Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Tuesday, 31 July 2012

University

When I was first looking at Universities, I didn't know where I wanted to go, or what I wanted to do. At the time I was enjoying ICT and thinking that I maybe wanted to do something creative to do with ICT. I couldn't find anything that grabbed my attention, so started to look at business courses. I was worried then about having a subject that would involve too much writing and essays, so I started to look for business courses that let me do other modules in maths.

In all this time I never thought of studying just maths. I always thought it would be too hard or too boring. But after looking at other courses, and not being particularly inspired by any of them, I settled on maths, simply because it's what I'm best at, and I didn't want to do anything else. But the more I thought about it, the more excited about the idea I got.

Maths has always been my favourite subject. It is logical, it doesn't involve many words, and it has a right answer. I hate it when you have to write an essay or something, and the teacher says "there is no right or wrong answer". How do you score points then? How do you know if what you have written is going to get you a good mark? I just don't, so that is why I like maths. I can usually tell if I've done well on a paper, just because there are methods to answering the questions.

My teachers were happy with the choice I'd made as well. They believed I was smart enough for it, and they knew that I enjoyed the lessons so they thought that I would be very interested in the subject at a degree level. With their support, I started to  believe that I could do it as well.

Last summer, a close friend of our family kindly bought me a couple non-academic maths books (one of which is the book I talked about in a previous post). I read The Number Mysteries by Marcus du Sautoy first. I found it surprisingly interesting and easy to read. I never realised how much more there is to maths than what you learn in the classroom. Reading this book made me think more about what I could learn at University, and how maths applies to the world, and I started to think more and more that it was something I wanted to do. Since then I have listened to a podcast and watched a TV series about the history of mathematics (both by Marcus du Sautoy as well). I have been given and bought a few more books that I have read various chapters from, and started this blog!

The next challenge, after getting exciting maths, was to choose which Universities to apply for. I had no idea where I wanted to go. I knew that I didn't want to go to Cambridge or Oxford, they just didn't seem right for me. Other than that, I didn't really know. My Step-dad suggested that I look at the Universities in the Russell group because they are below Oxford and Cambridge, but they are still high standing Universities. I ordered prospectuses from all the ones that weren't too far away, and looked at the courses they offered. The maths courses were all pretty much the same, as I decided I just wanted to do the basic mix of pure and applied mathematics, so I could find out what I like and specialise later on.

After a long time of still not really knowing what to do, my mum suggested a road trip to fit in with teaching my sister how to drive on motorways and to let me have a quick look at a University for myself. I'm not sure how we ended up choosing Warwick, whether it was purely because it was one of the closer ones, because the prospectus was on top of the pile, or because we used to go to Warwick castle a lot, but that is where we went. Although we only got an outside look at the Uni (it wasn't actually an open day), we had a drive around the town, and found a little shopping centre to get some lunch and we bought a couple of books. Mum kept saying things like "this could be where you do your shopping for food to survive!" which was a little bit scary, but kind of cool. We also decided that the maths course there would be ideal, because there was a lot of room for flexibility if I wanted to take a module of something else to break up the maths.

After that, everything got a bit easier. I applied to Warwick, Bath, Nottingham, Liverpool and Leeds, mostly based on the grades, and the flexibility of the courses. I went to open days to all of them (except Bath, because no one could take me and I wasn't ready to go on my own after having just gone to Liverpool by myself). I got offers from all five, which was very exciting, and I decided that I wanted Warwick as my first choice, and Nottingham as my back up.

I found out, at some point, that I would have to take an additional paper alongside my A-levels to get into Warwick. The STEP papers use A-level syllabuses, but the questions are designed to be more like the kind of questions you would see at University, so it tests you on how you would cope with degree level work. I surprised myself by not being put off by this. Even when I tried some questions and they were really hard, and I didn't really get them, I kept going with it. The day I got an answer (or close to part of an answer maybe...) I was so ridiculously excited. It was a really good feeling. I think it helped that my teacher thought that I could do it, and my mum said, even if I don't do very well in the exam, just the process of doing it will help me so much, and she is so impressed that I even had a go. The exams were hard. One of them more so than the other. But I gave it my best shot, and I was really quite pleased with what I did for one of them, so fingers crossed!

Even if I don't get into Warwick (the grades are really rather high!) I will still be happy to go to Nottingham. So from starting out with no idea what I want to do, I've managed to get to a position where, no matter what happens, I will be sorted for the next 3 or 4 years. After that, I have no idea what I'm doing!

Sunday, 22 April 2012

cos^2(θ) + sin^2(θ) = 1


I was doing an A-level maths paper when I found this question:
Use this triangle to prove cos2θ + sin2θ = 1. For what values of θ is this prove valid?
 My feelings about proof questions are conflicted. When I can see how it works, and I can find an elegant solution, I absolutely love them. Being able to show why something works cements the idea to me that there is a reason for why formulas work for any value you put in, or why you can substitute one formula for another and still come out with the same answer. However, when I can't see how all of the pieces fit together to make a proof, it really frustrates me. I guess that's part of being a mathematician, not knowing how the pieces fit together to start with, but persevering with it until they do.

As it happens, this is a proof question that I could fit together, so I am going to share it here. In my opinion, this is quite an easy one, but easy is subjective, so maybe someone will find it impressive anyway XD

1. Because this triangle is right-angled, I can use trigonometry to show that cosθ = b/c and sinθ = a/c

2. If I square both of them, I end up with cos2θ = b2/c2 and sin2θ = a2/c2

3. Next I add the two together to give cos2θ + sin2θ = (a2 + b2)/c2

4. Pythagoras says: a2 + b2 = c2 so (a2 + b2)/c2 = 1

5. Therefore cos2θ + sin2θ = 1

For the second part of the question, the proof is only valid for θ < 90o because if θ > 90o, ABC wouldn't be a right-angled triangle.

So that's all there is to the proof. Like I said, I think that one is quite nice. And now to leave you with an interesting fact about the Pythagoras Theorem. In China, it's named after mathematician Shang Gao, who came up with it independently from Pythagoras about 2,200 years ago (300 years after Pythagoras).