math

Dude, I still have one week of holiday.

Anybody up to solve Millennium Prize Problems?

Thanks for mentioning them. Because, as you probably know, they include the notorious Riemann hypothesis, my favorite and the most difficult open problem in math. It concearns the distribution of primes among integers. 1 is usually not counted as a prime, so: 2, 3, 5, 7, 11, ...Gauss was the first to conjecture the prime number theorem which says that the number of primes lesser than a given number (the so-called prime counting function) behaves asymptotically as the function x / ln x. ln stands for logarithmus naturalis, the natural logarithm, which has the number e as a base. (Logarithm can have as a base any real number greater than 0 and different from 1) It is the inverse function of the exponential function which has the same base. Exponential function started with powers. For example, if you take 2 as a base, you'll get powers of 2: f (x) = 2^x, 2^x is assigned to each number x. In the special case of positive integers, f (1) = 2^1 = 2, f (2) = 2^2 = 4 etc. It is rigorously defined like this:

Theorem. Let a > 0 , different from 1, be a given real number. There is one and only one function

f : R -> R with the properties:

1. f (x) > 0, for each x

2. f (x + y) = f (x) f (y), (f (x) multiplyied by f (y)), for each x, y

3. If a > 1 then f stricktly increases, if a < 1 then f stricktly decreases on R

4, For each y > 0 there's a unique number x such that f (x) = y

5. For each integer n, f(n) = a^n.

Definition. The function f from the theorem is called general exponential function of base a.

Inverse function of a given function is the one when composed with the given function, gives the identity function, f (x) = x for each x. The composition of functions f, g is the function h defined as h (x) = g ( f (x)).
Number e:

https://en.wikipedia.org/wiki/E_%28m...al_constant%29

Prime number theorem was proven independently by Charles de la Valee-Poussin and Jacques Salamon Hadamard in 1896. Riemann took it a step further, giving an explicite formula for the prime counting function, but it depended on the distribution of zeros (numbers which a certain function puts to 0) of the so-called Riemann Zeta function. If one could give the exact distribution of zeros, we would have the exact explicite formula. The Riemann hypothesis says that all so-called non-trivial zeros of the Riemann Zeta function have real part = 1/2.

I'd like to pose another challenge here.

Take two 4s and obtain 64. By that i mean you can do whatever you want, only out of all the numbers you can use only number 4, and use it only twice. The result has to be 64.

ok it's been a day, so I'll give you 5 more minutes.

In the meantime, I'll talk about the Gauss-Bonnet theorem, which I've been trying to understand the statement and proof this last century,and maybe I'm finally near. You'llbe able to understand cause it's from (differential)geometry which is always more intuitive and imaginable than algebra.

I'm talking about the general variation, that's in Lee's Riemannian Manifolds: Introduction to curvature.

So, we begin with manifolds. What's a manifold? The most common examples are curves and surfaces. curves are considered one-dimensional, no matter how many dimensions the ambient space they belong in has, because you get them by curving (segments) of lines. Surfaces are 2d cause you get them by bending or twisting (parts) of planes. But not just any parts. you can't take, for example, 15 points on a plane and bend them. The part has to have an area, be 2d itself, like discs or rectangles. There are two things to keep in mind here. First, by curving, bending, twisting I mean continuosly deforming - you can do whatever you want as long as it doesn't snap. Second, it has to be smooth. Curve can't have vertices and surface can't have edges (like a roof) or peaks (lik a cone).



An example of a surface.

The Gauss - Bonnet theorem applies for surfaces. It essentialy says that if you take a surface,and calculate its total Gaussian curvature, it equals 2 times pi its Euler characteristic.

In laymans terms: take a sphere, for example. You can continuously deform it, as long as it remains smooth,and somehow the total curvature of it remains unchanged. Because, if you stretch it, to form an ellipsoid, for example, it will be more curved at the stretched parts,but less curved elsewhere, so it cancels itself out.

A Maths thread.. I have a degree in maths

What kind of degree, may I ask? Have you graduated from college? What's your fave branch? mine's mathematical logic. Especially proof theory,polyvalent logic, and my spetialization , polivalent proof theory. Who's your fave mathematician? Mine's hilbert.

I have a BA hons. I am a trainee doctor, i took maths as i did like the idea of a teaching career, always best to have a Plan B

I didn't like it much, but I was always in the advanced classes... It was great to be able to check your work right on paper, and when you finally solved a quadratic equation, there was a feeling of accomplishment.

I still do math on paper and pen, but I've been telling myself to do a lot for fun, to work that part of my brain more.

I think it was Bhramaputra,an Indian mathematician from way back 1000 years ago who first gave the famous expression for the two solutions


There are two solutions, as you know, because you can either take the plus or the minus sign.

btw, if we ahve complex numbers, if we're in the complex or Gauss plane, and have a polynomial which thus has complex coefficients and a complex variable, there are always the same number of solutions (or zeros, asthey're called,because if you replace the variable with one of them the polynomial equals zero, like for quadratic equation, where you have a polynomialof degree 2 (the degree of the polynomial

a_n x^n + a_(n - 1) x^(n -1) + ... + a_1 x + a_0

is n)) as is the degree of the polynomial, if you count them with multiplicities (a number can be a multiple zero, like for

x^2 - 2x + 1 = (x - 1)^2 = (x - 1) (x -1) = 0, 1 is a double zero.

This result is known a s the fundamental theorem of algebra, first proven by Gauss in 1796, and published in his famous Disquisitiones Arithmeticae in 1799.

It was Cardano who first gave a general solution for the cubic equation, and there's one for quartic equation, but we cannot have one in the general case for the equation of any higer degree, this was first proved by the norwegian mathematician Niels Henrik Abel, and known as the Abel-ruffini theorem.

http://mymathforum.com/elementary-ma...64-two-4s.html

Has anyone figured out yet why this equals 64?

Not me. Science & math were my weakest subjects. And I do mean weak.

__________________
I’m here only on Mondays, Wednesdays & Fridays. That’s why I’m here now.

How come you're so smart, then? I'm willing to bet you have all the other intelligences. ok, no talent for math,we can't all have everything.

Does anyone know what that ! means?

! symbolzies the factorials.

For a natural number n,

n! = n (n - 1) (n -2) ...3 x 2 x 1.

so,

1! = 1

2! = 2 x 1 = 2

3! = 3 x 2 x 1 = 6

4! = 4 x 3 x 2 x1 = 24

etc.

http://mymathforum.com/elementary-ma...tml#post578889