-square (making the root an irrational), and if the non-square is either a prime number, or a composite number that does not have a square divisor (other than ...
20
votes
6answers
3k views

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Please prove that $\sqrt 2 + \sqrt 3$ is irrational. One of the proofs I've seen goes: If $\sqrt 2 +\sqrt 3$ is rational, then consider $(\sqrt 3 +\sqrt 2)(\sqrt 3 -\sqrt 2)=1$, which implies ...
2
votes
3answers
138 views

Do irrational numbers have equivalence classes the way rational numbers do?

Rational numbers are defined as equivalence classes of ordered pairs (less formally, "fractions") of integers, where $m_1n_2=m_2n_1$. This equivalence relation justifies the common practice of ...
5
votes
1answer
123 views

What is the sum of $4\sqrt{28}$ and $3\sqrt{7}$ ?

As far as I can simplify it - $$4\sqrt{7*4} + 3\sqrt{7} = 8\sqrt{7} + 3 \sqrt{7} = \sqrt{7} * 11$$ However , The options for the correct answer are - A) $ 8/3$ B) $ 16/3$ C) $ 18/3$ D) $24/3$ I ...
0
votes
1answer
206 views

Simplify : $\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} + \frac{3\sqrt{2}}{\sqrt{6 + \sqrt{3}}} - \frac{4\sqrt{3}}{\sqrt{6 + \sqrt{2}}}$

My exams are approaching fast and this question was in one of the sample papers . I have to simplify $$\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} + \frac{3\sqrt{2}}{\sqrt{6 + \sqrt{3}}} - ...
0
votes
1answer
239 views

Proving that any rational number can be represented as the sum of the each cube of three rational numbers

I found the following question in a book: Prove that any integer can be represented as the sum of the each cube of five integers. The answer : ...
5
votes
4answers
222 views

Can an irrational always be found by multiplying irrationals?

I was thinking about the function $\ f(a,b) = a/b $ where $a$ and $b$ where both irrational. It quickly stood out to me that the codomain of that function would include every rational number. But, ...
3
votes
2answers
675 views

How to evaluate $\sqrt{5+2\sqrt{6}}$ + $\sqrt{8-2\sqrt{15}}$ ?

My exams are approaching fast and I found this question in one of the unsolved sample papers. I tried squaring the whole term but couldn't work out the answer . I am a ninth grader so please try to ...
3
votes
2answers
648 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 ...
12
votes
1answer
252 views

Multiplying by an irrational number in combinatorial problems

Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$. It is also widely known that the $(n+1)$th Fibonacci number $F_{n+1}$ is the nearest integer to ...
0
votes
1answer
41 views

What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational. I can prove that ...
10
votes
4answers
566 views

How to show $\sqrt{4+2\sqrt{3}}-\sqrt{3} = 1$

I start with $x=\sqrt{4+2\sqrt{3}}-\sqrt{3}$, then $\begin{align*} x +\sqrt{3} &= \sqrt{4+2\sqrt{3}}\\ (x +\sqrt{3})^2 &= (\sqrt{4+2\sqrt{3}})^2\\ x^2 + (2\sqrt{3})x + 3 &= 4+ 2\sqrt{3}\\ ...
4
votes
5answers
495 views

How to know a irrationals never repeat?

How would you respond to a middle school student that says: “How do they know that irrational numbers NEVER repeat? I mean, there are only 10 possible digits, so they must eventually start repeating. ...
0
votes
2answers
50 views

Boundaries for Specific Sets with Ambient Space $\mathbb{R}$

I'm trying to find the boundaries for each the following sets: (a) $\begin{Bmatrix}\frac{1}{n}:n\in\mathbb{N}\end{Bmatrix}\overset{?}{=}\{1\}$ (b) $[0,3]\cup(3,5)\overset{?}{=}\{0,5\}$ (c) ...
6
votes
2answers
466 views

Sum of two irrational radicals is irrational?

If $a,b,m$ and $n$ are positive integers such that $\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational numbers, how can we prove that the sum $\sqrt[m]{a}+\sqrt[n]{b}$ is also irrational?
11
votes
1answer
362 views

Linear independence of the numbers $\{1,\pi,{\pi}^2\}$

Does someone know a proof that $\{1,\pi,{\pi}^2\}$ is linearly independent over $\mathbb{Q}$ ? The proof should not use that $\pi$ is transcendental. $\{1,e,e^2,e^3\}$ is linearly independent over ...
9
votes
1answer
490 views

Proving the irrationality of $e^n$.

Let $n$ be a positive integer. I know the traditional proof that $e$ is irrational. How do we show that $e^n$ is irrational in some sort of similar line? I am of course assuming it is but I would be ...
26
votes
2answers
512 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
-3
votes
1answer
188 views

The Irrationality of 2

I am sorry it is not 'research level'. A quick answer will do. When I attempt using the Square root of 2 method to prove the rationality of Square root of 4 according to how it was done in a book, 2 ...
1
vote
2answers
62 views

Integer outputs of $y=x^2$ , do their last digits form an irrational?

Let the domain of $y=x^2$ be the positive integers. I input consecutive positive integers from $[1, \infty)$ their last digits are $a, b, c, ...$ respectively. If I then make the number $z=\frac ...
3
votes
1answer
95 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
5
votes
3answers
510 views

property of real number system

"Between every two rational numbers there exist infinite irrational numbers and between every two irrational numbers there exist infinite rational numbers. Is this statement correct? If it is, then ...
0
votes
3answers
230 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. All this means that we can find out which digit exist in any position of rational number. But what about ...
4
votes
2answers
112 views

What irrational number has the simplest calculation in terms of computation?

I came across https://github.com/philipl/pifs which is a fancy way of storing data. And a thought struck my mind, is it so that Pi is the simplest irrational number to calculate? So the Question is. ...
1
vote
1answer
172 views

how do we know the BBP formula for $\pi$ is valid?

I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we ...