0
votes
1answer
18 views

Help proving a theorem in my textbook

If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational. For reference, an integer $n$ is a perfect square if $n=m^2$ for some $m \in \mathbb{Z}$. Any help proving this ...
2
votes
2answers
45 views

Help with a proof my professor gave my class

Let $x,y \in \mathbb{R}$ with $x<y$. There exists an irrational number $z$ such that $x<z<y$. My proof so far: Let $x,y \in \mathbb{R}$ and assume $x<y$. Then, by Theorem 11.8 (in ...
6
votes
3answers
111 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
2
votes
2answers
68 views

Prove that if $y,z \in Q$ then $y^z \in A$

Question : Prove that if $y,z \in Q$ then $y^z \in A$ My attempt: Definition 2.7.8 states that a number s is an algebraic number when there exists some $p \in Z[x]$ such that $p(s) =0$. Let us ...
1
vote
2answers
131 views

Let a, b, c, d be rational numbers… [closed]

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 $b=d$.
0
votes
1answer
117 views

Logic: Prove Log(base 9) 15 is irrational

Im having trouble with the following proof... Ill post what I have completed so far.. Prove $\log_915$ is irrational. Ill attempt by contradiction assuming $\log_915$ is rational. So, $\log_915 = ...
1
vote
2answers
674 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
754 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 ...
3
votes
4answers
212 views

What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$?

In this proof extracted from the Wikipedia A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it could be expressed as ...
2
votes
1answer
233 views

Geometric proof that if n is a non-perfect square, then √n is irrational.

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere. ...
15
votes
10answers
2k views

Critiques on proof showing $\sqrt{12}$ is irrational.

My only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ ...
3
votes
1answer
88 views

Why is this proof on rational set valid?

Prove that if $y$ is rational and $x$ is irrational, then $y + x$ and $yx$ (assume y $\neq 0 $) is irrational. I kinda guessed the proof and it turned out to be right with the key, but it doesn't ...
2
votes
4answers
1k views

Why doesn't the indirect proof of irrational roots apply to rational roots?

When trying to prove that a particular root (say $\sqrt{2}$ or $\sqrt{10}$) cannot be rational, I always see a particular indirect proof that goes something like this: Suppose $\sqrt{x}$ were ...
8
votes
2answers
1k views

Has Euler's Constant $\gamma$ been proven to be irrational?

I found a paper by Kaida Shi called "A Proof: Euler’s Constant γ is an Irrational Number" which claims to have proven the irrationality of $\gamma$. I know people have been trying to prove that ...
55
votes
4answers
6k views

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
3
votes
5answers
3k views

How can one prove that the cube root of 9 is irrational?

Of course, if you plug the cube root of 9 into a calculator, you get an endless stream of digits. However, how does one prove this on paper?