We know that $\displaystyle e^x$ is continuous, and so must take rational values. For which $\displaystyle x $, is $\displaystyle e^x$ rational/irrational??
(-> Thank/Blame MathGuru, for triggering this question. )
If $\displaystyle x$ is rational, then $\displaystyle e^x$ is not rational, except of course $\displaystyle x = 0$. A proof can be found on PlanetMath.Originally Posted by Rebesques
Τhanks for the answers
I 'll now collect distinct proofs about the irrationality of $\displaystyle e^x$, when $\displaystyle x$ is rational. Here 's a second one, based on the transcedental nature of e, tell me please if it is right:
Let $\displaystyle x=\frac{p}{q}, \ p,q$ a natural and an integer respectively. If $\displaystyle e^{\frac{p}{q}}$ were rational, so would be $\displaystyle (e^{\frac{p}{q}})^q=e^p.$ This implies that the polynomial $\displaystyle z^p-e^p$, has a zero at z=e. I have only a basic knowledge of Galois Theory, but recognize the contradiction here
Now, what about irrational $\displaystyle x$ and rational $\displaystyle e^x$ ??
WangTaoSG beat me to the punch so I chose not show my answer anymore. But your latest/added question made me show my solution anyway. But it is still basically that of WangTaoSG.
"Now, what about irrational x and rational e^x ??"
Why, is logarithm of a rational number not irrational?
Is ln(3), say, not irrational enough?
e^x = (rational number)
Take the natural log of both sides,
x*ln(e) = ln(rational number)
x = ln(rational number) -----------***
Again, is ln(rational number) not irrational enough?
x=ln(y), where y is rational, exp(x) is rational;
This does hold true, but it's only trickery (1)
I asked if we could determine forehand, which x provide rational/irrational e^x, and the irrationality has been quite dealt with.
(1) Addendum to this message: Say I 'd like to know if e^x is rational. With this reasoning, then, I would have to solve x=lny for y; Which means that I would have (again) to compute y=e^x, and see if it this is rational... This is why I called it a trick, 'cause it answers, but not unappealably
Yes, WangTaoSG just gave answers without explanations/solutions. He belongs to the group who thinks there is more prestige in giving only answers without enough explanations. I see that assumption a lot on Mathheads. Looks like, for them, it is demeaning if they show how they arrived at the answers. Maybe, for them, without explanations will maintain the "mystery" of their "mastery" of Math. Something for the students/lesser mortals to view these Mathheads in awe. ;-)
I don't belong to his group. In fact, even if I can get the answer my way but I cannot explain this way so that it will be understood by the seeker/poster of the question, then I will rather not give the answer.
It is more challenging, and rewarding, for me to show a good explanation/solution for the answer than in finding the answer. It is easier to just find the answer.
But then, hey, to each his own. Nobody is forced to copy anybody's style.
He belongs to the group who thinks there is more prestige in giving only answers without enough explanations. I see that assumption a lot on Mathheads. Looks like, for them, it is demeaning if they show how they arrived at the answers. Maybe, for them, without explanations will maintain the "mystery" of their "mastery" of Math. Something for the students/lesser mortals to view these Mathheads in awe. ;-)
You are kinda mean here
Those men, the mathheads, have given serious effort and precious time, to knowing math deeply. Their experience points out a way, when we just stand there gazing No need to be mad about it, maybe one day all math on this forum will seem elementary to us too.
...See ya! I 'm off for two weeks of sea and sun (...and bikinis hopefully )
Oh no, I am not mad at the Mathheads. Only at their attitudes towards us lesser mortals when it comes to Math. Like they saw/studied the Fountain of Knowledge on Math, and we did not, so we give them excuses when they don't explain Math at our level of understanding. Either we understand what they like to show us or they care not. If we don't get their lambdas and epsilons and phis, so much the better for them.
Most of them only, though. A few of them are still human enough.
Math, for me, will never be elementary. Math, in any of its branches, from lower than Arithmetic, (?), to whatever that I understand, will always be interesting and challenging. I get the same pleasure from any of the branches of Math that I understand. So I treat these equally.
If I don't practice, I will forget even pre-Algebra in no time.
That is why I join, and enjoy, Math forums like this one.