# Tagged Questions

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 ...
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 ...
142 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 ...
78 views

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

Question : Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$. My attempt: Definition 2.7.8 states that a number $s$ is an algebraic number when there exists some $p \in\Bbb Z[x]$ such that ...
159 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 $b=d$.
126 views