Let's discuss Mathematics

[ParadigmShifter]

It's more of a psychology question than a maths one.

 

[Mise]

Game theory, surely? But yes, best answered empirically...

It's such an odd question because if 1 person picked A, 2 picked B, 27 picked C and the rest picked D, everyone would be correct.

 

[ParadigmShifter]

Game theory is still very basic. We can only assume perfect information. Which makes the human answers equivalent to the computer answer...

 

[Mise]

Alright then, monte carlo simulations That's maths isn't it?

 

[Atticus]

I came across with a mention of "new theory of prime numbers" by Gauss. It was said that in this theory 5 isn't a prime number, because (1+2i)(1-2i)=5. So, does someone know more about this theory? Is it just a curiosity, or perhaps misinterpretation of the writer I read?

 

[ParadigmShifter]

5 isn't prime in the Gaussian integers.

Prime numbers in that ring are of the form a+ib, where either:

a² + b² is prime in Z

or

a = 0 and |b| == 3 (mod 4), and |b| is prime in Z

or

b = 0 and |a| == 3 (mod 4), and |a| is prime in Z

This follows from Fermat's 2 squares theorem.

It get's weirder though - not all rings of the form Z[sqrt(n)] are unique factorisation domains, e.g. in Z[sqrt(-5)] we have

6 = 2*3 = (1 + sqrt(-5))(1 - sqrt(-5))

sqrt(-5) is i*sqrt(5) of course. EDIT3: Z[sqrt(-5)] is all numbers of the form a + ib*sqrt(5), a, b integers.

That proved a major hitch proving Fermat's Last Theorem, someone claimed they had a proof but Kummer told them they had assumed unique factorisation when it didn't hold. This led to ideal theory.

EDIT: The Gaussian integers are a UFD though.

EDIT2: Several edits

 

[Atticus]

That's quite a mess to read

So it's just like ordinary number theory, but instead of Z they talk about Z x iZ ={a+ib: a,b\in Z} and consequently have some more oddities, right?

From the text I read (Concise history of the mathematics by Dirk J. Struik) I got the impression this was just an alternative definition for a natural number to be prime, which sounded odd. That's the price of reading concise history of maths, I guess...

 

[ParadigmShifter]

Depends on the ring.

Nothing is prime in a field since every (non zero) element has an inverse.

Definition of a prime element in a ring:

p is prime: if p|ab then either p|a or p|b (and p is not zero or a unit)

A unit is an element with an inverse.

The prime elements in Z are p, -p, where p is prime in N. The units in Z are +1, -1.

 

[Petek]

5 isn't prime in the Gaussian integers.

Prime numbers in that ring are of the form a+ib, where either:

a² + b² is prime in Z

a = 0 and |b| == 3 (mod 4), and |b| is prime in Z
b = 0 and |a| == 3 (mod 4), and |a| is prime in Z

This follows from Fermat's 2 squares theorem.

(snip)

You left out some Gaussian primes.

(1) 1 + i, and its associates, and

(2) All integers associated with either a + bi or a - bi, where a > 0, b > 0, a is even and a² + b² is a rational prime of the form 4m + 1,

where two elements are associates if they differ by a multiple of a unit in the Gaussian Integers. Such units are {1, -1, i, -i}.

 

[ParadigmShifter]

1) The norm of (1+i) is 2, so is prime by the first statement (a² + b² is prime in Z).

So is the second.

I guess I should have had an "or" inbetween the 3 cases

EDIT: Multiplying by a unit in Z doesn't change the norm.

 

[Petek]

OK, I misread your post.

 

[ParadigmShifter]

What's a "rational prime"? There are no prime numbers in Q (using the ring definition of a prime number), since it is a field, and every non-zero element has divisors. EDIT: Since every non-zero element is a unit.

EDIT: Wiki says it's just a term for an ordinary prime That's confusing terminology.

 

[DMOC]

I miss this topic. To add to discussion, let's talk about Russell's Paradox.

 

[ParadigmShifter]

Well it's just a variant on the barber paradox (A male barber shaves all the men in the village who don't shave themselves - who shaves the barber?).

It did show the limits of axiomatic set theory however.

 

[Atticus]

Did you mean: naive set theory?

 

[ParadigmShifter]

Oh yeah

 

[Atticus]

Speaking of set theory, I saw Axiomatic Set Theory and the Continuum Hypothesis by Smullyan in bookstore, relatively cheap, and thought of buying it.

It's Von Neumann–Bernays–Gödel set theory though, which I've heard is a lot nicer than ZFC, but I hear it rarely mentioned. People are always talking about ZFC, so I suppose it wouldn't be a bad idea to learn that set theory.

So, can anyone offer advice, which one to learn?

 

[pboily]

I really enjoyed Halmos' Naive Set Theory as an undergrad. http://en.wikipedia.org/wiki/Naive_Set_Theory_(book)

It's ZFC, which is still really nice when you get down to brass tracks. I'm not sure that there is any real advantage to learning Von Neumann-Bernays-Godel set theory instead of ZFC, although who's to say you wouldn't also find it interesting?

 

[Atticus]

Thanks for the suggestion! I probably would have passed that book on a glance due to it's name.

 

[Japanrocks12]

I'm having a little trouble understanding how to do arithmetic with complex numbers, the derivation to Euler's formula (the one with es and cosines), and what u(t) stands for in the situation that the Laplace transform of F(s) = 1/s is e^t times u(t).