This is the English Version of my Post from yesterday. This is not a translation though, I hate doing translations and I really don't want to auto-translate my posts. That said it is hard for me to talk about math in English. For politics that's no problem, I am used to that, but when I think about Math I think completely in German. So I might use some of my Germans thoughts as a blueprint for this post, I will definitely use the same proofs because those are very relevant to me.
Why does Math matter?
Most people know mathematic as the basis for many sciences and while that is all really nice and dandy, I don't care about it too much. There was always that misconception that science should be really my thing, but it is not, I can understand the math behind every science quite easily, but I don't really care too much why my graphic card functions or even worse how my own body functions. It's cool that it works I don't really want to be bothered by the details.
Mhh, I didn't expect to start out with an anti science rant, but being on the topic, if you ever want to have a highly sophisticated discussion with any scientist in any field you at least need to understand the basics of higher Mathematics. I would even go so far and argue that the Math I am about to show you actually enhances critical thinking in fields that you previously thought got nothing to do with mathematics.
What I am going to talk about are mathematical proofs. I am not sure how familiar you are with the concept but in math you can proof something against any possible doubt. Take the sentence of Phytagoras for example: we don't have a theory about a²+b²=c², that seems to work as a model and might be disproven with new discoveries in the fields of mathematics. No! We know for sure that a²+b²=c² will be true for all eternity.
Here Math is completely different from any science, there is just one truth, not some dialectic-the-truth-lies -in-the-middle-bs. Mathematicians know everything with 100% certainty. There are some arguments about definitions, like if 0 is a natural number. You can have an opinion on that or just agree THAT 0 IS NOT A NATURAL NUMBER! Sorry, I get heated on this one.
Let's look at our first easy proof 0.999...=1 . I like this one cause you might think that 0,999... is smaller and the number closest to 1 but it is easy to proof it is actually the same as 1:
0.333...=1/3 ---> 0.999...=3/3=1
To me the beauty of a proof can be divided in the following categories: The impressivness or importance of the rule that is proven . The length of the proof, shorter is better. And lastly how understandable your proof is. These criteria are subjective but once you have done some mathematical proofs, it can be really nice to agree sometimes that proofs are either really beautiful or sometimes really ugly.
I present you my favorite: The Proof that there is an unlimited amount of Prim Numbers.
Lets assume we know all Prim Numbers. We can now multiply them all to the Product we shall call P. Now we look at P+1, it can not be divided by any Prim Number because P can divided by all of them. Therefore P+1 has to be a Prim Number itself that is not part of all known Prim Numbers. Therefore our assumption that we can know all Prim Numbers is wrong. Therefore the number of Prim Numbers has to be infinite.
Please comment if you can or can not understand my proof, I would really like know how digestible I made it for the average consumer.
For the Pros
There is one proof I failed in a test. Maybe you can help, I still can't get the solution. If you look at n^4, with n being any natural number, you always get a result that ends on 1,6,5 or 0. In other words, why do you only have two rest classes for n^4 modulo 5?