Reply by Abe Kohen September 25, 20052005-09-25
"kiki" <lunaliu3@yahoo.com> wrote
> "quasi" <quasi@null.set> wrote > > On Sun, 24 Jul 2005 11:14:52 -0700, "kiki" <lunaliu3@yahoo.com> wrote: > > > >>... I am wondering if anybody has seen > >>a collection of mathematical tricks ranging from high school math up to > >>graduate school math? Any website, Internet resources, books that have > >>these > >>kind bags of tricks? If not, I may want to start collecting and compile > >>one > >>such resources.
For HS math: The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk. Best, Abe
Reply by ToddSmith July 30, 20052005-07-30
Yeah you're right. It's just a regular wiki linked to a regular phpBB.
It's still the best place to go for graduate level math, though!

www.exampleproblems.com - Graduate level math
-Todd

Reply by Jane July 27, 20052005-07-27
"ToddSmith" <elliptic1@gmail.com> wrote in message 
news:1122342726.367323.304240@f14g2000cwb.googlegroups.com...
> Yes the code you use to write equations is basically the same as TeX. > Thanks for going to it. Post some stuff! > > -Todd Smith > www.exampleproblems.com >
Hi Todd, Surely I will frequent you forum... how did you put math into the PHPBB forum? I also want to luanch a forum that supports math... can I licence your invention? Thanks a lot!
Reply by Jane July 26, 20052005-07-26
"ToddSmith" <elliptic1@gmail.com> wrote in message 
news:1122342726.367323.304240@f14g2000cwb.googlegroups.com...
> Yes the code you use to write equations is basically the same as TeX. > Thanks for going to it. Post some stuff! > > -Todd Smith > www.exampleproblems.com >
Hi Todd, When I went to the forum session, http://www.exampleproblems.com/phpbb/ The equation editing feature was gone. So the forum itself does not support Latex equations, right? Thanks a lot!
Reply by Henry July 26, 20052005-07-26
On Tue, 26 Jul 2005 00:16:51 -0700, quasi <quasi@null.set> wrote:

>On 25 Jul 2005 23:22:08 GMT, rusin@vesuvius.math.niu.edu (Dave Rusin) >wrote: > >>The intuition with limits may >>be important in some contexts but there is a priori nothing wrong >>with a definition like >> >> 1. x^0 =1 if x > 0. >> 2. 0^x = 0 if x > 0. >> 3. 0^0 = 17 > >Sure the definition of 0^0 is arbitrary but the only natural choices >are "undefined", 0, or 1. As for 0 or 1, even disregarding the >continuity argument, why should 0^x have priority over x^0 or vice >versa.
What is unnatural about: For any a>1 (a^(-1/x)) ^ x = 1/a if x>0?
Reply by Jerry Avins July 26, 20052005-07-26
David Kastrup wrote:
> quasi <quasi@null.set> writes: > > >>On 25 Jul 2005 23:22:08 GMT, rusin@vesuvius.math.niu.edu (Dave Rusin) >>wrote: >> >> >>>The intuition with limits may >>>be important in some contexts but there is a priori nothing wrong >>>with a definition like >>> >>>1. x^0 =1 if x > 0. >>>2. 0^x = 0 if x > 0. >>>3. 0^0 = 17 >> >>Sure the definition of 0^0 is arbitrary but the only natural choices >>are "undefined", 0, or 1. As for 0 or 1, even disregarding the >>continuity argument, why should 0^x have priority over x^0 or vice >>versa. > > > Because 0^x is discontinous at 0 even if you set 0^x=0? Setting it at > 0 only for one-sided continuity seems to be less useful than setting > it to 1 for total continuity of x^0 everywhere on C.
Have we looked at exp(-1/x) lately? Jerry -- Engineering is the art of making what you want from things you can get. &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
Reply by Dave Seaman July 26, 20052005-07-26
On Tue, 26 Jul 2005 14:45:18 +0200, David Kastrup wrote:
> Dave Seaman <dseaman@no.such.host> writes:
>> There are some who prefer to say that 0^x = 1 whenever the exponent >> is viewed as an integer or a rational number,
> Cough, cough.
Make that x^0 = 1. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Reply by Shmuel (Seymour J.) Metz July 26, 20052005-07-26
In <1h09nx3.19tuzby1lbqkqoN%mh@michael-hoppe.de>, on 07/25/2005
   at 07:18 PM, mh@michael-hoppe.de (Michael Hoppe) said:

>The proof of
> sin(a + b) = (sin a) * (cos b) + (sin b) * (cos a)
>according to E. Schmidt runs as follows:
> f(x) := sin(a + b - x)*cos(x) + cos(a + b - x)*sin(x)
>is constant on R (as well on C), because f' = 0. From
> f(0) = f(b)
>follows the desired theorem.
There's an easier trick. cos(a+b) + i*sin(a+b) = e^i*(a+b) = e^i*a * e^i*b = (cos(a) + i*sin(a)) * (cos(b) + i*sin(b)) = (cos(a)*cos(b) - sin(a)*sin(b)) + i*(cos(a)sin(b) + sin(a)*cos(b)) It's fast and easy to work out just about any common trig identity with this tool. -- Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel> Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org
Reply by David Kastrup July 26, 20052005-07-26
Dave Seaman <dseaman@no.such.host> writes:

> There are some who prefer to say that 0^x = 1 whenever the exponent > is viewed as an integer or a rational number,
Cough, cough. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
Reply by Dave Seaman July 26, 20052005-07-26
On Tue, 26 Jul 2005 00:16:51 -0700, quasi wrote:

> Sure the definition of 0^0 is arbitrary but the only natural choices > are "undefined", 0, or 1. As for 0 or 1, even disregarding the > continuity argument, why should 0^x have priority over x^0 or vice > versa.
Because x^0 = 1 for all x. That's a theorem of ZF according to Suppes, _Axiomatic_Set_Theory_. The x in that case is an arbitrary cardinal number (finite or infinite), but it is easily extended to other values of x. The theorem in Suppes is based on the definition of cardinal exponentiation, which is that if A and B are cardinals, then A^B is the cardinality of the set of mappings from B into A. In the case A = B = 0 = {}, there is exactly one such mapping, hence 0^0 = 1. According to Lang's _Algebra_, if M is a multiplicative monoid with identity e, then m^n makes sense for each m in M and each natural number n, and furthermore m^0 denotes an empty product, which evaluates to e (the identity in M). Notice that nothing I have said above has anything at all to do with limits. In fact, there isn't even a topology involved. There are some who prefer to say that 0^x = 1 whenever the exponent is viewed as an integer or a rational number, but that 0^0.0 remains undefined if the exponent is regarded as real. <http://db.uwaterloo.ca/~alopez-o/math-faq/node40.html#SECTION00530000000000000000> -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>