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The good things about maths is that you can argue about anything and maths progress through debates. Great mathematicians in the past also made many mistakes and their theories either refuted or become theorems. Euler's conjecture was refuted when someone found the counter-examples which was not possible to be calculated during his time. Lame made a mistake in his attempt to proof Fermat's Last Theorem. We know for sure the summation of 1-1+1-1.. is either one or zero for odd and even terms respectively but this Grandi's series could be 1/2 and it depends on how people interpret it. Of course, 0.99999..recurring could be 1 but I beg to differ even when 0.999999..recurring=3x0.3333333..recurring or 3 x 1/3=1. By the way God particle has being found and scientists are 99.99995% sure about finding it at CERN. Maybe they need to be 99.9999%..recurring sure before it could be accepted:) http://www.dailymail.co.uk/sciencetech/article-2167188/God-particle-Scientists-Cern-expected-announce-Higgs-boson-particle-discovered-Wednesday.html
Well! If you draw a curve of y=1/x, you would see that y would be infinite as x approaches 0 but it would never touch y-axis indicating infinite value of y. Same with the so called 0.99999..recurring, if you draw a curve of y=0.9999..recurring..you would see, even if x approaches infinity y will never intersect line y=1. I bet to differ with many people regarding infinity. For example, in mathematics, to make sure it is true not enough to prove it up to billions or trillions terms but should be all term even up to infinity. This is why mathematicians developed method of induction. When you prove x=0.9999..recurring=1, you simply multiply it with 10 and then minus the original one. Before we talk it to the infinity why not we talk it on a finite scale. Lets say n=decimals points, when you have like n=1 trillion decimals and multiplying 0.999..recurring(up to 1 trillion decimals after 0) with 10, you will get 9.99..recurring( 1 trillion minus 1 point of decimals after first digit 9). Then, by subtracting the ten fold of x with original x, you will get 9x=8.999..(recurring 9 and ends with digit 1). Solving the equation, you will never get x=1. Now, consider infinite recurring 9, when you multiple x with 10, you take infinity as an integer yet you contradict the basic fundamental of math dealing with numbers. When you multiple a number with 10, you simply shifted a decimal point forwards and leaving the trailing digit as zero. For example, when you have x=0.9999 and multiply by 10, you will get 10x=9.9990 because a decimal point had being shifted forwards yet in proving 0.9999..recurring=1 you simply abuse the basic mathematics as if x=0.9999 and multiply by 10=10x=9.9999 (The last digit should be zero and not 9). The same thing happens in the infinite series, if you want to take it as a number not as a concept then you have to obey the fundamental of mathematics operations. If you obey this fundamental you will get 10x-x=9x=8.9999999..(recurring) ends with 1. Thus x/=1. When people talk about infinite series, they simply abuse it. To them infinity-1 is equal to infinity even though it is one less. Yet they simply multiply infinite series with integer and do basic mathematical operation without considering it could be false. This is how Euler linked his prime numbers with Riemann's Zeta Function or how Ramanujan got 1+1+1....=-1/12 or 1-1+1-1+1...=-1/12 because people don't believe in even or odd infinity yet they believe infinite series could be multiplied by integers to suit their ideas. If they could abuse it, then why not I say, Riemman's Zeta function C(2)=4 not (pi^2)/6. The value of 1/0 could be defined on certain circumstances because the creation of this universe after the big bang tells us a different thing, at t=0, the volume of universe is zero with infinite density or density=mass/volume=>density=mass x 1/0 from this relationship we get the existence of this entire universe. Indicating that 1/0 is not undefined because it created something which we could see or perceive now as observable universe! Machines have simple lives, they only knows 1 and 0 and when they see 1/0, the simply say error:)
Hi Testu
Basically nobody knows whether this universe is finite or infinite but many thinks it is infinite as the universe still expanding and the birth or stars still occurring. If you look into the formation of this universe, in the beginning, the volume of this universe is zero with infinite density (singularity). From zero to something after t=0, since the density is infinite, the formation of mass of particles should be also infinite. http://en.wikipedia.org/wiki/Gravitational_singularity . In other words, all the things or masses in this entire universe could be confined into a single piece of thing which has the volume near zero after t=0s. If this is a singularity, then the universe is infinite or maybe so big that we couldn't simply comprehend because there is a place in this universe where the space expanded further away way before a light could reach it. How big this universe is, perhaps as big as 1/0 (division by zero).
Hi anonimnystefy
The derivation is quite lengthy:
By using my formulation for sums of power for integers and non-integers power, you would get this relationship,
Setting p=-s
Taking limit n to infinity and simplifying the coefficients yields,
I agree bobbym, sometimes the next prime just jumps into gigantic number out of blue. Sometimes they are just next to each other. You can never understand prime.
Hi Bobbym!
I like all numbers:). I am currently working of sums of power for non-integers. There is a lot of things making us confused when dealing with infinity. Here is my Riemman's Zeta Function derived using my sums of power formulation.
