In order to keep up my maths and leave my options for next year as open as possible, I’ve been teaching myself the modules for Further Maths A-Level. So far, it’s going really well and I’m doing 1 module per month, meaning I should have completed all 6 by the end of February and can focus on past papers from that point until the exams at the end of May/throughout June (it’s a sad thing there are no January exams any more!).
Anyway, having done Decision D1 (in my mind, not mathematics, but helpful for anyone wanting to go into computer science of logic) and Mechanics M1 as my applied modules last year, I had to discover Statistics for the first time this year!
Now Stats was always going to have a hard act to follow, as I absolutely ADORED mechanics! My mechanics revision is what you will see written all over the window both still in the Maths Workroom of my school last year, and also as my cover photo if you wander over to my Google+ Profile Page (+María Norlov)! I LOVED it. It was just so much fun. So then Stats… well, I’m sorry statisticians, but it just doesn’t have the same draw for me. I mean, FP01 has by far been the best module I’ve done so far and I am definitely more interested in Number Theory type questions than applied mathematics, but Stats just isn’t so fun even within that bracket.
Or that was until (dun dun duuuuuuuuun) we reached Chapter 3 of S2 and we hit integration!!! I was like, what on earth is integration doing here? Turns out continuous random variables give us probability density functions and cumulative distribution functions which give us a graph, the area under which, of course, equals 1 (as it is probability), but also gives us the opportunity to integrate to obtain the area for specific sections, or to integrate and equal to one to find any constants we don’t yet know. And it all go me to thinking; ‘What actually IS integration? How does it work?!’
 |
| Right, Left, Maximum and Minimum methods for calculating Riemann sums as shown in the link below! |
I mean, we integrate an equation of a line between two points and it gives us the area under the curve. So I thought back to when I was first introduced to integration, and sadly - I think now - I believe it was just as a “Fundamental Theorem of Calculus”. Just as the opposite to differentiation really. Which is sad, considering it is actually such a beautiful mathematical phenomenon. Integration gives us the area under the curve, right? Well this is because it is like splitting the area under the curve into infinitely many and infinitely small strips which we can then find the area of and add to get the area under the entire curve. That’s why we integrate ‘with respect to x’ or ‘dx’. Imagine ‘dx’ as relating to ‘x’ meaning a ‘small part ofx’. And this is a small part of x where the width of this strip is getting infinitesimally close to 0.
And then just think about that! How amazing is integration?!?!?! That it can contain within it infinitely many strips to get an accurate estimate for the area under the curve! The coloured picture above actually shows estimates made by various ways of calculating ‘Riemann sums’ [another wonderful thing that came from the world of Bernhard Riemann - his famous hypothesis is one of the things that inspired me to a love of mathematics in the first place!!!]. And it shows how these converge to give closer and closer to the right answer, obviously, the more strips the space is divided into. But if you look at all the complexity that goes into Riemann sums (http://en.wikipedia.org/wiki/Riemann_sum), you will very quickly begin to see that the relative simplicity and beauty (and contained-ness if I might push the limits of English vocabulary!) of the humble integration. I mean, can you IMAGINE having to calculate Riemann sums every time you wanted the area under a curve?!?!?! And apparently it is defined by being merely the opposite of differentiation.
And then just think about that! How amazing is integration?!?!?! That it can contain within it infinitely many strips to get an accurate estimate for the area under the curve! The coloured picture above actually shows estimates made by various ways of calculating ‘Riemann sums’ [another wonderful thing that came from the world of Bernhard Riemann - his famous hypothesis is one of the things that inspired me to a love of mathematics in the first place!!!]. And it shows how these converge to give closer and closer to the right answer, obviously, the more strips the space is divided into. But if you look at all the complexity that goes into Riemann sums (http://en.wikipedia.org/wiki/Riemann_sum), you will very quickly begin to see that the relative simplicity and beauty (and contained-ness if I might push the limits of English vocabulary!) of the humble integration. I mean, can you IMAGINE having to calculate Riemann sums every time you wanted the area under a curve?!?!?! And apparently it is defined by being merely the opposite of differentiation.
I encourage you, see integration for what it is today - it’s not merely the undoing of differential equations, it is something all of itself which contains within it the mystery of infinity. And that shouldn’t be something that is over-looked! Having not used integration for a while, I forgot how immensely wonderful it is; thankfully, today I rediscovered it! :D
No comments:
Post a Comment