Blogs, Mathematical Reasoning, Proofs

Beautiful Proofs

One of the difficulties in teaching mathematics is that it takes a long time to learn the background knowledge needed to be able to understand the true nature, beauty and creativity involved in the discipline.  Throughout elementary and secondary school we are taught the key skills that will be need as we go into the world or enter academia, but we very seldom see the how these can all come together.  A thought that frequently comes to my mind when pondering this is a quote from Love and Math: A hidden reality

What if at school you had to take an “art class” in which you were only taught how to paint a fence?  What if you were never shown the paintings of Leonardo da Vinci and Picasso?

-Edward Frenkel

One of the reasons I started this blog was to try to share the beauty of mathematics with everyone I can.  In the Symmetry series I really tried to share the beauty and creativity I see in the mathematical process, while focusing on symmetry.  Throughout the coming posts, I hope to share this beauty by looking at some “pretty proofs”.

By “pretty proof,” I mean proofs that when you get done with them, you take a step and back and just revel in what you have just shown.  While some of the most interesting theorems may require extremely long and complicated proofs and can leave you feeling like this, I will focus on simple proofs.  Things that, when you’re done, you can marvel not only in the result, but also in the ease that the theorem was shown.

In this series, I will therefore try to only give a general overview of what the proofs look like so that the topics will be more accessible.  For those of you that want to see all the details laid out, I will also include a separate write up providing these details.  While the details aren’t necessary to see the overall beauty, looking at these is much like focusing of the dabs and dashes of a Monet painting.  An understanding of the details and processes adds to the enjoyment of the piece.

Woman in the Garden Sainte-Adresse

The first of these proofs I will provide is a proof by picture.  Here we prove the Pythagorean Theorem, the picture is provided below and as the featured image.


I hope you will continue to join us as continue these proofs.  In the next post we look at the irrationality of √2 and √3.  If you’d like to know when new posts are provided, please follow the blog.

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