tag:blogger.com,1999:blog-3722233.post4565358997401984776..comments2022-01-27T21:41:51.434-06:00Comments on Computational Complexity: either pi is algebraic or some journals let in an incorrect paper!/the 15 most famous transcendental numbersLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-3722233.post-89978261911339456762017-09-12T01:30:19.065-05:002017-09-12T01:30:19.065-05:00Sorry, abort the previous comment :)
Phi is irrati...Sorry, abort the previous comment :)<br />Phi is irrational but not transcendental...Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-55828134146654991072017-09-12T01:25:27.789-05:002017-09-12T01:25:27.789-05:00The golden ratio phi from Fibonacci numbers, which...The golden ratio phi from Fibonacci numbers, which should probably take the third position for its importance: https://en.wikipedia.org/wiki/Golden_ratioAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-37424344247564179372017-09-10T21:59:02.566-05:002017-09-10T21:59:02.566-05:00Log(2) is famous because it is the sum of the alte...Log(2) is famous because it is the sum of the alternating harmonic series.NP Slaglehttps://www.blogger.com/profile/06322388966706601689noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1711228947909149882017-09-02T05:40:39.267-05:002017-09-02T05:40:39.267-05:00i^i = exp(i Log(i)) = exp(i(log(1) + iArg(i))) = e...i^i = exp(i Log(i)) = exp(i(log(1) + iArg(i))) = exp(-pi/2) = exp(pi)^(-1/2) is transcendental by 11.Greghttps://www.blogger.com/profile/03676035237635094814noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-46876658223179748732017-09-01T10:38:16.904-05:002017-09-01T10:38:16.904-05:00I think it's impossible to not be convinced th...I think it's impossible to not be convinced that γ is interesting. Consider f(n)=log(n!), and g(n)=1+1/2+...+1/n. Take the derivative of f at any point (with n! understood in terms of gamma function). The difference between the result and g(n-1) is ALWAYS equal to γ. Euler's constant is all over the place in the theory of special functions. It's arguably as fundamental as e or pi.Mahdihttps://www.blogger.com/profile/12382401759237060260noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-30423246963995978302017-08-29T04:43:46.626-05:002017-08-29T04:43:46.626-05:00"(Actually, this is one of many possible valu..."(Actually, this is one of many possible values for i to the i, because, for instance, exp(5i*Pi/2)=i. In complex analysis, one learns that exponentiation with respect to i is a multi-valued function.)"<br /><br /><br />From: https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-27950066883263417332017-08-28T15:08:51.627-05:002017-08-28T15:08:51.627-05:00What is the definition of i^i? It seems that you h...What is the definition of i^i? It seems that you have to give up some property of the power function to properly define it. For example i^5=i, so if we define t=i^i, then t^5 = i^{5i}=(i^5)^i = i^i = t, which in particular means that t is algebraic. If you want to remain consistent with Euler's formula then you're in bigger trouble since i=exp(i*(pi/2+2pi*k) for any integer k, raise this to the power i and you get an inconsistent value.Anonymousnoreply@blogger.com