As for integers I am done with them, I had formulated enough formulation for them on my capability. Now I am working on non-integers but not like whole numbers, what you get from non-integers is approximation. I have developed numerical analysis for non-integers power like how to add this equation
, the method not that amazing because sometimes it takes you to infinity to calculate a simple value.Of course you are right! I didn't mean all numbers are real of course there are imaginary numbers etc. But in sense of human capability of understanding it, even when it is real number it doesn't looks real to our understanding like the value of pi, we can have exact circle yet the pi in it is irrational, it is a mystery. I just pointing that, that even the so call imaginary numbers does exist in many real things or coexist with real numbers like roots of equation. Not people like Ramanujan formulated 1+2+3+4...=-1/12 got carried away? Maths were based on thinking and opinions of people and there are some maths which were considered fringe science and later on were proven to be true. Words or languages evolved through times and we human made them, English didn't exist 50,000 thousands years ago, same with the word GOOGLE, this word wasn't there before us. Unless you do believe that human was from Adam and Eve and they already made with language embedded in them. Mathematics is the only thing that you can make all people to agree 100% that something is true or not, our opinions differ but not the mathematics. Pythagoras Theorem doesn't need our opinion to be true, itself is true, we give opinions and the true opinion will fill in the gap eventually. This is why you can give your opinions what ever you like because restricting opinions would make Mathematics static. From garbage opinions there could be something true.
Basically, I don't think I am too much to say if there is the One who invented the mathematics the one must be super genius etc. Because not the same thing applies in the quest of God particle or Higg Boson? They had built multi billions facility to find this mystery thing which is is thought to be the most fundamental subatomic particles which could be the fundamental building block of this universe. http://www.guardian.co.uk/science/2011/nov/07/massive-hunt-god-particle-ian-sample-review . They built 7.5 billions Euros Hydra Collider to find God Particle, shouldn't they keep the word God out of this too? Perhaps Euler, one of the greatest mathematicians shouldn't even ventured further to proof the existence of God http://en.wikipedia.org/wiki/Leonhard_Euler#cite_note-41 because is it not ethical to link anything to one's belief? Our knowledge evolved from people opinions regardless what they are believing in and I stand on this.
Don't give up anonimnystefy! The largest Prime nearly 13 million digits! 100k digits is not that big in comparison.
To be honest we are not real too:) Nothing is real actually in this world, our intelligence is about comparing things, the unit of Energy Joule is not exactly a unit of energy in real sense but representation of comparison what this guy name could do to a temperature etc. Even a meter length is not real but a comparison of a metal bar stored somewhere in France. Numbers act the same ways too. They are there for us to compare but there are numbers we can't compared like irrational, imaginary and infinity numbers. Basically, astrophysics got to do with mathematics, like singularity 1/0, it doesn't exist in our limited knowledge yet it does exist in the formation of this universe or black holes for examples. The 0.99999999.... is the same thing as playing with singularity, to get 0.99999..you need to do this 1/(1000000..recurring zeros until infinity and at the end of infinity you need to finish it with digit 1). Maybe all numbers are real, even the irrational, imaginary,infinity etc but our brain capacity couldn't cope with it. Like we could have a perfect circle yet pi is irrational. We could draw a perfect square with a unit sides yet we failed to measure its diagonal with 100% accuracy. We can see it yet we can't measure accurately 100%, it is sad somehow:). There is limitation in us to understand all mathematics but if there is the One who invented mathematics that we found in this universe He must be endless genius and He is either infinite or more than it.
Sorry anonimnystefy..not mass but density. Like the birth of supernova, in the beginning of its creation, the mass of 1 miligram of it could be several times or perhaps million times more dense than our entire solar system masses combined together. I think imaginary numbers are like the images in the mirror, they look real but just images and they need real things to exist.
*Pythagoras was not the world's first mathematician. * I didn't say he was the first mathematicians but I gave his name as example. Mathematics was there even before human were first on earth. This universe is governed by mathematics billions years ago, I think the first concept of singularity ever known to us is the creation of this universe, when t=0, the volume of universe is zero but with infinite density. If you read the Hindus manuscripts you would find that they had done earlier many of the mathematics found in Europe or Middle East. Yet Hindus civilization not even older than the Sumerian in Iraq, Sumerians had Pythagorean triple ages ago written on stone tablet. Where ever you found wheel or circle, then you can find pi and Pythagoras theorem. Now they found a new civilization in Turkey dated back 12,000 years and they predated the invention of wheels. http://www.youtube.com/watch?v=TZ0ViMVxKZA&feature=related
You can perceive 3 because it is part of decimal system. Human developed numbers according to our fingers. How many fingers do you have? 10...so what is 3? ..one two three fingers..can you calculate 1/2? yes..0.5 or half fingers..can you quantify 1/3..no way..until forever 3333333 would go on. There is no absolute value you can quantify from 1/3 using our finger system or decimal! Infinity is a concept but numbers is real..if you play with infinity then you compromise many things. Therefore, anything deals with the infinity like irrational numbers decimal places..you can never actually write it down, the fraction we representing them is more like a symbol not a value. I can prove you that 0.999999999.. can never be 1 because (1-0.99999999999..)x infinity not equal to zero. Yet (1-1) x infinity=0! Can you give me three apples? Yes you can but can you give me exactly
apples? I hope you could:)I think we human didn't invent numbers because it is already there even before we existed. We discovered mathematics and we don't invent anything dealing with mathematics. For example, Pythagoras didn't invent his theorem but he discovered it. This is because the movement our our galaxy, sun, earth, planets etc involved Pythagoras mathematics long before we existed on this planet.
