I recently discovered there’s a matrix representation of conic sections, and I was wondering if we can ‘connect’ two different areas of mathematics in a similar way?
Like, for example, this connects algebra and calculus:
- the derivative of a quadratic function = ±√(its discriminant) (don’t know how to handle complex numbers using this, though)
- Cramer’s rule, n×n-determinant method for solving a system of linear equations, etc.
- Calculus and geometry? Apparently setting ∇Q = [0,0] for any quadratic curve gives its center
Please keep in my mind that I’m still studying in school, so if you talk about fields and Lorentz transformations and fifth dimensions and eigenvalues, I’ll lose you.
I’m desperately in need of some way to crack the maths section in my entrance exam that comes in a week, and I don’t want to screw my life over, so I’d like the trusty tricks of mathemagical veterans.
Thanks! Your help is appreciated!
EDIT: Formatting, and added the ∇Q bit
The site is setup for people who already know the math so they can more or less do it like a crossword puzzle. The subjects are:
- Vector Calculus
- Linear Algebra
This is the first time I have really “advertised” the site so you will be the first users and obviously not everything will be perfect. But I’d love it if some of you tried a few problems. The site is here.
I want to define a field over ℝ². I am free to choose the elements 0 and 1 and the operations + and *
The most common way is 0 to be (0,0), 1 to be (1,0), (a,b) + (c,d) = (a+c,b+d) and (a,b)*(c,d) = (ac-bd, ad + bc). This is the way we define complex numbers.
Is the complex numbers the only possible field? Are their any other way to choose 0, 1, + and * so that it is a field?
A poem I wrote inspired by the documentary series ‘Dangerous Knowledge’.
Welcome to the world of mathematics
With laws that seem universally static
from calculus to geometry to the quadratic
ruled by a genius few, rather erratic
laid down this world with strange schematics
these pupils pure souls, rather unpragmatic
seized by its beauty, they turned fanatic
their life dramatic, their end traumatic
but to this day they remain enigmatic
their theories, their ideas rather problematic
ostracized by many, they remained dogmatic
and exiled themselves to a dark dingy attic
sometimes distressed, sometimes ecstatic
their work paradigmatic seemed axiomatic
which in the end, turned them into a lunatic.
I’m taking a year long course in Abstract Algebra and Real Analysis, it’s partitioned into two parts. I always find that the analysis concepts come really easy to me, but the abstract algebra concepts are always a up hill battle. It seems like in Abstract Algebra I have to memorize a bunch of theorems and definitions in order to do well on Exams, but in Real analysis I can get by with just using my analytic abilities and memorizing very little. I know you should not memorize, but how do you except me to properly apply a theorem if I don’t even know the statement and edge cases? Abstract Algebra to me seems like a dull subject that’s filled with jargon and symbol pushing, I know that’s not true but that’s how I feel because everything I learn about rings, fields, and such seem to lack intuition of any kind. I don’t see why someone would define certain things in a proof, the whole thing is just a blurry mess for me.
Speaking specifically about Eulers Sum of Powers Conjecture, which states that if you add n integers, each taken to the kth power, that there will never be a solution if k≥n, for n>2.
I understand that the most general version of it has been disproved, but does anything remain of it? It looks like from Wikipedia that counterexamples have been shown for k equals 3, 4, 5, 7 and 8, but has it been shown that counterexamples exist for all combinations k and n, other than where n=2? Or is there some new modified conjecture stating that this limitations may exist for certain classes of k and n?
It just seems hard to believe that when n=2, there are zero solutions, but when n>2>n, there are always infinite solutions? What’s the current state of our understanding here for all ks and ns?