User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Exponent Combination Laws
I realized that the proofs of the Laws of Logarithms don't work according to the 2nd proof of Derivative of Natural Logarithm Function. Since I'm the one who added the second proof after the laws of logarithms were put up, I'm trying to add a proof for the laws that aren't circular. While Googling it I found someone who encountered the same problem on a homework assignment, and (lucky for me?) he didn't post his resolution. That is, he found a source for the the definition using $\exp$ as the inverse of the natural log, whereas he wanted to prove it using the definition


 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$

I might wait until next beginning of next year when I do sequences and series and limits at infinity in detail. At the very least, I think I know enough to prove laws of exponents for integer indices, should be a simple proof by induction?, but I don't want to clutter PW with extraneous proofs. Thoughts? My semester just ended so I have some free time until Spring semester. --GFauxPas 12:09, 23 December 2011 (CST)


 * "Extraneous proofs": If it's a result which is interesting enough to document, then it's worth putting up on ProofWiki. One of the first questions you have when putting a site together like this is: "Is this statement too trivial to mention?" If you find yourself asking that question, then you will probably find yourself, sooner or later, making the wrong decision. Thus a vital step in a proof is glossed over, as its proof is "too trivial", and then a mistake creeps in because this point is insuffuciently understood. There's a biblical quote that applies here: something like "[The stone that the builders cast aside has become the keystone]." Hence my (own personal) decision that: there is no mathematical proof which is so simple and trivial it's not worth putting up here. I even believe we can put up a category of "examples" and/or "exercises" (but I haven't bent my brain cell in that direction yet).


 * As for the question about:
 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$
 * Other way about, btw, it should be:
 * $\displaystyle \exp x := \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$
 * (it means: "the $\exp x$ function is defined as being the limit of ..."


 * The aim is to prove that it follows strictly from the definition as given in these pages. Then we can confirm we have no circularity. --prime mover 16:35, 26 December 2011 (CST)
 * It is actually easy to prove that above definition coincides with the unique solution definition (the unique $f$ s.t. $f' = f, f(0)=1$) as, given uniform convergence (which is there on closed intervals), limits and differentiation may be swapped, yielding the desired immediately. I think it will be enough to just show equivalence of the definitions, after which all steps are allowed. Note that proving this equivalence of definitions as I sketched above builds on uniqueness theorems in differential equation theory and elementary calculus, so no dependency on logarithms required. The remaining equivalences I can't pull out of my hat at will.


 * P.m, I agree on your assessment of the exercises and such, compare the three example pages on Definition:Convex Set. Lastly, your quote originates from Psalms 118, verse 22. Also, this is quoted again in the New Testament. Just for completeness' sake. --Lord_Farin 16:57, 26 December 2011 (CST)
 * L_F: Examples: Yes, that page cited is an instance of how examples work - I may spend some time thinking about how to template-ize this. I'm on vacation all this week and my wife's at work, so (if I'm not spending the whole week watching DVDs she doesn't like) I may well think about this. On the other hand I've just got my hands on and, so, a-ha-ha, both are doubtful. --prime mover 17:18, 26 December 2011 (CST)

Proof for integer powers
Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Exponent Combination Laws
I realized that the proofs of the Laws of Logarithms don't work according to the 2nd proof of Derivative of Natural Logarithm Function. Since I'm the one who added the second proof after the laws of logarithms were put up, I'm trying to add a proof for the laws that aren't circular. While Googling it I found someone who encountered the same problem on a homework assignment, and (lucky for me?) he didn't post his resolution. That is, he found a source for the the definition using $\exp$ as the inverse of the natural log, whereas he wanted to prove it using the definition


 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$

I might wait until next beginning of next year when I do sequences and series and limits at infinity in detail. At the very least, I think I know enough to prove laws of exponents for integer indices, should be a simple proof by induction?, but I don't want to clutter PW with extraneous proofs. Thoughts? My semester just ended so I have some free time until Spring semester. --GFauxPas 12:09, 23 December 2011 (CST)


