How many kinds of infinity are there ?

mart

Wolfetone said:

selfobsessednewguy said:

1 + 1/2 + 1/4 + 1/8 + ... = 2

Surely that does not equal 2 as following the progression you will never be able to reach 2 as the number will become infinitely close to 2 but never reach it. 

And that's the magic and mystery of those three dots. If you stop the sum at any finite point, say 

1 + 1/2 + 1/4 + ... + 1/32768

then, indeed, you won't get 2.

But you get closer and closer to 2 as you include more terms, and the ellipsis (...) is a mathematical short-hand for saying that 2 is the number that this sum is heading towards.

You cannot literally add infinitely many numbers together, so this is the best way to ascribe a meaning/value to an infinite sum.

dogload

Two, Infinity and beyond.



See what I did there? 


I'll get me coat.

Nitefly

A mathematician named Hall
Had a hexahedronical ball.
When doubled, its weight
Plus its volume, times 8,
Was two thirds of four fifths of fuck all.

PolarityMan

mart said:

Wolfetone said:

selfobsessednewguy said:

1 + 1/2 + 1/4 + 1/8 + ... = 2

Surely that does not equal 2 as following the progression you will never be able to reach 2 as the number will become infinitely close to 2 but never reach it. 

You cannot literally add infinitely many numbers together, so this is the best way to ascribe a meaning/value to an infinite sum.

Chuck Norris totally can.

thecolourbox

ChrisMusic said:

How many kinds of infinity are there ?

I think they mostly do Strats and Teles and a few basses, but might be best to go for a Classic Vibe or Vintage Modified as they are substantially better guitars

selfobsessednewguy

Wolfetone said:

selfobsessednewguy said:

1 + 1/2 + 1/4 + 1/8 + ... = 2

Surely that does not equal 2 as following the progression you will never be able to reach 2 as the number will become infinitely close to 2 but never reach it. 

i like the way that you think.

you are of course correct. however you're also not.

in mathematics when we talk of infinite series, we always mean 'limit'. so when i say

1 + 1/2 + 1/4 + 1/8 + ...

what i actually mean is

'which is the smallest number which cannot be exceeded by this series?'
or equivalently 'what is the limit of this infinite series?'

this series is said to 'converge to 2', and so we say 'the limit of this series as it continues off to infinity is 2' because we can actually get as close to 2 as we like.. so actually the 'dot dot dot' notation is unanimously taken to mean 'the limit as x tends to infinity of the sum from n=1 to x of 1/(2^n)' but i didn't write that and neither do mathematicians because there's no need... we all know what we're talking about. limits.

so how fascinating is it that the series

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... = PI SQUARED OVER 6 ?????????

this is the inverse squared series. the Basel problem back in the day was 'what is the limit of the infinite inverse squares series?' because it was used in physics but german physicists got tired of approximating it using that sum and they wanted to know what the limit is in some kind of closed form. and euler came along with a beautiful and very unconventional but relatively simple proof that the limit is exactly pi ^ 2 over 6.... so this series can not exceed that precise value. what the fuck is pi doing here? well i dont know. maths.

TheBlueWolf

Sausages!*


*The decent ones mind, not the cheap value filth, which are neither sausages or edible

pintspiller

PolarityMan said:

monquixote said:

There are infinite kinds of infinity. 

Well that's a given...but what *kind* of infinite kinds of infinity?

The kind that go on for ever and ever. Continuously.

ChrisMusic

selfobsessednewguy said:

so how fascinating is it that the series

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... = PI SQUARED OVER 6 ?????????


what the fuck is pi doing here?

Pi ?   and can I have custard with that, please ?     :-B

ChrisMusic

selfobsessednewguy said:

so how fascinating is it that the series

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... = PI SQUARED OVER 6 ?????????

this is the inverse squared series. the Basel problem back in the day was 'what is the limit of the infinite inverse squares series?' because it was used in physics but german physicists got tired of approximating it using that sum and they wanted to know what the limit is in some kind of closed form. and Euler came along with a beautiful and very unconventional but relatively simple proof that the limit is exactly pi ^ 2 over 6.... so this series can not exceed that precise value. what the fuck is pi doing here? well i dont know. maths.

more to the point defining one unresolvable number in terms of another unresolvable number !

I think all this points a finger in the direction of a fundamental misconception at the heart of our understanding of maths.

Crack the intrigue and paradoxes and maybe you have the next layer of understanding of the reality which the language maths describes ?
(maybe we don't have the full alphabet / lexicon yet?)

It would be interesting to come back and view the progress in this, in say 500 years time, once computing and algorithmic power has come of age and is roped in to enhance our conceptual understanding, rather than build reality based on weird and almost prehistoric approximations as we seem want to do now.  Really, we are better than that...

axisus

You want me to watch a 14.55 video?? talk about bloody infinity.

axisus

Scientists are still trying to find the opposite of infinity - outfinity. It's a never starting conundrum that has baffled egg-heads for centuries.

PolarityMan

ChrisMusic said:

selfobsessednewguy said:

so how fascinating is it that the series

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... = PI SQUARED OVER 6 ?????????

this is the inverse squared series. the Basel problem back in the day was 'what is the limit of the infinite inverse squares series?' because it was used in physics but german physicists got tired of approximating it using that sum and they wanted to know what the limit is in some kind of closed form. and Euler came along with a beautiful and very unconventional but relatively simple proof that the limit is exactly pi ^ 2 over 6.... so this series can not exceed that precise value. what the fuck is pi doing here? well i dont know. maths.

more to the point defining one unresolvable number in terms of another unresolvable number !

I think all this points a finger in the direction of a fundamental misconception at the heart of our understanding of maths.

