Linked Questions

42
votes
9answers
66k views

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

I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. ...
22
votes
6answers
7k 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 that ...
3
votes
6answers
18k 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: $a^2/b^2 = \sqrt{40}$.
5
votes
3answers
2k views

Proving/disproving that √7 - √2 is irrational [duplicate]

It's been proven that √7 and √2 are irrational. However, I am not sure how to go about proving that √7 - √2. Is it an acceptable proof to just solve the equation which would prove/disprove the ...
4
votes
3answers
818 views

The sum of square roots of non-perfect squares is never integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
1
vote
3answers
266 views

Prove that $\sqrt{7}+\sqrt{3}$ is irrational [duplicate]

Is there a method by which we can prove that $$\sqrt{3}+\sqrt{7}$$ is irrational. It's obviously an irrational number, but I want to prove that mathematically.
1
vote
3answers
1k views

Proving that $\sqrt{3} +\sqrt{7}$ is rational/irrational [duplicate]

I took $\sqrt{3}+\sqrt{7}$ and squared it. This resulted in a new value of $10+2\sqrt{21}$. Now, we can say that $10$ is rational because we can divide it with $1$ and as for $2\sqrt{21}$, we divide ...
2
votes
6answers
271 views

Is $\sqrt{2}+\sqrt{3}$ rational or irrational? [duplicate]

I have proven that the sum of any two irrational numbers is not always irrational and that a rational plus an irrational is irrational, however I am not sure how to prove this specific case. I was ...
2
votes
3answers
221 views

proving $ \sqrt 2 + \sqrt 3 $ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
3
votes
2answers
139 views

$\sqrt{m}+\sqrt{n}\notin\mathbb{Q}$ Irrationals, Roots Proof [duplicate]

Let $n,m\in\mathbb{N}_0$ and $\sqrt{n}\not\in\mathbb{Q}$. Conclude that $\sqrt{n}+\sqrt{m}\not\in\mathbb{Q}$ What would you recommend me to read to be able to solve this? Because apart from the case ...
11
votes
0answers
146 views

Is my proof for the sum of two square roots being irrational correct? [duplicate]

Prove that the following number is irrational: $$\sqrt { 5 } +\sqrt { 3 } $$ Steps I took: Proof by contradiction: let us assume that $\sqrt { 5 } +\sqrt { 3 } $ is rational. If $\sqrt { 5 } +\...
-1
votes
1answer
80 views

Both 2^(0.5) is irrational. [duplicate]

Isn't 2^(0.5) rational? Method for proving: Contradiction. So show not P:2^(0.5) is rational.
1
vote
0answers
70 views

Sum of radicals is rational only for perfect squares [duplicate]

How can we prove that if $a_1\sqrt{n_1}+a_2\sqrt{n_2}+\dots+a_k\sqrt{n_k}\in\mathbb{Q}$ with $a_1,a_2\dots, a_k\in\mathbb{Q}^{*}_{+}$, then the natural numbers $n_1,n_2,\dots, n_k$ are perfect squares?...
26
votes
5answers
5k views

Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.
55
votes
1answer
3k views

Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a ...

15 30 50 per page