 * "Extraneous proofs": If it's a result which is interesting enough to document, then it's worth putting up on ProofWiki. One of the first questions you have when putting a site together like this is: "Is this statement too trivial to mention?" If you find yourself asking that question, then you will probably find yourself, sooner or later, making the wrong decision. Thus a vital step in a proof is glossed over, as its proof is "too trivial", and then a mistake creeps in because this point is insuffuciently understood. There's a biblical quote that applies here: something like "[The stone that the builders cast aside has become the keystone]." Hence my (own personal) decision that: there is no mathematical proof which is so simple and trivial it's not worth putting up here. I even believe we can put up a category of "examples" and/or "exercises" (but I haven't bent my brain cell in that direction yet).


 * As for the question about:
 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$
 * Other way about, btw, it should be:
 * $\displaystyle \exp x := \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$
 * (it means: "the $\exp x$ function is defined as being the limit of ..."


 * The aim is to prove that it follows strictly from the definition as given in these pages. Then we can confirm we have no circularity. --prime mover 16:35, 26 December 2011 (CST)
 * It is actually easy to prove that above definition coincides with the unique solution definition (the unique $f$ s.t. $f' = f, f(0)=1$) as, given uniform convergence (which is there on closed intervals), limits and differentiation may be swapped, yielding the desired immediately. I think it will be enough to just show equivalence of the definitions, after which all steps are allowed. Note that proving this equivalence of definitions as I sketched above builds on uniqueness theorems in differential equation theory and elementary calculus, so no dependency on logarithms required. The remaining equivalences I can't pull out of my hat at will.


 * P.m, I agree on your assessment of the exercises and such, compare the three example pages on Definition:Convex Set. Lastly, your quote originates from Psalms 118, verse 22. Also, this is quoted again in the New Testament. Just for completeness' sake. --Lord_Farin 16:57, 26 December 2011 (CST)
 * L_F: Examples: Yes, that page cited is an instance of how examples work - I may spend some time thinking about how to template-ize this. I'm on vacation all this week and my wife's at work, so (if I'm not spending the whole week watching DVDs she doesn't like) I may well think about this. On the other hand I've just got my hands on and, so, a-ha-ha, both are doubtful. --prime mover 17:18, 26 December 2011 (CST)

Proof for integer powers
First I'll deal with $x,y > 0$

Sum of Powers
...


 * Funny, but I thought all the above was already in ProofWiki. Is there a need to overhaul it all, or something? --prime mover 14:28, 27 December 2011 (CST)


 * The proofs on PW use logarithms, which means they don't hold for negative bases. I think that with the right constraints, you can combine negative bases raised to integer exponents, and I'm trying to figure out how. Or is there a page out for integer power properties I wasn't able to find? --GFauxPas 15:06, 27 December 2011 (CST)


 * Regardless of all that: Please, save the hassle and write $a^{x+0} = a^x = a^x\cdot1 = a^xa^0$. It saves space. --Lord_Farin 15:20, 27 December 2011 (CST)

...

Soo.. is this already on PW? --GFauxPas 16:22, 27 December 2011 (CST)


 * The first paragraph of Definition:Power (Algebra), under "... and is defined recursively by ..." gives the definition of the zero and negative power. Presumably $a$ and $b$ here are real, so Multiplicative Group of Real Numbers holds and so you can use Powers of Group Elements. Job done. --prime mover 16:44, 27 December 2011 (CST)


 * Oh okay! I think it's a distinct enough result that it deserves its own page though?, even if the proof is just "follows directly from huiaehjebgthjbqw".
 * If you feel like putting it up, go for it. Transclude it into the page that does the rest of the power laws is the way I'd do it. --prime mover 17:04, 27 December 2011 (CST)
 * Also, do you have any thoughts of how to un-circularfy the proof for $D_x\ln x$? What do you think about LF's idea? I'm not learning convergence in depth until 2012, so I can't do that proof myself yet--GFauxPas 16:51, 27 December 2011 (CST)
 * Not tonight, Josephine, I've got a headache. --prime mover 17:04, 27 December 2011 (CST)