Subject: Successive Square Roots of i [S/MIME] Date: Sun, 25 May 1997 01:45:03 -0400 Signed From: "David C. Manchester" Organization: David Manchester To: "David C. Manchester" http://xp9.dejanews.com/getdoc.xp?recnum=%3c4g0b7l$3gp@news.continuum.net%3e&server=db96q1&CONTEXT=864535317.31376&hitnum=30 -- PĂ --------------------------------------------------------------------- [Previous] [Next] [Current Results] [Get Thread] [Author Profile] [Post] [Post] [Reply] ------------------------------------------------------------------------ Article 31 of 36 Subject: Successive Square Roots of i From: Vead@q.continuum.net (D. Manchester) Date: 1996/02/15 Message-Id: <4g0b7l$3gp@news.continuum.net> Newsgroups: sci.math,sci.physics,sci.fractals [More Headers] A number of years ago (Nov. 1988) I lived in NC and my good friend Chris was about to graduate from RIT way up in NY. As we are both from SE CT, I was concerned about losing touch, but happily that hasn't happened. At that time Chris wrote me a letter on a subject of mutual interest, the square root of -1. (He was, after all, a Math major, and was aware of my interest in building a mathematical Pulsor (discontinuously pulsating Twistor) model of Space-Time. As fortune would have it, I promptly lost the letter. :( But a few years later, it turned-up in my canvas "bag 'o math" :) Chris has graciously given his ok for me to post it (so long as I do the data entry work). Any errors in transcription are mine. -dcm (Vead@q.continuum.net) [my comment] =================================================================== [11-88] Hey Dave! How's it sliding? I am here in Rochester, basking in the hedonistic glow of vacation, and have had some Free time to diddle with mathematics. Being thus vacated and diddling, I happened upon Some Identities that caused me to grin uncontrollably. I thought I'd share them with you, in the hopes you might incorporate them into a new theory of consciousness-time-space, and that I might subsequently steal all the credit. The First identity is based on the Familiar 2 2 a - b = (a+b)(a-b). We don't really _need_ a difference of _squares_, however, we could just as well write a - b = (sqr(a) + sqr(b)) (sqr(a) - sqr(b)) [in the letter, Chris used radical signs...prettier-dcm] We can repeat the process and Factor the BLUE term [on right above] similarly: a - b = (sqr(a)+sqr(b)) (sqr(sqr(a))+(sqr(sqr(b))) (sqr(sqr(a)) - (sqr(sqr(b))) We will always have a difference term left, so we can repeat this process indefinitely. After taking the limit and throwing around a lot of calculus, we get: 1 1 ( ___ ) ( ____ ) I. infinity n n _____ 2 2 a - b = (ln(a) - ln(b) | | ( a + b ) | | --------------- n=1 2 [this look better in longhand. the exponent above should be 1/(2**n)..dcm] This is what we get if we carry on the Factorization infinitely many times. I'll spare you the proof. (I can't find it!) Notice the 'log ' popped up. e Now consider a special case of I. where b=1: infinity (1/2**n) a - 1 _______ (a ) + 1 -------- = | | ------------------ ln(a) | | 2 i=1 [sorry about the notation switch...this goes on...dcm) or, II. ln(a) = 2 2 2 (a-1)(-----------------)(-----------------)(-----------------) ... 1/2 1/4 1/8 a + 1 a + 1 a + 1 A new expression For log , aside from the usual Taylor Series! e Now, the _natural_ thing to take the log of is -1, since ln(1)=i*pi. e So let's plug '-1' in for 'a': ln(-1)=i*pi= 2 2 2 (-2)(--------------)(--------------)(--------------) ... 1/2 1/4 1/8 (-1) + 1 (-1) + 1 (-1) + 1 or, III. 2 2 2 i*pi = (-2)(--------------)(--------------)(--------------) ... 