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irrational_numbers_data_compression [2018/04/21 03:42] (current)
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 +====== Irrational Numbers & Data Compression ======
  
 +By Tony Ganino Bach Science Info Tech.
 +
 +[[treating_virus_with_radiation|]]
 +
 +Irrational numbers are infinite algorithms with variable patterns where the logic of all the data ever can be communicated in several kilobytes. For example if I transmit the symbol π in 1 byte, most would know what I mean if that happened to be the bitstream for your fav blu ray movie then we would have a pretty good compression ratio, but a mere blu ray movie is really nothing for Pi as it never ends.
 +
 +==== We are gonna need some big computers... ====
 +
 +Their are limits to data compression,​ what is ultimately possible is open territory for anyone intelligent enough. Conveying a bit stream in relation to where it can found in π or a derivative irrational number will in practice take as long as transmitting the data. Theoretically one need only convery the starting (decimal place) and ending (decimal place) co-ordinates along with the irrational number used, __a block of data__ will coincidently be in a portion of pi eventaully. One does not trasmit data, one transmits the co-ordinates within the irrational number that signifies the data, a small packet of information can represent a truely vast amount of data.
 +
 +Their are different types of computers that excel at working out problems like these.
 +
 +==== How to work out combinations... ====
 +
 +Simply by multiplying all the numbers that come bfore it e.g 5 items = 5x4x3x2 = 120 total ways or grouping them, pairing, 3 of a kind, tall to smallest and all the rest of the possible ways without any rules except you can't repeat. Its a lot of combinations for a few items.
 +
 +Most home computers will go into endless loops at a 5 combination loop. Because computers are base 2 and not base 10, for 6 items we instead 128,​32,​16,​8,​4,​2 which are at 268,435,456 combinations.
 +
 +Thanks boss but this communicates an unbreakable condition by very act of concatenating the same symbols. So the regime of college will lock people into a way of thinking that does not allow advancement and probably why we cannot determine pi with the current age of numeracy.
 +
 +==== What if we can't find a match... ====
 +
 ++ or - approximation,​ so you have gotten a new blu ray DVD and you want to put it on the internet and away you go, convert the blu ray to hexadecimal and you begin searching base 16 pi for matches, searching all day and night you come up with close matches but no identical blocks, then you can also transmit + add or - subract to make it match. ​
 +
 +So now you will need to transmit more, the irrational number used, the starting point and the ending point and how much to subtract or add to get the data back on the other side.
 +
 +==== Irrational numbers candidates... ====
 +
 +=== π ===
 +
 +The most famous mathematical constant is pi, denoted by the Greek letter π, is the ratio of a circle'​s circumference to its diameter. It is used in geometric formulas for circles, spheres, cones, cylinders, and other shapes with circular cross-sections. Pi is a transcendental number. Pi approximates to **22/7** as a fraction. People have calculated Pi to over a quadrillion decimal places and still __there is no pattern__. ​
 +
 +π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989...
 +
 +to 1000 decimal places but it keeps going what might be forever...
 +
 +=== e ===
 +
 +The constant e, (Euler'​s Number) named after the famous mathematician Euler, is defined several ways. It is the limit of the expression (1 + 1/n)^n as n goes to infinity. It also the convergent value of the sum of factorial reciprocals:​
 +
 +1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ...
 +
 += 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...
 +
 += e
 +
 +In calculus, e is the base of the natural logarithm, and appears in a number of other curious identities and formulas. Like pi, it is a transcendental number. People have also calculated e to lots of decimal places without any pattern showing.
 +
 +e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952605956307381323286279434907632338298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537423454424371075390777449920695517027618386062613313845830007520449338265602976067371132007093287091274437470472306969772093101416928368190255151086574637721112523897844250569536967707854499699679468644549059879316368892300987931277361782154249992295763514822082698951936680331825288693984964651058209392398294887933203625094431173012381970684161403970198376793206832823764648042953118023287825098194558153017567173613320698112509961818815930416903515988885193458072738667385894228792284998920868058257492796104841984443634632449684875602336248270419786232090021609902353043699418491463140934317381436405462531520961836908887070167683964243781405927145635490613031072085103837505101157477041718986106873969655212671546889570350354...
