0
votes
2answers
43 views

Help With a proof (Irrational Number)

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational. Its contrapositive will be: If $r^\frac{1}{5}$ is not irrational, then $r$ is ...
0
votes
1answer
19 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 ...
0
votes
6answers
117 views

Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational.

The sum of a rational and irrational number is always irrational, that much I know - thus, if n is a perfect square, we are finished. However, is it not possible that the sum of two irrational numbers ...
2
votes
2answers
77 views

Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
0
votes
3answers
85 views

Prove that there is no rational number solution for an equation.

Prove that there is no rational number solution to the equation $x^2-3x+1=0$. (Note, we do not assume that we know all the solutions of $x^2-3x+1=0$ are given by quadratic formula)
0
votes
4answers
143 views

Help me to Prove that log2 3 is irrational. [closed]

seemingly simple homework assignment, help? Was never the best with logarithms, how would I go about proving? Sorry the question read IRrational!
0
votes
1answer
119 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 = ...
27
votes
9answers
3k views

Rational + irrational = always irrational?

I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + ...
4
votes
2answers
268 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
vote
1answer
155 views

Proof by contradiction that irrational numbers conform to $f(x)=x^2$

I am having some difficulty with this proof for my Real Variables class. I know that $f(x)$ is a continuous function defined on $R^1$, and $f(x)=x^2$ for any rational $x$. I also have the definition ...
3
votes
5answers
166 views

Please explain this step in proving the square root of 3 is irrational

Assume that $$3 = \frac{p^2}{q^2}$$ So, $$ 3 q^2 = p^2$$ So $p^2$ is divisible by $3$. How we can conclude this?
2
votes
3answers
369 views

Constructing the proof that $\sup \{x:x\in\mathbb Q \wedge x<\sqrt 2 \}$ doesn't exist.

I know there is a very well known proof that for any rational such that $$x=\frac p q < \sqrt 2$$ there exists another rational $y=\dfrac mn$ such that $$x=\frac p q < \frac m n <\sqrt 2$$ ...
6
votes
1answer
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 ...