Math

I use this tag for when the topic is about pure math. So whenever I dive into proofs, explanations of math topics or news on the research this is where you should look.

18
May
2011

Improving my mathematical rigour

I have reached a point in my mathematical journey where I feel the need to learn how to make sound arguments for the validity of a mathematical claim. Or in other words, I want to learn more on how to prove things.

The path I took through the Danish educational system has never dealt much with mathematical proofs, but rather on how to apply the mathematics we have learned. I have developed an intuition for mathematics in some areas. But I lack mathematical rigour, so I often time have to resolve to hand waving instead.

The usual approach to learning proving techniques is through a taught topic where you are presented with some proofs. Through that you will expand your toolbox and learn how to do proofs. However, I would through a series of blog posts dabble into how to prove mathematical things and study different techniques.

Ben Tilly pointed me through his blog – random observations – to a document he wrote on how to do proofs. It has a flow chart which you can also see to the below, which I think is a very thorough way to ensure that you get through the proof. It doesn’t say anything about how to actually make your arguments, but it helps you break down the problem.

Let me spend the rest of this blog post to go through the flow chart and interpret it. Continue reading →

Posted by Kristian in Math, 0 comments
04
May
2011

GeoGebra – a cool geometric tool

GeoGebraAt work I use many fairly advanced tools that can do a lot of things such as Matlab, which in all it’s glory is a very nice tool. However, for things such as Project Euler which does contain small problems, where I try to get a feeling for the problem in a whole other sense than I usually do Matlab is not very good, let alone that I could never afford it for private use.

I have been pointed towards a tool called GeoGebra which is a free tool written in Java, where you can manipulate lots of different geometric objects such as functions, points and lines. You can also use tools such as intersection and getting the slope shown. You can even define your own tools if you like.

I think it has been made as a teaching tool for high school students, but I have been playing around with it for a short while now and I can definitely see some useful things in it when I try to grasp a problem. It wont work on abstract problems, but small geometric problems will be a thrill to work with from now on.

It promotes it self as

GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. It has received several educational software awards in Europe and the USA.

The best thing about the tool is the readily available material. There are a tutorial and manual right at the GeoGebra help site, there are tons of You tube videos from GeoGebra let alone the ton of user made videos and there is the user forum.

Math and Multimedia has a list of the 10 best tutorials for the tool to get you started. and GeoGebra Applet Central has some applets which shows the use of different parts of GeoGebra and I especially like the applet showing how to approximate Pi by increasing the number of sides in a polygon inscribed in a circle.

I am very impressed so go ahead and take a look at it at GeoGebra. You will definitely see some illustrations from me using this tool in the future, since it is so easy to illustrate the functions and so on.

Posted by Kristian in Math, 7 comments
27
Apr
2011

Fermat’s Last Theorem

Fermat’s last theorem is one of the best known mathematical puzzles ever posed. It is very easy to understand yet it eluded a proof for 350 years. Fermat stated in the margin of Arithmetica that he had the most marvellous proof of the conjecture, but it was too long to fit in the margin. It has always been known as Fermat’s last theorem even though it has only been a conjecture for 350 years. Pierre de Fermat stated that

it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.

In other words

\displaystyle a^n + b^n = c^n

does not have solutions for n > 2. Continue reading →

Posted by Kristian in Math, 25 comments
20
Apr
2011

Tricks for multiplying large numbers

Recently I saw a video with a small trick for easily squaring a 2-3 digit number. Unfortunately I haven’t been able to find it again, but if you have seen the video with the trick somewhere feel free to link it in the comments or tell me about it if you just know the trick. It was really neat.

What I did find when I searched for it was another small trick for graphically multiplying large numbers, so this is what you get for now.

If you know some good tricks for making calculations easier when you are standing with a pen and a piece of paper, or just mental calculation. Please share them with me, it is so much easier than having to find a calculator for everything.

Posted by Kristian in Math, 3 comments
16
Mar
2011

Video on Difference Quotient

A few weeks ago I stumbled on a few videos on YouTube showing me something valuable. It is three videos telling about derivatives, one of the fundamentals of calculus.

The videos come from Mathtv.com which is a website with a ton of good explanatory videos on different math topics. I have only watched a few of them, but I am really impressed at the way they explain it. They have a channel on YouTube where there are a lot of videos, some which I haven’t found on the website, so check out both places. I think you will share my enthusiasm about the site once you see these videos. They also have a YouTube channel which seems to have a slightly different set of videos in them. Continue reading →

Posted by Kristian in Math, 0 comments
07
Feb
2011

Course in Linear Algebra by Gilbert Strang

Within the field of mathematics I handle every day linear algebra plays a vital role. Linear algebra is a field of mathematics that studies vectors and vector spaces. On common use of linear algebra is to solve a set of linear equations. Personally I learned it in university using a book by David C. Lay called Linear Algebra and Its Applications. It is a perfectly good book, and I can recommend it if you want to have a book on the topic.

The reason why I bring up the topic, is that I rediscovered a video version a MIT course in linear algebra taught by Gilbert Strang. I found the videos when I first studied to my exam in linear algebra. I think he teaches it in a very understandable manner.  All of the videos can be found at Academic Earth. However, I think that the videos there are poor quality, so I have compiled a list of the videos representing the course in a better quality. Continue reading →

Posted by Kristian in Math, 6 comments