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.

Proof methods: Proof by mathematical induction

Proof methods: Proof by mathematical induction

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.

I have been reading quite a few blog posts recently and they all seem to be witty and clever, so I really wanted to add a joke right here on induction. But I honestly couldn’t find one I think was funny enough. So if you could just laugh or smirk for a few seconds before reading on, my day is saved. Continue reading →

Posted by Kristian in Math, 4 comments
Khan Academy – a great source of math knowlegde

Khan Academy – a great source of math knowlegde

This post is shameless promotion of what I think is a great site – Khan Academy. If you don’t like shameless promotions, you should stop reading now.  The site features a whole set of instructional videos along with a good amount of exercises. All topics are what I consider to be some basic skills of mathematics. Not basic because they are simple skills, but because they are fundamental for you to build upon. Continue reading →

Posted by Kristian in Math, 3 comments
Contrapositive proofs – part 2

Contrapositive proofs – part 2

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 →

Posted by Kristian in Math, 3 comments
Proof Method: Contrapositive proof

Proof Method: Contrapositive proof

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 →

Posted by Kristian in Math, 10 comments
McNugget numbers – The Answer

McNugget numbers – The Answer

Last Sunday I posted a question I had asked my self about McNugget numbers with a promise that I would actually post an answer to the problem as well. So here we go.

McNuggetsThe McNugget numbers are fairly well described on the Internet and both Wolfram and WikiPedia describes the problem. Continue reading →

Posted by Kristian in Math, 2 comments
McNugget numbers – The Question

McNugget numbers – The Question

This post is about McNugget numbers and a small mental math exercise I gave my self this week while driving on the high way.

McNuggetsTo start with the beginning of the story. I was driving for quite a while this Friday – actually for the better part of 4 hours. Which I in many ways consider a waste of time and energy, but I had to for several reasons. At one point I stopped at McDonald’s to get a cup of coffee and stretch my legs.

While waiting in the line I was watching the prices for the many delicious offerings…. And I saw they had McDonald’s Chicken McNuggets in the four quantities 4,6,9 and 20. That’s when I realised that I had bumped into the curious little phenomenon called McNugget numbers at some point and decided I wanted to think this through my self. Continue reading →

Posted by Kristian in Math, 3 comments
Proof method: Proof by contradiction

Proof method: Proof by contradiction

I was first presented with a proof by contradiction while I was studying Discrete event systems in Canada. And I was puzzled about it most day. I came to really like it though.

When we want to prove something by contradiction we assume that the statement we want to prove is false and then show that it leads to a logic contradiction at some point, therefore the statement must be true. Don’t be confused just yet. I will come to the examples.

Proof by contradiction is not limited to conditional statements like the the direct proof is. So we don’t need to have a proposition on the form if Q then P. Continue reading →

Posted by Kristian in Math, 11 comments
News on proving the Collatz Conjecture

News on proving the Collatz Conjecture

A few weeks ago Gerhard Opfer posted a preprint of a paper titled An analytic approach to the Collatz 3n+1 problem. The paper claims the proof of the Collatz conjecture. In it self that isn’t something really interesting as there are probably several hundred people every year who think they have proven the Collatz conjecture. However, there is one difference here as the paper comes from a research institute. It might sound a little arrogant and imply that non-mathematicians don’t understand math. That is not what I imply, just that the sender packs a bit more punch by being from a research institute. Continue reading →

Posted by Kristian in Math, 9 comments

Theorems, Lemmas and Other definitions

I was asked by an avid reader (I always wanted to write that), to cover the different terms in mathematics regarding proofs, so here is a post which covers some of the terms which I think we will see a lot more of. Continue reading →

Posted by Kristian in Math, 4 comments

Proof method: Direct Proof

In my first post on my journey for improving my mathematical rigour I said that I would go through a few different techniques for conducting proofs.

The first one I want to dabble into is direct proofs.  This is the “simplest” method and sometimes it can seem that the proof isn’t there at all.

It will often go something like “if a then b”. So using some definition of a, we can show that b follows as a direct consequence through an unbroken line of logical arguments such that

a -> … -> b

Lets try it out on some sample problems Continue reading →

Posted by Kristian in Math, 3 comments