Linked Questions

0
votes
3answers
526 views

Show that there is no rational number $r=m/n$ such that $r^3=3$ [duplicate]

How do I solve this by prime factorization? I came across a similar problem on MSE just recently, but I can't find it and I thoroughly searched for it. If anyone can find it, please post it in the ...
2
votes
1answer
285 views

Prove that $\sqrt 3$ is irrational [duplicate]

I have to prove that $\sqrt 3$ is irrational. let us assume that $\sqrt 3$ is rational. This means for some distinct integers $p$ and $q$ having no common factor other than 1, $$\frac{p}{q} = \...
0
votes
0answers
37 views

Why is $\sqrt3$ not an element of $\mathbb Q$? [duplicate]

I'm bad at maths. Why is $\sqrt3$ not an element of $\mathbb Q$? (A book states this)
75
votes
24answers
12k views

What is your favorite application of the Pigeonhole Principle?

The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ...
2
votes
7answers
7k views

How to prove that $\sqrt 3$ is an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational number?
8
votes
8answers
4k views

Proof that $\sqrt{5}$ is irrational

In my textbook the following proof is given for the fact that $\sqrt{5}$ is irrational: $ x = \frac{p}{q}$ and $x^2 = 5$. We choose $p$ and $q$ so that the have no common factors, so we know that $p$ ...
3
votes
2answers
2k views

If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
4
votes
4answers
507 views

Density of a set. Exercise from Spivak.

I'm trying to do a series of exercises from Spivak's Calculus, in chapter 8, Least Upper Bounds. I'm trying to tackle these two exercises, $5.$ and $^*.6$ From $5.$ I have proven the first claim ...
2
votes
7answers
707 views

What is the most rigorous proof of the irrationality of the square root of 3?

I am currently trying to self-study Stephen Abbott's Understanding Analysis. The first exercise asks to prove the irrationality of √3, and I understand the general idea of the contradiction by finding ...
3
votes
2answers
138 views

Show that $\sqrt{2} \notin \mathbb{Q}(i)$ using infinite descent

Standard exercise is to show $\sqrt{2} \notin \mathbb{Q}$ (e.g. Wikipedia). There are examples on Math.SE such as [1, 2, 3, 4]. If we adjoin an element to $\mathbb{Q}$ does the same proof by ...
0
votes
2answers
167 views

Prove that $\sqrt{2} + \sqrt{17}$ is irrational. Is my proof correct?

$2+17 = a^2/b^2$ $19b^2 = a^2$ ($a$ is divisible by 19) $19b^2 = (19k)^2$ $19b^2 = 361k^2$ $b^2 = 19k$ ($b$ is divisible by 19) Since both numbers are divisible by 19,it means they have a ...
1
vote
1answer
119 views

Irrationality proof trick with Mod [duplicate]

You will see here: Bill Dubuque's Slick $\sqrt{3}$ irrationality proof What is the trick with modulus for proving irrationality? What about $\sqrt{2}$ Can you prove this is irrational by that ...