Assignment 1 Reflection

 Pythagorean Triples have been a favourite of mine for many years, since we learned about them in high school. It just seemed to be an incredibly simplistic solution. In high school, we didn't learn anything about the history of Pythagorean Triples though, so this was a cool opportunity to dive deeper. For example, I didn't know that Pythagorean Triples went as far back as Babylonia, this shows the depth and complexity that the Babylonians were able to accomplish, very impressive. It was discovered that the Babylonians knew a form of this theorem because it was found on a tablet and the numbers were so large that it stands to reason that trial and error could not have been used. It is a shame that one side of the tablet was cut off however, it would provide great context on how and why this tablet was used. Aside from the history of Pythagorean triples, this assignment was also a fantastic opportunity to put myself in the shoes of a teacher. I am one of the few non-teaching majors in the class so this experience likely taught me more than it did most of the other students. I was able to learn about a subject, then think about how it could be most effectively taught, which was a lot of fun for me. Being a finance major, excel is always my go-to-method, but this also made me think about how other students would want to solve the equations. Overall, working with my teammates on this assignment was an extremely fulfilling experience, and one I hope to have again (excited for assignment 2!). 

Dancing Euclid Proofs

Euclid is iconic not only for the collection of proofs that he has combined into a series of textbooks, but also for the beauty of his work. The first thing that jumped out to me was just how easy it was to communicate Euclid’s proofs into dance form. The simplicity of Euclid’s proofs makes it so easy to understand and therefore easy to communicate. There are not many other mathematicians that come to mind when thinking about interpretive dance, which I think really speaks to the artistry behind his work. Secondly, I was really taken by the way these shapes symbolized connections between two people. Despite the size of people’s arms not being the same, the dancers didn’t let that stop them from using their limbs to create connections in this dance. When I see lines of a triangle, I have always thought of them as just that, lines, now I will be able to see the symbolic connection of two people. Lastly, this article and video opened my eyes to the math and patterns that surround us in nature. This can be in the form of trees, shells and waves, all things that I have never thought of as being math related but truly, they are. Altogether, this article and video has made me think of math more as a living, breathing thing and less like numbers on a board. 

Euclid Alone Has Looked on Beauty Bare

 Admittedly, I have never been very good at understanding or writing poems. As far back as middle school, poems have never made a ton of sense to me. This poem was no different, it wasn't until I did further research that I began to understand this poem. Euclid was a Greek mathematician, often referred to as the "founder of geometry". His elements are often thought of as one of the most influential works in the history of mathematics, often being used as the main textbook for teaching math, especially geometry. 

My interpretation of these poems is that Euclid has seen beauty in shapes that no one else at that point was able to see. His geometric proofs have given beauty and grace to shapes that were once thought of as just "shapes". In "The Euclidean Domain", David Kramer questions this notion, asking "Has no one else seen hide or hair?". This is basically Kramer saying "we have all seen shapes, how is Euclid the only one that sees this beauty?". Kramer may have a point, during the current day and age I'm sure a lot of people see the beauty and complexities of geometry. Is this only because of the discoveries and proofs of Euclid though? I guess we will never know, but I think even Kramer would acknowledge how important Euclid's proof are still to this day.   

The Eye of Horus and Its Significance in Ancient Egypt

I felt that the above diagram was very helpful to explain the Eye of Horus and its significance with unitary fractions.  Each of the "sacred unit fractions" were attributed to six parts of the eye of the god Horus. These fractions, all with powers of two in their denominators, were used to represent the fractions of hekat, which is the unit measure of capacity for grains. The most interesting aspect of the story for me was that according to legend, the pieces were lost in a battle and were restored by the god Thoth. 

For special numbers in my life the closest I can relate to is my hockey number. I wear the number 8 because my birthday is February 8, 1998 and my favourite player growing up was Alexander Ovechkin who also wears the number 8. 

The Magic Square

As a finance major, I am most comfortable doing math and solving puzzles on excel. That is why it was an easy decision to tackle this problem with my trusty friend. With no rhyme or reason, I decided to start with trial and error. For optimal problem solving, I made sum formulas at the end of each row to see if I was on the right track and a formula at the bottom right that would tell me if I have solved the puzzle or if I should keep trying. For attempt one I tried to start with the largest number (9) then fill in the adjacent squares with the smallest numbers, then go from there. This attempt eventually proved to not work. On my second attempt, I chose a diagonal approach with the three largest numbers, knowing that no two of them could be on the same row (horizontally or diagonally). I then chose to fill in the empty corners with the numbers that made the most sense to me. After that step, I filled in the boxes accordingly and think I got extremely lucky. I enjoy puzzles and because of that I enjoyed this blog post the most out of all the homework assignments thus far. 

The History of Babylonian Word Problems

 I used to think that word problems were a new concept that had been developed to assist children in learning math. Little did I know that these word problems have been used for hundreds of years. The word problems look pretty similar to current day word problems, with the exception of some obvious differences, like the use of drachmas and the purchasing of livestock as shown in the problem below. 

That is where practicality comes in, back then in 900 AD that was a practical way of learning math, these are actual situations that someone could be caught in. These days word problems look somewhat different and are possibly less practical. Now this question would likely involve dollars and common grocery store items. The difference is that in current-day calculators are incredibly accessible, being on every smartphone. This means for the most part that the majority of calculations do not need to be done by hand. This also means that word problems are less practical and necessary in my opinion. 

Then there is the issue of "pure" vs. "applied" mathematics. Pure mathematics is in its own world, a world that has its own language and can be extremely daunting at times. Applied mathematics takes these over-complicated concepts and applies them to real-world problems that need solving. As a child or a student of math, I also believe it is easier to understand when it is placed in real-world situations. For example, as a child, it is easier to divide twelve apples between your three friends than just dividing twelve by three. As a person that is not overly fond of math, I appreciate it when applied mathematics simplifies a concept for me. 

Assignment 1

1) When x=100, there are 30 solutions

2) When x=210, there are 54 solutions

3) When x=420, there are 162 solutions

4) When x=35, there are 18 solutions

Extension: See if you can find out the number of solutions for x=5 within the next two minutes

Answer: When x=5 there are 2 solutions; y=12, z=13 & y=0, z=5