Basically x^0=1 because
and Mathematics is a knowledge field that depend on proof because if anything could contradict it, would prove it wrong. Pythagoras believed that numbers were not only the way to truth, but truth itself. http://www.ancient.eu.com/article/213/ . While a concept not always true, could be wrong sometimes!0.9999999.... or 3x (1/3) doesn't exist in our human limited understanding. It is like partitioning a unit into 3. The divided units (1/3) only exist in the mathematics and not in the real world. This is what we call irrational numbers, same with the diagonal length of a square with a unit sides. We can see these irrational numbers in our life even though they actually don't exist on their own. Can you measure 1/3 or square root of 2? Square root of 2 does exist in a square but can you measure it? Never ever because they don't exist in the real world. In other words, irrational numbers only exist as complementary to others but when you take them out on their own, they do exist as a form which can't be quantify exactly into decimal system because they are endless into infinity. This is why you can never say 0.99999.....=1 because somewhere in the infinity there is a residue left when 1-0.9999....it is endless.
Hi anonimnystefy!
Can you get us P9?:)
P7=235711131719232931374143475359616771737983899710110310710911312713113713914915115716316717317918119119319719921122322722923323924125125726326927127728128329330731131331733133734734935335936737337938338939740140941942143143343944344945746146346747948749149950350952152354154755756356957157758759359960160761361761963164164364765365966167367768369170170971972773373974375175776176977378779780981182182382782983985385785986387788188388790791191992993794194795396797197798399199710091013101910211031103310391049105110611063106910871091109310971103
11091117112311291151115311631171118111871193120112131217122312291231123712491259127712791283128912911297130113031307131913211327136113671373138113991409142314271429143314391447145114531459147114811483148714891493149915111523153115431549155315591567157115791583159716011607160916131619162116271637165716631667166916931697169917091721172317331741174717531759177717831787178918011811182318311847186118671871187318771879188919011907191319311933194919511973197919871993199719992003201120172027202920392053
20632069208120832087208920992111211321292131213721412143215321612179220322072213222122372239224322512267226922732281228722932297230923112333233923412347235123572371237723812383238923932399241124172423243724412447245924672473247725032521253125392543254925512557257925912593260926172621263326472657265926632671267726832687268926932699270727112713271927292731274127492753276727772789279127972801280328192833283728432851285728612879288728972903290929172927293929532957296329692971299930013011301930233037
This number has 1543 Digits, ok Bobby, get me P8.
Kool Bobbyn:)..then P7 could be at least 2000-3000 Digits..I guess
I think P6 would be having 800-1000 digits or even bigger. Would try finding it tomorrow.
You can check them using ECPP http://www.ellipsa.eu/ (Primo) or ECM/SIQS/Rabin-Miller http://www.alpertron.com.ar/ECM.HTM
I did check them all but in case I missed any one of them, hope people can update it.
Just found P5 (499 Digits).
P5=2357111317192329313741434753596167717379838997101103107109113127131137139149151157163167173179181191193197199211223227229233239241251257263269271277281283293307311313317331337347349353359367373379383389397401409419421431433439443449457461463467479487491499503509521523541547557563569571577587593599601607613617619631641643647653659661673677683691701709719727733739743751757761769773787797809811821823827829839853857859863877881883887907911919929937941947953967971977983991997100910131019102110311033
So, can someone find P6?
What is the biggest Prime Number you can get by arranging consecutive prime numbers side by side? Few examples:
P1=2
P2=23
P3=2357
P4=2357111317192329313741434753596167717379838997101103107109113127131137139149151157163167173179181191193197199211223227229233239241251257263269271277281283293307311313317331337347349353359367373379383389397401409419421431433439443449457461463467479487491499503509521523541547557563569571577587593599601607613617619631641643647653659661673677683691701709719 (355 Digits) Hope I didn't miss any prime number in this sequence!
Can you get P5?
All equations are governed by their digital root cyclic order for all n-term (digital root 1..9). For the power series (i.e x^n), their digital root would repeat forever. From the plots of digital root for the prime numbers less than 2000, we can conclude that even at prime number less than 2000 it would be almost impossible to get an equation that could governed them. Therefore, as prime numbers go into infinity, there would be no equation could governed them. If there is an equation that could governed them it would definitely having an infinite variables.
Ok..I got how to do it..
My blog is here: http://stangerzv.wordpress.com/2012/05/31/using-digital-root-to-verify-equations-especially-the-prime-distribution/
Both I guess..from my machine and my blog.