Posts Tagged with "Proofs"

News on the ABC conjecture

Sunday, October 7, 2012

0 Comments

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...
Proving the fundamental lemma of Arithmetic

Proving the fundamental lemma of Arithmetic

Tuesday, November 1, 2011

0 Comments

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...
Why are proofs so frustrating?

Why are proofs so frustrating?

Wednesday, October 12, 2011

0 Comments

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...

A Short Treatise on a Vectors Concept

Tuesday, October 4, 2011

1 Comment

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.

Continue reading...
The pigeon hole principle

The pigeon hole principle

Wednesday, September 14, 2011

0 Comments

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...
Proof by mathematical induction

Proof method: Multidimensional induction

Wednesday, August 24, 2011

0 Comments

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...
Proof by mathematical induction

Proof method: Strong Induction

Wednesday, August 10, 2011

10 Comments

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...
Proof by mathematical induction

Proof methods: Proof by mathematical induction

Wednesday, August 3, 2011

4 Comments

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.

Continue reading...
Proof Method: Contrapositive proof

Contrapositive proofs – part 2

Wednesday, July 13, 2011

3 Comments

Once I finished up the post on contrapositive proofs I spend the better part of an hour feeling I wasn't quite finished with the topic. I still had a couple of things to explore. The first one is a contrapositive proof that puzzled me, the other thing is De Morgan's Laws which tells us how to negate a statement.

Continue reading...
Proof Method: Contrapositive proof

Proof Method: Contrapositive proof

Tuesday, June 28, 2011

8 Comments

On my journey to improve my mathematical rigour I have covered direct proofs and Proof by Contradiction. In this post I will cover the third method for proving theorems.Reading up on different methods for proving things I realized that a contrapositive proof is a really clever thing to used and often a better way to prove things than a proof by contradiction. However, I love the proof by contradiction so much that I wanted to cover it first.

Continue reading...