# Tagged Questions

77 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 ...
51 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 ...
79 views

### Let a, b, c, d be rational numbers…

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 $b=d$.
136 views

### Proof Involving Rational Numbers [duplicate]

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
227 views

### Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...
1k views

### Prove that $x^3 + x^2 = 1$ has no rational solutions?

Is this enough for a proof?: $$x^3+x^2 = 1$$ I would factor and get: $x^2(x+1) = 1$ I would show that $x = \sqrt1$, which is irrational but then do I have to show more? $x+1=1$ which gives me $x=0$ ...
291 views

### Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?

Suppose I have a rational number $a$ and $a^2 < 2$. Can I find another rational number $B$ such that $a^2<B^2<2$? Based on the answer to this question, I thought of doing the following: ...
2k views

### Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that ...
83 views

### In what circumstances can $\dfrac{aA+b}{cA+d}$ be rational?

I am working on the chapter one practice problems in Hardy and cannot seem to figure it out. My attempt has actually left me with a result contrary to what the question is looking for. The Question ...
195 views

### Conditions that $\sqrt{a+\sqrt{b}} + \sqrt{a-\sqrt{b}}$ is rational

Motivation I am working on one of the questions from Hardy's Course of Pure Mathematics and was wondering if I could get some assistance on where to go next in my proof. I have attempted rearranging ...