Crack the intrigue and paradoxes and maybe you have the next layer of understanding of the reality which the language maths describes ?
(maybe we don't have the full alphabet / lexicon yet?)

It would be interesting to come back and view the progress in this, in say 500 years time, once computing and algorithmic power has come of age and is roped in to enhance our conceptual understanding, rather than build reality based on weird and almost prehistoric approximations as we seem want to do now.  Really, we are better than that...

The thing with Maths is that because it is an abstract system and based on proof nothing that we have now is going to be incorrect in 500 years time. In fact the speed of progress in the field is substantially slower than it was 500 years ago so we might not even have gained that much new understanding when compared to the existing body of work.

57Deluxe

isn't thinking that the expansion of the universe is slowing and at some point will contract back. In which case infinity does not exist at all.

ICBM

57Deluxe said:

isn't thinking that the expansion of the universe is slowing and at some point will contract back. In which case infinity does not exist at all.

No, current thinking is that the mass of the universe is far too small - even including the dark matter - to do that and the universe will continue to expand forever into emptiness and darkness. Which is a cheery thought.

selfobsessednewguy

ChrisMusic said:

selfobsessednewguy said:

so how fascinating is it that the series

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... = PI SQUARED OVER 6 ?????????

this is the inverse squared series. the Basel problem back in the day was 'what is the limit of the infinite inverse squares series?' because it was used in physics but german physicists got tired of approximating it using that sum and they wanted to know what the limit is in some kind of closed form. and Euler came along with a beautiful and very unconventional but relatively simple proof that the limit is exactly pi ^ 2 over 6.... so this series can not exceed that precise value. what the fuck is pi doing here? well i dont know. maths.

more to the point defining one unresolvable number in terms of another unresolvable number !

I think all this points a finger in the direction of a fundamental misconception at the heart of our understanding of maths.

Crack the intrigue and paradoxes and maybe you have the next layer of understanding of the reality which the language maths describes ?
(maybe we don't have the full alphabet / lexicon yet?)

It would be interesting to come back and view the progress in this, in say 500 years time, once computing and algorithmic power has come of age and is roped in to enhance our conceptual understanding, rather than build reality based on weird and almost prehistoric approximations as we seem want to do now.  Really, we are better than that...

well actually, we have been working with pi for much longer than 500 years. approximations of the ratio of a perfect circle to its diameter have roots going back more than 1000 years. we are very advanced in our development of mathematics, and in fact we know that pi is not an 'unresolvable' or somewhat mysterious number as many view it, it is what we call an irrational number, which is a number which cannot be expressed as a fraction because it is non-repeating. 

many numbers are irrational. that doesn't make them unresolvable or mysterious. square root of 2 is irrational, which means if we have a right angle triangle with sides of length 1, then the longest side has length sq(2), which is an irrational number. in fact, if we measure anything in the real world to a very high accuracy, you will find that any measurement will be non-repeating as the real world is not as perfect as mathematicians describe... so irrational numbers are everywhere.

the mystery here with the infinite sum is how the inverse squared series, if we take finitely many terms of ths series (making it resolvable) then it will be always under pi^2/6. however, i am saying mystery but it's not really a mystery to any mathematician who has read and understood the proof... but it still is.

however this is a very small thing when we compare it to some of the most beautiful results in mathematics. another one of my favourites is the fact that

https://en.wikipedia.org/wiki/Euler's_identity

(euler's identity) here we have an irrational number e which related to exponential growth and another irrational number pi, and also we have a complex identity called i which is defined as the 'imaginary number' used in complex analysis, defined mathematically as sq(-1). if we raised e to the power i*pi, we get fucking -1, an integer. AN INTEGER.





A FUCKIN INTEGER

selfobsessednewguy

i dont know how to delete comments

ChrisMusic

The "say 500 years" had no significance other than to moot the potential exponential growth in the trilogy of the power of computing, the increasing sophistication of algorithms, and the advent of 'proper' artificial intelligence.  Once those systems are left to develop themselves, then I think there is the potential for a quantum shift in our understanding of the inner workings of the universe.  IMHO, we have nowhere near the sophisticated understandings we like to credit ourselves with as a species, but we are developing the tools to furnish the never-ending curiosities the human race is blessed with.  

ChrisMusic

Oh, and thanks for your posts @selfobsessednewguy , that's the most stimulation I've enjoyed with my clothes on for a very long time  

selfobsessednewguy

ChrisMusic said:

The "say 500 years" had no significance other than to moot the potential exponential growth in the trilogy of the power of computing, the increasing sophistication of algorithms, and the advent of 'proper' artificial intelligence.  Once those systems are left to develop themselves, then I think there is the potential for a quantum shift in our understanding of the inner workings of the universe.  IMHO, we have nowhere near the sophisticated understandings we like to credit ourselves with as a species, but we are developing the tools to furnish the never-ending curiosities the human race is blessed with.  

yess.... we still dont know very much cuz.


wanna see the most beautiful relationship in mathematics?



wait... the fuck is this? ok so there are two huge symbols here. one on the left, means 'SUM 1/n^s from n=1 to n=infinity', and one on the right, meaning 'take  PRODUCT from p= 2 to p= infinity, where we ONLY RUN OVER PRIME NUMBERS'

this is the euler product formula... one of the greatest mindfucks of our generation. a relationship between natural numbers n=1,2,3,4,5... and PRIME NUMBERS, the mysterious p=2,3,5,7,11,13,........... how the fuck are they related like this hmm?

the riemann hypothesis is a million dollar prize problem which extends from euler's product formula, when riemann constructed his Zeta function. there are things very deeply misunderstood about mathematics. im doing a masters in pure maths right now, and every day i leave each lecture thinking... what just happened?