Prove it! S2-E4

The History of Math is the best type of Math don't @ me!

In working on my independent study class, The History of Mathematics, I have now experienced the traditional way math has been taught for the past 2,000 years. Math has had one common textbook all students of liberal arts and mathematics education have in common, Euclid's Elements. This book has been the backbone of mathematics up until the 1800s when the modern textbook was born. The Elements has taught the likes of Thomas Hobbes, Baruch Spinoza, Isaac Newton, Abraham Lincoln and now myself. 

I recently completed a proof that took me a while to understand but after working through it I have a whole new tool set up thinking mathematically and abstractly. I now ask myself why I wasn't taught this earlier! Writing a proof is like writing an essay its all a process with lots of justification. I first had to do research in this case what I was proving. I set out to prove that Euclid's GCF algorithm (Book VII Proposition I) worked and that when the algorithm produced an answer of one then the two numbers I tried to find the GCF of are relatively prime. I read the proof from Euclid's Elements and then I looked at other versions of the proof that are easier to understand. 

After that, I had to write an outline with help from my teacher and this article here. I learned the logic behind writing a proof. Whenever I write anything I tend to talk to myself. I do this to walk myself through the process and figure out what I am trying to say. I have learned that this is a great tool for writing proofs. Many times I have caught myself having a full-blown conversation because I need to self justify why I am jumping to a certain conclusion. 

Outline completed I needed to hedge out my ideas. I did this by using a whiteboard. I started out by writing my axioms and rules for my proof what was allowed, what was true, and what I am trying to prove. Then I took my outline which had my key steps and I filled in the steps so someone reading the proof for the first time would understand. 

Now I needed to edit my rough draft. I took my proof to my Professor and we edited out our draft. This can be a demeaning process depending on one's mindset but it essential in all writing to have someone edit one's work. This is the only way to improve. We spend in total two hours adding verbiage and solidifying my logic but it the result is a full-fledged proof! The final piece to the process is having his colleagues look at our final draft and critique it. 

For those who are mathematics aficionados, I have included my final draft to my proof here.

Here is a short video that explains the algorithm:

Note the only difference between this video and the proof is that our remainder is one. So the only GCF our two numbers can be is 1 and nothing more. And what this means is that these two numbers are relatively prime to one another. And this is exactly what we are trying to prove.