Assignment 3 topic: Mathematics behind the guitar fretboard

Reference List:

Fretboard.com - The History Of The Guitar. (n.d.). Www.fretboard.com. Retrieved December 4, 2023, 

            from https://www.fretboard.com/guitarhistory.html

Garofalo, J., Corum, K., & Rutter, J. (2022). engineering - a context for learning mathematics: the case of 

            guitar fret spacing. Technology and Engineering Teacher, 82(2), 14.

Language Log» The Musical Origin of the Seven-Day Week. (n.d.). Retrieved December 28, 2022, from 

            https://languagelog.ldc.upenn.edu/nll/?p=54528

Passy. (2012, September 20). Guitar Mathematics. Passy’s World of Mathematics.       

            https://passyworldofmathematics.com/guitar-mathematics/

Schulter, M. (1998). Pythagorean tuning and medieval polyphony. Web source: www. medieval. 

            org/emfaq/harmony/pyth. html.

The Guitar Fretboard’s Mind-Blowing Mathematics. (n.d.). Www.youtube.com. Retrieved December 4, 

            2023, from https://www.youtube.com/watch?v=WxAf3NgF4jk

Lugg, O. (2021). How Pythagoras Broke Music. YouTube. Retrieved December 10, 2023, from https://www.youtube.com/watch?v=EdYzqLgMmgk.

Artistic Format: Undecided

Medieval Islam Mathematics

 1) One stop for me was the mention of the introuction of the concept of 0 in the Islamic world. I was surprised because I thought it was India who introduced the number zero and brought it to the rest of the world. I did not know that the Islamic world also came up with the concept of the nunmber zero independent of India or other nations.

2) Another stop for me was the construction of a geometric representation of complete the square (first one). I found it so interesting that Al-Khwārizmī was able to come up with an alternate method for complete the square than we are used to. This shows that mathematics has room for creativity and imagination, and that there are multiple ways to get to the same solution. This concept that there are multiple ways to get to the same answer in mathematics is something that is not taught in schools nowadays. 

3) I thought that the construction of a parabola using circles on a grid was very interesting and a concept that I have not seen before. This method offers a unique perspective on how geometric shapes and lines intersect and converge to form the distinct curve of a parabola. It blends geometry rules and creativity, showing how math and visuals connect. The step-by-step process involving circles, perpendicular lines, and intersections illustrates a precise yet visually engaging method to depict the elegant structure of a parabola on a grid.

Maya numerals

1. Consider the so-called Hardy-Ramanujan number 1729, and the story of Taxicab Numbers, retold on Wolfram Mathworld at the link above. Hardy is quoted as saying of Ramanujan that "each of the positive integers was one of his personal friends". What do you make of this in terms of Major's paper?

Ramanujan's story tied to the Hardy-Ramanujan number 1729 aligns well with Major's exploration of the personalities linked to numbers. His attachment to numbers, seeing each as a personal friend, resonates with Major's insights into Ordinal Linguistic Personification (OLP).

Major delves into how some people attribute human-like traits or personalities to numbers through OLP. Ramanujan's statement about integers as personal friends showcases a deep and intimate relationship with these numerical entities.

2. Is this something that you might want to introduce to your secondary math students? Why or why not? If you would use these ideas in your math class, how might you do so?

I think that personifying numbers is a pretty cool way to incorperate the First People's Principle of storytelling. Teaching exponents, for example, through personifying numbers can help students remember the rules and methods better. For example, I can tell my students that the "negative" 4 in the exponent is being very negative, grumpy, and sad, and ask them how we can make it positive and happy. They will answer my question by instructing me to bring it to the denominator.

3. Do numbers have particular personalities for you? Why, how, or why not? What about letters of the alphabet, days of the week, months of the year, etc.?

I think as we learn mathematics, we innately label numbers with distinct personalities to help us remember content. The author calls this phenomenon OLP. For example, certain numbers are "nice" numbers and thus "happy" numbers as well. For example, any number that ends with a zero is a nice and happy number to me, most likely because they are nice numbers in the base 10 system. For letters, I guess common letters like vowels, s, t, etc. are nice and happy letters? Regarding other things like days of the week and months of the year, I often associate them with my mood. Obviously, Mondays are sad and depressing because it is often the first day back to school/work, and Fridays are all happy. For years, I love the summer so June and July are like happy months while January and Febuary are sad months.

trivium & quadrivium

 I found it extremely interesting that back in the day, priests could not be ordained if unable to use arithmetic to compute the day of Easter and teach it to others. I have always wondered that without technology how history and dates were able to be retained, and it nbever occurred to me that mathematics was involved.

In class, we learned that multiplication and multiplication tables date back all the way to the Babylonians. However, it was interesting to read that methods of division was only developed during the medieval times. This begs the question of what took so long for scholars to find the counterpart of multiplication. 

Recorde's method of multiplication really intrigued me. At first glance, I was confused and did not know what he was doing. It is really interesting as to why and how he has came up with such a method, that at first glance, looks like he is going in circles. It was also interesting how the text mentioned that the cross he does when multiplying could be the first glance of the multiplication symbol.