Homework: Babylonian 'Algebra' from Crest of the Peacock

Before the development of algebra and algebraic notation, words are used to represent the variables/geometric term. 

For example, ush is used to represent length, and sag is used to represent breadth. The product of ush and sag gives us the geometric quantity area. 

The operations were also described verbally, where "add to the two-thirds of my two-thirds" means   $\frac{2}{3} \times \frac{2}{3} $

Mathematics is not all about generalization or abstraction, but understanding. Of course, proper symbols and tools for abstraction save a lot of brain space when we try to understand a mathematical problem, but the end goal is still to be able to decode the math syntax and connect to real world applications. 

Take calculus for example, the function integration can be interpreted with discrete steps. Instead of calculating  $\displaystyle \int_{x=1}^{10} f(x) dx$, we can use summation: $\displaystyle \sum_{x=1}^{10} f(x) \Delta x$ to approximate it, where the symbol $\sum$ can be further broken down into 10 addition steps, and the babylonians had alreay established verbal rules for describing addtion and multiplication.

Base 60 Multiplication Table for 45

 I have written a MATLAB code to generate such base 60 numbers. I am not a programmer by any means ,but you are welcome to check it out! 

Number 1	Number 2
4	11,15
18	2,30
40	1,7,30
24	1,52,30
16	2,48,45
37,30	1,12
6,15	7,12
2,24	18,45

How to use Latex in blogger

For my own record and my fellow math people: we can use Latex in blogger! Toggle the HTML view when editing a post, copy and paste the following codes, and Latex is ready to use!

 

Examples: $\pi$, $\sqrt{\frac{8}{3}}$, $\displaystyle \lim_{x \rightarrow \infty} a^x$

Enjoy!

Response to "The Crest of the Peacock - Chapter1: The History of Mathematics: Alternative Perspectives"

Upon reading George Gheverghese Joseph's justification of this book, I was immediately attracted to his view on the Eurocentric approach to history. While the development of the world is viewed holistically in generally with more focus on European contribution, I never realized that the field of modern mathematics is regarded as entirely European-origin. Joseph points out that "the contributions of the colonized peoples were ignored or devalued as part of the rationale for subjugation and dominance", and "the development of mathematics before the Greeks suffered a similar fate". 

What's surprising is that the ancient Greek did acknowledged their learning from the Egyptians, and yet this affirmation was devalued by the descendants of the Greek. The same thing also happened to the naming of "Arabic numerals", where the original Indian inventors were credited but not translated (page 11). These biased events really made me think how deep the Eurocentric ideas are embedded in modern documentation of the history of mathematics. 

Another interesting point raised by Joseph is how all the mathematical knowledges are transmitted among cultures without much barriers (page 13). While this chapter talks a lot about mapping the transmission, not much emphasis is put on how our ancestors overcame the cultural and language barriers. Investigating this could also provide insights to the development of mathematics history as a whole. 

Base 60 - Babylonian Mathematics

Speculative phase:

When talking about base 60, the first thing that occurs to me is the notion of time: 60 seconds constitute a minute, 60 minutes constitute an hour, etc. However, it is unlikely that the Babylonian people had the technology to use such a precise and complicated time-telling system. 

I then suspect that they used base 60 in combination with base 10 to reduce the amount of writing needed on the clay tablet, as it was difficult to store clay tablets of larger sizes. 

In other times, sexagesimal is used not only in the time-telling system as mentioned above, but also in a variety of practices. For example, ancient Chinese started to use the sexagenary cycle to represent dates in cycles of sixty terms. 

Research phase: 

There have been numerous studies on why the Babylonian used sexagesimal counting system. 

The Greek mathematician Theon of Alexandria (335-405) proposed that 60 was the smallest number divisible by 1, 2, 3, 4 and 5 so the number of divisors was maximized. While some people do not agree with this reasoning by saying that a base 12 system would be more efficient in terms of the number of divisors. I personally like Theon's theory because the number 5 is an important component of both the base 10 and base 60 systems. Moreover, humans have five fingers on each hand, making counting in five's a very practical methodology. 

Otto Neugebauer (1899-1990) suggested that the adoption of sexagesimal by the Babylonian to divide weights and measures into thirds. Again, this theory does not explain why the base 12 system was not chosen instead. 

Reference: https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/

Response to "Integrating History of Mathematics in The Classroom: An Analytic Survey" by Constantinos Tzanakis and Abraham Arcavi

I always think that history is an important part of mathematical education and it should be incorporated into the current curriculums. As a subject with a long history, the knowledge of math builds on itself. Learning about the history behind math can help students to better understand the development of mathematics. Demonstrating the historical and practical side of mathematics also engages students, reliefing them from repetitive arithmetic and rule-memorization. 

I totally agree with the author that "mathematics is often regarded as a discipline which is largely disconnected from social and cultural concerns and influences". (page 212) Many people see mathematics beyond basic arithmetic as useless, especially after the popularization of smart phones with built in calculators. But, in reality, the applications of mathematics penetrate every aspect of people’s daily life. (example, the book GEB)

My understanding of the history of math was limited to the people who made breakthroughs in this field, and their contemporary environment. However, after reading about "taking advantage of errors, alternative conceptions, change of perspective, revision of implicit assumptions, intuitive arguments" from chapter 7.4.6 (page 219), I realized that history is not composed of only scattered pieces of event — the sequence of those events and their connections also constitute history. The trials and errors of our pioneers provide tremendous insights to mathematical education.