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.

Everyday I (and probably everyone else) am faced with decisions that we have to make, some of them are small, some of them are big. At least for me there are two aspects governing the decisions I want to make. I want to make the best decision and I am making the decisions without knowing […]

Continue reading...7 October 2012

Here is some news of the possible breakthrough of the ABC conjecture. It might already be commonly known, but it is something I only recently discovered was going on. There are rumors that Shinichi Mochizuki from Kyoto university has solved the abc conjecture. You can find his proof here. Not that I understood anything of it, but I though a […]

Continue reading...6 September 2012

I got a fun little question from Jean-Marie by email the other day. With the disclaimer that I don’t really have the time to solve that many puzzles and therefore probably wont be able to give you a solution to puzzles like this one. This one triggered my curiosity and I managed to find a […]

Continue reading...1 November 2011

Last time I blogged I wrote about why proofs are so difficult. This time we should try to prove something a little more complicated and see if we can get some of the thoughts behind the proof while doing it to see if it really requires that amount of insights or part of a proof can be done by using more or less common sense. My guess is that we can come far with common sense, but we at some point need to rely on some insights or methods we have seen before.

Continue reading...12 October 2011

I have often been sitting there swearing when I couldn't figure out to how to prove something and I know for certain I am not the only one. But why are proofs so difficult when they seem so easy when you read them? This post is about the softer side of proofs, to encourage you to not loose faith when you meet something you can't prove.I think the answer to the question lies in the fact that what we see when other presents a proof is the result. That doesn't mean the person constructing the proof didn't make them, but there is no reason to leave it in the final version, since it doesn't help in the argument that the statement is valid

Continue reading...4 October 2011

I have been in contact with Frederick Koh from Whitegroupmaths.com who kindly agreed that he would write a guest post for the blog to promote what he has to offer - Tutoring in A level maths in Singapore. So without further ado let me present you with the real content.I have been asked on numerous occasions by students to provide a short effective mathematical proof verifying the fact that obtaining the vector product of the normals characterising two separate non parallel planes in 3 dimensional Cartesian space produces **the direction vector of the line arising from the intersection** of the above mentioned planes.

14 September 2011

The pigeon hole principle is a counting argument stating something as simple as "if you have n items which are to be put into m < n boxes then there is at least 2 items in one of the boxes". My first comment on that was "well duh!", that is obvious, but it can actually be used to prove some less intuitive things.So an example if you have 10 balls which are to be put into 9 boxes then at least 1 of the boxes contains at least 2 balls (unless you drop one of them...). But what can we use it for. I will show one of my favourite examples of it which I do not find intuitive at first glance.

Continue reading...24 August 2011

For the last couple of posts we have been dealing with mathematical induction, I will pick up where we left and continue on this topic one more time. Many introductions to proof by induction covers only the one dimensional case, here is an introduction to multidimensional induction. I will treat the two dimensional case in this post, but the expansion to n dimensions should be relatively easy. So for now we will assume that we have a statement P(m,n) which we need to prove.

Continue reading...10 August 2011

As I promised in the Proof by induction post, I would follow up to elaborate on the proof by induction topic. Here is part of the follow up, known as the proof by strong induction. What I covered last time, is sometimes also known as weak induction.In weak induction the induction step goes: Induction step: If P(k) is true then P(k+1) is true as well.Strong induction expands that to: Induction step: If P(b), P(b+1), P(b+2)... P(k) is true then P(k+1) is true as well for some k > b.

Continue reading...3 August 2011

It has been a while since I last posted something about proof methods, but lets dig that up again and take a look at a fourth method. The first three were direct proof, proof by contradiction and contrapositive proofs. Proof by induction is a somewhat different nature.Induction usually has it's force on statements of the type "For all integers k greater than b, P(k) is true". For some statements we could prove it with some of the already covered methods, but for others it would mean that we had to prove an infinity of cases.The analogy I see everywhere and which I find quite fitting is to compare induction to dominoes (and I don't mean the pizza thing) which are lined up. As soon as you knock the first one over it knocks all the remaining once over one by one. Induction works in much the same way.

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25 May 2013

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