 +
 +to 1000 decimal places but it keeps going what might be forever...
 +
 +=== Φ ===
 +
 +The Divine Ratio or the Golden Ratio (aka Golden Mean or Golden Section), denoted by the Greek letter φ (phi), is equal to [1 + sqrt(5)]/2. Geometrically,​ the Golden Mean is the proportions of the Golden Rectangle. If you cut a square from the Golden Rectangle, the remaining rectangular piece has the same side ratios as the original rectangle.
 +
 +The Golden Ratio is also the limiting ratio of consecutive Fibonacci numbers. For example, the 20th and 21st Fibonacci numbers are 6765 and 10946, and 10946/6765 = 1.61803399852. The 30th and 31st Fibonacci numbers are 832040 and 1346269, and 1346269/​832040 = 1.61803398875. As you can see these two ratios are very close to the actual value of φ. The larger the consecutive Fibonacci numbers, the better the accuracy.
 +
 +The Golden Ratio also satisfies the curious algebraic identity φ - 1 = 1/φ, and φ + 1 = φ^2. Because φ is the root of the polynomial x^2 - x - 1 = 0, it is an algebraic number.
 +
 +phi = 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203408058879544547492461856953648644492410443207713449470495658467885098743394422125448770664780915884607499887124007652170575179788341662562494075890697040002812104276217711177780531531714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782977834784587822891109762500302696156170025046433824377648610283831268330372429267526311653392473167111211588186385133162038400522216579128667529465490681131715993432359734949850904094762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561943208275051312181562855122248093947123414517022373580577278616008688382952304592647878017889921990270776903895321968198615143780314997411069260886742962267575605231727775203536139362...
 +
 +to 1000 decimal places but it keeps going what might be forever… ​
 +
 +=== Roots ===
 +
 +Many square roots, cube roots, etc are also irrational numbers.
 +
 +√3 1.7320508075688772935274463415059 (etc)
 +√99 9.9498743710661995473447982100121 (etc)
 +
 +=== sqrt(2) ===
 +
 +The square root of 2 is the length of the hypotenuse in an isosceles right triangle whose legs have length 1. In other words, if you take a square with sides equal to 1 and cut it in half along the diagonal, the length of the diagonal is sqrt(2). It was one of the first numbers proven to be irrational by the ancient Greeks.
 +
 +The square root of two is one of the roots of the equation x^2 - 2 = 0, thus it is an algebraic number.
 +
 +sqrt(2) = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237235288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836986836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847208969...
 +
 +to 1000 decimal places but it keeps going what might be forever...
 +
 +=== sqrt(3) ===
 +
 +The square root of three is the height of an equilateral triangle whose sides have length 2. Like sqrt(2), sqrt(3) is an algebraic number since it is the root of the polynomial equation x^2 - 3 = 0. The tangent of 60 degrees is sqrt(3).
 +
 +sqrt(3) = 1.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756756261414154067030299699450949989524788116555120943736485280932319023055820679748201010846749232650153123432669033228866506722546689218379712270471316603678615880190499865373798593894676503475065760507566183481296061009476021871903250831458295239598329977898245082887144638329173472241639845878553976679580638183536661108431737808943783161020883055249016700235207111442886959909563657970871684980728994932964842830207864086039887386975375823173178313959929830078387028770539133695633121037072640192491067682311992883756411414220167427521023729942708310598984594759876642888977961478379583902288548529035760338528080643819723446610596897228728652641538226646984200211954841552784411812865345070351916500166892944154808460712771439997629268346295774383618951101271486387469765459824517885509753790138806649619119622229571105552429237231921977382625616314688420328537166829386496119170497388363954959381...
 +  ​
 +to 1000 decimal places but it keeps going what might be forever...
 +
 +==== History of irrational numbers ====
 +
 +Hippasus (one of Pythagoras'​ students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn'​t write the square root of 2 as a fraction and so it was irrational.
