Quintic Functions & the Abel-Ruffini theorem

We are all familiar with functions such as these:



We all remember from high school that they are called quadratic equations and can be solved easily using the quadratic formula. Formally, a quadratic equation is a polynomial of second degree that has the form:



such that a =/= 0.

Similarly, we remember polynomials of three degrees, which are called cubic equations and follow the form ax^3 + bx^2 + cx + d. Clearly we all remember how to find the roots of these functions using various formulas. A polynomial of degree four is called a quartic function and it to can be solved in a relatively straightforward manner.

But a problem arises when trying to solve the dreaded quintic function, which has the form:



Quintic equations have graphs that have a shape something like this:




It is a something that is never mentioned back when we are first learning to solve polynomial equations, but there exists a theorem known as the Abel-Ruffini theorem (published in 1824) which states that there are no general solutions to polynomial equations of degree 5 or higher.

But not all hope was lost!

In 1885, a class of quintic functions were discovered which can be solved. These are irreducible quintics of the form:



where all coefficients are rational. This equation is solveable by radicals <=> it is in the form:



where μ and ν are rational.

To get an even better idea of how terribly convoluted the solutions are to quintic functions, realize that in order to properly solve these equations, one has to use the tools of linear algebra to construct a companion matrix and then find the eigenvalues of that matrix!

So next time you are walking down the street, finding the roots of a second, third, or fourth degree polynomial functions, realize how lucky you really are!! Take the time to appreciate the formulas that are in place for them!

Poor function...

This poor function is as crazy as they come:



Do you know what makes it so special? It's very unique actually. This function was discovered by Karl Weierstrass in 1872 and has an incredible property. This function is continuous at every point, but it's derivitave does not exist at any point.

Poor function... it probably has no friends!! Didn't anyone ever tell it to lay off the crack?

If you like math... this will blow your mind

OK, so my mind has officially been blown. In a completely good way, however.

Everyone knows basic properties of algebra. We know for example, that whether you write out 2X3 or 3X2 you will still get a product of 6. We know that whether you write 3 + 2 or 2 + 3 you will still get a sum of five.... or will you?!? mwahaha not always!

What? Yeah, that's what I said too. You might be shocked to find that when dealing with sums of an infinite number of terms, commutitivity of algebra doesn't hold. This means that, you will get a different sum based simply on the order of the terms you are adding.

When dealing with infinite series, one is forced to deal with this problem. For some background, an infinite series generated by some sequence X is itself a sequence, where partial sums are taken for each element in the sequence.

To make matters easier, it is simply a summation, like what is pictured in the graphic in the next paragraph to the left.

Consider a summation of the form:
1 - 1 + 1/2 - 1/2 + 1/3 - 1/3....
this is an alternating harmonic series, a very common series to come across in mathematics. This sequence converges quite obviously to 0, however that convergence is conditional, rather than absolute convergence.

You will see a simple rearrangement of terms, perhaps:
1 - 1 + 1/2 + 1/4 - 1/3 + 1/5 + 1/6 + 1/7 + 1/8 - 1/4.....
will not converge to 0 at all, in fact it will diverge to positive infinity.

Think about it... every term is still represented. 1/2 is there, - 1/2 is there, 1/4 is there as well as 1/80000 and -1/80000, however they are just ordered differently.

In fact, any conditionally convergent series (meaning that a rearrangement can change the limit of the sequence) has an interesting property. For any number L, there exists some rearrangement of terms such that you can force the series to converge to that number.

So there you go. I hope it blew your mind as much as mine.

Thoughts on life: African countries have the best flags

Zimbabwe is bitchin.