Each problem that I solved became a rule, which served afterwards to solve other problems

Hello Class!

We have made it to the last blog post for this month’s mathematician! You all have had some wonderful thoughts about both the life of Descartes and his work in mathematics. Much of Descartes work that we have discussed so far connected topics you have learned in previous math classes to problems in this class. This week, we are going to use Descartes to introduce a major concept in this class, proofs. I am sure that many of you have heard about proofs from your friends who have already taken geometry, but before we get started I want you to think about and respond in the comments below to the following “I Wonder” questions and then read more about how the idea of proofs came to be.

I Wonder
1. What have you heard about proofs before this class?

2. Do you think you have ever used the concept of proof before geometry? If so, how? If not, why not?

While Descartes studied at La Fleche, he had the opportunity to work with many other famous mathematicians who contributed greatly to the field. During this time he made great contributions to the concept of geometric proofs, especially when it came to identifying curves. The addition that Descartes was able to provide to geometric proof was the intense need to justify and explain what was happening in the math. For those of you who admitted to using the concept of proof before in you life, my guess would be that you provided an example in which you had an opinion or stance on an issue and then proved that your opinion was valid. In order to do this, you had to explain the situation, support your stance with facts, and present the material in a clear and orderly way. Descartes used this same process many years ago. He had a deep understanding of both geometry and algebra and sought to connect the two by using one to justify things that were happening in the other. To get a better understanding of this, try to reason through the algebraic proof below:

This idea of having to justify every step of the problem is something that Descartes believed in. He spent the later part of his life unmarried and uninvolved in his one daughters life because he was dedicated to understanding and explaining why things were occurring in the then booming subject of geometry. Despite Descartes early death due to health complications from his difficult birth, he spent his life trying to apply this idea of proving things to mathematics and the world around him.  

Keeping in mind the spirit of Descartes desire to find explanations for things occurring in the world around him, I would like you to pick one of the topics below and prepare a convincing argument, or proof, for your stance on the topic for class tomorrow. You should be prepared to present your argument to the class.

Topics:

1. Apple vs Android
2. The legal driving age (should it be 16 or 18)
3. Twitter vs Facebook
4. Climate Change

Remember that Descartes worked hard to apply factual information to his opinions and explanations on topics. Your arguments may require some research and should be delivered in a clear and concise way!

Domski, M. (2015, December 11). Descartes' Mathematics. Retrieved March 01, 2018, from https://plato.stanford.edu/entries/descartes-mathematics/

Perfect Numbers like Perfect Men are Very Rare

Hello Class!

Thank you for all of your insightful comments last week about how someone’s culture can impact the work that they do. We had such a great conversation about how everything going on in our lives impacts what we do at school everyday, how we perform in our sports and clubs, and even what we want to do after high school. Hopefully we can keep those things in mind as we continue through this class.

Now we have been using a lot of basic algebra in class recently as we factor expressions for various reasons. Many of the polynomials that we have factored so far have been quadratics. Have you ever wondered who decided we would write a little 2 in above, and slightly to the right of a number or variable to signify that we wanted to multiply it by itself twice? I would imagine that many of you never thought to questions why we write things that way. If your teacher told you thats how its done, then that’s all that matters right? Well, not anymore! This week we are going to talk some about how the mathematician we are studying impacted exponent notation. Before reading the rest of the blog post about Descartes, please answer the following “I Wonder” question in the comments below.

I Wonder

If we did not have the exponent notation () that we are used to today, how would you denote the following:

Create your own exponent notation for the expression and place it in the comments below. It MUST be different than the notation that is widely used today.

Before we discuss Descartes contributions to exponent notation, let's discuss other mathematicians that impacted this area of mathematics. There are records that trace the use of exponents all the way back to Euclid (our first mathematician that we studied in September). This shows that the idea of exponents was not new to math when Descartes started studying math. In fact, Nicolas Chuquet used a raised number to represent an exponent in the early 1500s. Several other mathematicians made contributions to the topic over the next 100 years until Descartes added to their work in 1637 by providing the notation of superscript that we are used to seeing today.

Your homework for class tomorrow is to answer the following question and provide two citations to support your claims. Be prepared to discuss your responses with the class.

1. In the blog post this week, I mentioned that several other mathematicians made contributions to the topic of exponents and their notation over a couple hundred year time period. Pick on mathematician that made a contribution to the topic and explain why you this that contribution was significant to how we use exponents today.

**You may not use Descartes or Chuquet because I already told you their
contributions.

Hay, K. (n.d.). History of Exponents. Retrieved February 26, 2018, from https://www.sutori.com/story/history-of-exponents