 +
 +However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus'​ “irrational numbers” and so Hippasus was thrown overboard and drowned!
 +
 +==== Match the base to the bitstream... ====
 +
 +If you compute π using a Mac, does that make it an apple pi? 
 +
 +Ok, much of the effort to determine digits of π has been focused on the decimal expansion of the value. ​ However, it is interesting to see how π appears in different number bases. The values below are the first 100 digits. ​ Keep in mind that it would take different numbers of digits to express the same precision. ​ As an example, the 100 digits in binary are approximately equivalent to 30 digits in the decimal expansion. ​
 +
 +<​html>​
 +<​table>​
 +<​tbody><​tr valign="​top">​
 +  <td width="​140"><​b>​Binary</​b></​td><​td nowrap=""​ width="​90">​base = 2,</​td><​td>​digits:​ &nbsp; {0,​1}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">​11.00100 10000 11111 10110 10101 00010 00100 00101 10100 01100 00100 01101 00110 00100 11000 11001 10001 01000 10111 00000<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Ternary</​b></​td><​td nowrap="">​base = 3,</​td><​td>​digits:​ &nbsp; {0,​1,​2}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">​10.01021 10122 22010 21100 21111 10221 22222 01112 01212 12120 01211 00100 10122 20222 12012 01211 12101 21011 20022 01202<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Quaternary</​b></​td><​td nowrap="">​base = 4,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;​3.02100 33312 22202 02011 22030 02031 03010 30121 20220 23200 03130 01303 10102 21000 21032 00202 02212 13303 01310 00020<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Quintary</​b></​td><​td nowrap="">​base = 5,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;​3.03232 21430 33432 41124 12240 41402 31421 11430 20310 02200 34441 32211 01040 33213 44004 32444 01441 04233 41330 11323<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Sextary</​b></​td><​td nowrap="">​base = 6,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4,​5}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;​3.05033 00514 15124 10523 44140 53125 32110 23012 14442 00411 52525 53314 20333 13113 55351 31233 45533 41001 51543 44401<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Heptary</​b></​td><​td nowrap="">​base = 7,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4,​5,​6}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;​3.06636 51432 03613 41102 63402 24465 22266 43520 65024 01554 43215 42643 10251 61154 56522 00026 22436 10330 14432 33631<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Octal</​b></​td><​td nowrap="">​base = 8,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4,​5,​6,​7}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;​3.11037 55242 10264 30215 14230 63050 56006 70163 21122 01116 02105 14763 07200 20273 72461 66116 33104 50512 02074 61615<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Decimal</​b></​td><​td nowrap="">​base = 10,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4,​5,​6,​7,​8,​9}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3"><​b>&​nbsp;&​nbsp;​3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679</​b><​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Duodecimal</​b></​td><​td nowrap="">​base = 12,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4,​5,​6,​7,​8,​9,​A,​B}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;​3.18480 9493B 91866 4573A 6211B B1515 51A05 72929 0A780 9A492 74214 0A60A 55256 A0661 A0375 3A3AA 54805 64688 0181A 36830<​br>&​nbsp;</​td>​
 + </​tr>​
 + <​tr valign="​top">​
 +  <​td><​b>​Hexadecimal</​b></​td><​td nowrap="">​base = 16,</​td><​td>​digits:​ &nbsp; {0,​1,​2,​3,​4,​5,​6,​7,​8,​9,​A,​B,​C,​D,​E,​F}</​td>​
 + </​tr>​
 + <​tr>​
 +  <td class="​s"​ colspan="​3">&​nbsp;&​nbsp;<​b>​3.243F6 A8885 A308D 31319 8A2E0 37073 44A40 93822 299F3 1D008 2EFA9 8EC4E 6C894 52821 E638D 01377 BE546 6CF34 E90C6 CC0AC</​b><​br>&​nbsp;</​td>​
 + </​tr>​
 +</​tbody>​
 +</​table>​
 +</​html>​
 +
 +see https://​libraryofbabel.info/​
irrational_numbers_data_compression.txt · Last modified: 2018/04/21 03:42 (external edit)