Here is the question and i have no idea how to go about it any help would be hugely appreciated.
Prove no rational satisfies
$\displaystyle 2^x = 3$
Many thanks
On the contrary, 1.584962501 certainly IS a rational number because it is a terminating decimal! $\displaystyle log_2(3)$,which is NOTequal to 1.585962501 is not rational but I suspect it would be necessary to prove that to get credit for this problem!
This is in the prealgebra/algebra section, so i am not sure if a proof like this (a proof by contradiction) will be too complicated. if it is, tell me, we can come up with another.
Assume, to the contrary, that there is a rational number $\displaystyle x$ such that $\displaystyle 2^x = 3$.
Then $\displaystyle x = \frac ab$, where $\displaystyle a$ and $\displaystyle b$ are integers, with $\displaystyle b \ne 0$. So we have
$\displaystyle 2^{\frac ab} = 3$
$\displaystyle \Rightarrow 2^a = 3^b$
However, this is absurd, since $\displaystyle a$ and $\displaystyle b$ are integers, this equation contradicts the uniqueness of prime factorization (since the left side is always a product of 2's and the right is always a product of 3's).
note that a = b = 0 makes the equation make sense... but b cannot be zero!
oh, you're from the UK, hehe, I was worried. you have a US flag up
i suppose you are also familiar with the uniqueness of prime factorization theorem (you might know it under a different name), but it talks about how each integer can be uniquely expressed as a product of primes with integer powers
Yes, we're studying this whole semester on these sorts of theorems about rational numbers (the Completeness axiom etx) and with these come a whole host of proofs...
Do you know any good proof resource where i can learn as many types of proof as possible? (not including Induction since i am fairly happy with my ability to manipulate these sorts of problems)
completeness axiom? ...this is prealgebra?!!
i am sure there are resources online that have what you are looking for. i guess wikipedia is always a good place to start, at least for quick reference. but i have never really searched for any such site, as i have several textbooks that cover such material and am fairly happy with them, so i don't know. sorry