The fascination of Maths thread

[sasguru]

Do current models of the universe still have 11 dimensions?

I don't think I'm intelligent enough to answer that. Ask NotAllThere.

[NotAllThere]

I've not reached that chapter yet. There were theories about that - all to do with M-theory. But I'm hazy about the hazy details.

[BrilloPad]

Can't remember details but isn't it to do with degrees of freedom?

https://www.statsdirect.com/help/bas...of_freedom.htm

So if we had two parameters that were estimated, it would divide by n-2?

No wonder I gave up most stats at the end of my second year - I fact my final year stats was accountancy/economics!

And no wonder I only got a Attila the Hun (two-one) instead of a Geoff Hurst.

[BrilloPad]

I don't think I'm intelligent enough to answer that. Ask NotAllThere.

They did 5 years ago. These things move on to quickly - or am I getting old?

[BrilloPad]

123456789*8+9=987654321

Hth

[NotAllThere]

I've just read "e: the story of a Number" by E. Maor. Highly recommended.
I get the strong feeling that maths is universal, in the sense that if there is another civilisation/intelligence in the universe the same rules of maths would apply.
And I'm not religious but if there is a God, it is in the strange beauty of maths.

Someone once said "God invented the integers, all else is of man". So the idea is that once you've got the integers, then you can derive all else. Penrose points out that going from Integers to the Reals is actually a far bigger conceptual jump than going from the Reals to the Complex. For that, once you've got the idea of the square root of -1, you get everything else turning up for free.

Here's something cool:

The series 1 + x^2 + x^4 + ... converges to 1 / ( 1 - x^2 ) when -1<x<1 For other values of x it is divergent.

The series 1 - x^2 + x^4 - ... converges to 1 / ( 1 + x^2 ) when -1<x<1. For other values of x it is divergent.

It turns out if you take the two convergent functions, and make them complex functions, they both have two points where there value is undefined - i.e when the value under the 1 / goes to zero. So for the first, that's when x = 1 and -1, and for the second when x = i and -i. It turns out, that if you've a power series that can be represented by a convergence function, then, if there are values for those functions that are undefined, then all values lying within the cricle defined between the origin and the nearest point where the function is undefined, do converge (for the power series), and all those that lie outside the circle do not converge.

Not quite as cool as e raised to the power of pi times i = -1

[sasguru]

Someone once said "God invented the integers, all else is of man". So the idea is that once you've got the integers, then you can derive all else. Penrose points out that going from Integers to the Reals is actually a far bigger conceptual jump than going from the Reals to the Complex. For that, once you've got the idea of the square root of -1, you get everything else turning up for free.

Here's something cool:

The series 1 + x^2 + x^4 + ... converges to 1 / ( 1 - x^2 ) when -1<x<1 For other values of x it is divergent.

The series 1 - x^2 + x^4 - ... converges to 1 / ( 1 + x^2 ) when -1<x<1. For other values of x it is divergent.

It turns out if you take the two convergent functions, and make them complex functions, they both have two points where there value is undefined - i.e when the value under the 1 / goes to zero. So for the first, that's when x = 1 and -1, and for the second when x = i and -i. It turns out, that if you've a power series that can be represented by a convergence function, then, if there are values for those functions that are undefined, then all values lying within the cricle defined between the origin and the nearest point where the function is undefined, do converge (for the power series), and all those that lie outside the circle do not converge.

Not quite as cool as e raised to the power of pi times i = -1

Hmmm. Didn't understand most of that. I'm going to work it out tomorrow, without the benefit of a few glasses of wine. Problem is my maths stopped at A-level and that was more than 20 years ago. I'm getting the feeling that it's worth it to make the effort though.

[NotAllThere]

I'm well into second bottle - but as I mentioned, I did do a degree in pure maths, and some of it seems to have stuck. In chapters 7 to 12 of Penrose's book, I occasionally see something that links to what I learned at university. It makes me think - ah, now that's what they were on about...

it really is amazing stuff - if only they'd told us about this when we were at school... which is why I'm happy that my 16 year old son is working through my university calculus book because he's bored with the set curriculum.

( btw - if you need help on the above stuff about convergence, let me know, and I'll try to explain it sober... )

[BrilloPad]

Someone once said "God invented the integers, all else is of man". So the idea is that once you've got the integers, then you can derive all else. Penrose points out that going from Integers to the Reals is actually a far bigger conceptual jump than going from the Reals to the Complex. For that, once you've got the idea of the square root of -1, you get everything else turning up for free.

Here's something cool:

The series 1 + x^2 + x^4 + ... converges to 1 / ( 1 - x^2 ) when -1<x<1 For other values of x it is divergent.

The series 1 - x^2 + x^4 - ... converges to 1 / ( 1 + x^2 ) when -1<x<1. For other values of x it is divergent.

It turns out if you take the two convergent functions, and make them complex functions, they both have two points where there value is undefined - i.e when the value under the 1 / goes to zero. So for the first, that's when x = 1 and -1, and for the second when x = i and -i. It turns out, that if you've a power series that can be represented by a convergence function, then, if there are values for those functions that are undefined, then all values lying within the cricle defined between the origin and the nearest point where the function is undefined, do converge (for the power series), and all those that lie outside the circle do not converge.

Not quite as cool as e raised to the power of pi times i = -1

I am sure I did convergent series in 2nd year pure maths. There are several rules for proving if a series is divergent or not?

I left Exeter over 23 years ago!

[BrilloPad]

I did do the maths of the fusion taurus in 3rd year applied - that was fun.