# Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$ (where $T$ = transcendental numbers)

Question: Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$.

This is Exercise 2.7.13(a) from Mark E. Watkins, Jeffrey L. Meyer: Passage to Abstract Mathematics.

I'm currently practicing on proof writing and I want to know if I did this correctly. If it's right then I'm wondering why some proofs with easy to understand definitions or one line definitions are easy to endure while other proofs that require over three definitions are so hard that it requires deep thinking skills just to understand and write a correct proof.

Attempt: We are going to disprove this statement.

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 denote the set

$A = [x \in C:$ x is algebraic]

By Proposition 2.7.9 all rational numbers are algebraic.

Moreover, the set $Q$ of rational numbers is $Q = [ \frac{a}{b} a,b \in Z$ and $b \neq 0]$

A number that is not algebraic is called transcendental. Thus, the set $T$ of transcendental numbers satisfies:

$T = [x \in C : x \notin A]$

For $t \in T$, $t$ is transcendental, so it's not algebraic.

. However, if $q \in Q$ it's rational, then by proposition 2.7.9, $q$ must be algebraic. Therefore, $q$ isn't transcendental. As a result, $t \in T$, but $q \notin T$, so $qt \notin T$. [What I'm trying to say is that since $t$ is transcendental and $q$ is rational, q doesn't belong in T because q is rational and algebraic. We can have $t \in T$, but $q \notin T$ due to to proposition 2.7.9 and definition 2.7.8]

• Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Martin Sleziak Apr 27 '14 at 9:27
• In this particular case, I think that a title like: "Is product of a transcendental number and a non-zero rational number again transcendental?" (or something similar) would help the reader to get the basic idea what the question is about without the need to read the whole question and looking for the place where the notation for $T$ is introduced. – Martin Sleziak Apr 27 '14 at 9:29
• huh? but I've been leaving questions like this for months... I never had problems before. – usukidoll Apr 27 '14 at 9:30