The Infamous Number PHI

I just heard about this incredibly number tonight, can anyone give me any interesting facts about it?




Sorry if this isn't in the right forum, I just didn't know where to put it.

 

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Knock yourself out:

Special Numbers

Phi is listed as "Golden Ratio" on this page.

This page is also good. Better, actually:
MathWorld - Constants

 

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Mgswini:

Goody- queer numbers.

Phi- I new of Phideas and the parthenon but never knew this golden ratio, found all over nature as is the Fabonocci sequence, was called Phi. It was also rumored among Greeks that this ratio on the male body makes it aesthetically beautiful.

I vaguely remember "charmed" numbers and the story behind the imaginary ones which I'm too tired or moody to remember now, but intersting all the same.

Perhaps you've heard of them, don't know.

 

And the book is "The Magical Story of (the radical sign with the i beneath)"

 

I know this may shock you, but Phi is my favorite number. Before I clicked on the link in the second post, I didn't know that phi and Phi were two different things, otherwise I would have made my username "Phi." Anyway, I'll try to tell you all the cool stuff I know about Phi.

Phi, which is equal to (1 + 5^(1/2))/2 (approximately 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203408058879544547492461856953648644492410443207713449470495658467885098743394422125448770664780915884607499887124007652170575179788341662562494075890697040002812104276217711177780531531714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782977834784587822891109762500302696156170025046433824377648610283831268330372429267526311653392473167111211588186385133162038400522216579128667529465490681131715993432359734949850904094762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561943208275051312181562855122248093947123414517022373580577278616008688382952304592647878017889921990270776903895321968198615143780314997411069260886742962267575605231727775203536139362107673893764556060605921658946675955190040055590895022953094231248235521221241544400647034056573479766397239494994658457887303962309037503399385621024236902513868041457799569812244574717803417312645322041639723213404444948730231541767689375210306873788034417009395440962795589867872320951242689355730970450959568440175551988192180206405290551893494759260073485228210108819464454422231889131929468962200230144377026992300780308526118075451928877050210968424936271359251876077788466583615023891349333312231053392321362431926372891067050339928226526355620902979864247275977256550861548754357482647181414512700060238901620777322449943530889990950168032811219432048196438767586331479857191139781539780747615077221175082694586393204565209896985556781410696837288405874610337810544439094368358358138113116899385557697548414914453415091295407005019477548616307542264172939468036731980586183391832859913039607201445595044977921207612478564591616083705949878600697018940988640076443617093341727091914336501371576601148038143062623805143211734815100559013456101180079050638142152709308588092875703450507808145458819906336129827981411745339273120809289727922213298064294687824274874017450554067787570832373109759151177629784432847479081765180977872684161176325038612112914368343767023503711163307258698832587103363222381098090121101989917684149175123313401527338438372345009347860497929459915822012581045982309255287212413704361491020547185549611808764265765110605458814756044317847985845397312863016254487611485202170644041116607669505977578325703951108782308271064789390211156910392768384538633332156582965977310343603232254574363720412440640888267375843395367959312322134373209957498894699565647360072959998391288103197426312517971414320123112795518947781726914158911779919564812558001845506563295285985910009086218029775637892599916499464281930222935523466747593269516542140210913630181947227078901220872873617073486499981562554728113734798716569527489008144384053274837813782466917444229634914708157007352545707089772675469343822619546861533120953357923801460927351021011919021836067509730895752895774681422954339438549315533963038072916917584610146099505506480367930414723657203986007355076090231731250161320484358364817704848181099160244252327167219018933459637860878752870173935930301335901123710239171265904702634940283076687674363865132710628032317406931733448234356453185058135310854973335075996677871244905836367541328908624063245639535721252426117027802865604323494283730172557440583727826799603173936401328762770124367983114464369476705312724924104716700138247831286565064934341803900410178053395058772458665575522939158239708417729833728231152569260929959422400005606266786743579239724540848176519734362652689448885527202747787473359835367277614075917120513269344837529916499809360246178442675727767900191919070380522046123248239132610432719168451230602362789354543246176997575368904176365025478513824631465833638337602357789926729886321618583959036399818384582764491245980937043055559613797343261348304949496868108953569634828178128862536460842033946538194419457142666823718394918323709085748502665680398974406621053603064002608171126659954199368731609457228881092077882277203636684481532561728411769097926666552238468831137185299192163190520156863122282071559987646842355205928537175780765605036773130975191223973887224682580571597445740484298780735221598426676625780770620194304005425501583125030175340941171910192989038447250332988024501436796844169479595453045910313811621870456799786636617460595700034459701135251813460065655352034788811741499412748264152135567763940390710387088182338068033500380468001748082205910968442026446402187705340100318028816644153091393948156403192822785482414510503188825189970074862287942155895742820216657062188090578088050324676991297287210387073697406435667458920258656573978560859566534107035997832044633634648548949766388535104552729824229069984885369682804645974576265143435905093832124374333387051665714900590710567024887985804371815126100440381488), is actually the solution to a quadratic equation. The cool thing about Phi is that its inverse or reciprocal is equal to Phi minus 1. Therefore, you can write:
x^-1=x-1 multiply both sides by x
1=x^2-x subtract 1 from both sides and flip around
x^2-x-1=0
a=1, b=-1, c=-1
x=(-b[+or-](b^2-4ac)^(1/2))/2a quadratic formula
x=(-(-1)[+or-]((-1)^2-4(1)(-1))^(1/2))/2(1) substitute in a, b, and c
x=(1[+or-](5)^(1/2))/2
Thus, we see that two answers are obtained: Phi and phi. Cool, huh?

But what is Phi squared? It's Phi plus 1. What about Phi cubed? It's Phi plus 1 plus Phi. In fact, my good friend Hector Berlioz expressed this interesting phenomenon in something he calls "The Power Sequence": Phi^n=Phi^(n-1)+Phi^(n-2).

That's not all; did you know that Phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...)))))...

Did you know that the five arms of a pentagram are golden triangles? That is, the ratio of the short side to the two long sides is 1 to Phi (or phi to 1). This is how Pythagoras first calculated an approximation of Phi. There are actually two kinds of golden triangles; one is acute and the other is obtuse. If you draw a line connecting the tips of two of the arms of a pentagram, you reveal the obtuse kind, in which the ratio of the the short side to the two long sides is Phi to 1 (or 1 to phi).

 

Oh, those are a couple of great links. Thanks, Pete!

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Let's not for whatever reason confuse the number φ (phi) with the number phive.
In fact, let us destroy the number phive; it should not exist.