1/2 1/4 i + 1 (i +1) (i +1) But what the hell is sqr(i)? Or sqr(sqr(i)) ? And so on? This question brings me to my prettiest set of equations, the successive square roots of i... *************************************** * * * The Successive Square Roots of i: * * * *************************************** What happens when we take sqr(i)? Do we get super-imaginary numbers? Hallucinogenic numbers? No, it turns out we just get ordinary complex numbers back. Take a look at this: [lovely recursively drawn mutant smiley omitted ')...dcm] i = sqr(-1). [i**(1/1)] sqr(i) = 1/2 ( sqr(2) + sqr(2)i ). [i**(1/2)] sqr(sqr(i))= 1/2 ( sqr(2+sqr(2)) + sqr(2 - sqr(2))i ). [i**(1/4)] sqr(sqr(sqr(i))) = 1/2 ( sqr(2+sqr(2+sqr(2))) + sqr(2 - sqr(2+sqr(2)))i ). [i**(1/8)] sqr(sqr(sqr(sqr(i)))) = [i**(1/16)] 1/2 ( sqr(2+(sqr(2+sqr(2+sqr(2)))) + sqr(2 - (sqr(2+sqr(2+sqr(2))))i ). ... ... ... [as I said, the original scrawl was done ] [ using radical symbols...rewrite it using ] [ them for a clearer picture of the ] [ recursive dynamic here....dcm ] Again, I'll spare you the proof (I'll send it to you if you like.) ___________ Interesting how the [radical]' / ' signs keep \/ 2 + ( ) re-entering themselves; G. Spencer-Brown would like it, I think. [G. Spencer Brown wrote THE LAWS OF FORM, a pre- mathematical treatment of symbolic logic...dcm] ---------------------------------------------------------------- (Sharpened my pencil ... much better.) Now that we can handle sqr(i), sqr(sqr(i)) and the like, we can go back to equation III and write a new equation for Pi in terms of i ... 2 4 4 i*Pi = (-2)(-------)(----------------)(----------------------------------) 1 + i 2+sqr(2)+sqr(2)i 2+sqr(2+(sqr(2)))+sqr(2-(sqr(2)))i [continued...fifth factor below..dcm] 4 x (--------------------------------------------------------------------) 2 + sqr(2+(sqr(2+sqr(2)))) + sqr(2 - (sqr(2+sqr(2)))) ... ... ... or, rearranged a little [God help me...], * 2 1-i 2+sqr(2)+sqr(2)i 2+sqr(2+sqr(2))+sqr(2-sqr(2))i ----- = (-----)(-----------------)(-------------------------------) Pi 2 4 4 [continued...fourth factor below..dcm] 2 + sqr(2 + sqr(2 + sqr(2))) + sqr(2 - sqr(2 + sqr(2)))i x(----------------------------------------------------------------)... 4 [ the 'x' above marked continuation in the original letter ] This equation connects Pi, i, and 2, while avoiding e. Oh, yea... I found the following identity completely by accident: * Pi - (-----) + (infinity)i = 2 (sqr(2)-sqr(2)i) (sqr(2 + sqr(2)) - sqr(2 - sqr(2))i) [continued...third factor below...x marks continuation..dcm] x ( sqr(2+(sqr(2+sqr(2)))) - sqr(2 - (sqr(2+sqr(2)))) i ) ... ... ... Well, that's all. Hope you Find these Formulas amusing. I'm sending you a copy of a book (undeserving old goon that you be...) called "Cognizers: Neural Networks and Machines that Think". It gives me an instant contact high. Also, I'm sending you a Basic program that allows you to experiment with imaginary numbers, supporting +, *, /, -, e**x, ln(x), plus trig. Functions. Let me know how the job in Jersey works out, and specifically, which exit? Unmistakeably yours, Chris Reiss. [ I hope this is of some use to someone...I found the forms of these equations quite aesthetic. I feel quite lucky to be blessed with such brilliant and witty friends.....dcm (Vead@q.continuum.net)] ========================================================================= "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."- Amendment I to the Constitution of the United States. D.Manchester Vead@q.continuum.net =========================================================================== ------------------------------------------------------------------------ [Previous] [Next] [Current Results] [Get Thread] [Author Profile] [Post] [Post] [Reply] ------------------------------------------------------------------------ Home Power Search Post to Usenet Ask DN Wizard Help Give us feedback! | Advertising Info | Press Releases | Jobs | Policy Stuff Copyright © 1995-97 Deja News, Inc. All rights reserved.