Magic Square

 This is my approach:

Eye of Horus and unit fractions in ancient Egypt.

After researching the Eye of Horus, I am fascinated about how the ancient Egyptians linked daily life to mathematics, and then to religion or beliefs. The complete eye is composed of number signs, which were used to describe hekat. Hekat is used for measuring amounts of grain or flour. 

There are many such examples in our life. For example, the number seven has its unique place in many religions and cultures.

In Buddhism, there are seven factors of awakening. 

In Christian teachings, there are seven deadly sins. 

In Chinese culture, we use "七七四十九天“ (which means" seven times seven gives forty-nine days") to describe a task that is very difficult. This is because the number seven is considered as "complete". By saying this task took one person 7x7=49 days to finish (exaggerating, of course), it shows that the task in complex, and the person spent a lot of time to perfect every aspect of the task. 

Ancient Egyptian 'algebra': The method of 'false position'

Response to "Was Pythagoras Chinese?"

The article by Ross Gustafson provides excellent insights on the naming of mathematical theories, which had never occurred to me before reading it. As an ethnic Chinese, I appreciate such discussions which contribute to the acknowledgment of non-European sources of mathematics. Similar to other STEM subjects, mathematics is extremely objective, rational, and universal in the sense that different mathematical sources give the same conclusion to the same problem. Acknowledging non-European sources likely will not change the foundations of modern day Mathematics. Nevertheless, such acknowledgement does makes a difference in enriching math education and reducing cultural marginalization. Inclusion of math works from non-Greek origins exposes students to a variety of cultures, which might engage students in learning mathematics. The marginalization of what traditional Europeans think of as "peripheral cultures" does not only occur in the attribution of mathematical development. Introducing non-European sources of mathematics to the nowadays "global citizens" can be the first step towards the elimination of the long-established cultural injustice. 

 The article talks about how the mathematical theories are named in western societies, and I can provide a perspective of the naming issue from within a non-European society. Through my K-12 education in China, I learned Pythagorean theorem, as well as Pascal's Triangle under the name of their Chinese origins, Gou Gu theorem and Yang Hui Triangle respectively. At the same time, I was also taught the Vieta's Formula and many other Greek-origin concepts. From my understanding, the naming of these mathematical concepts was determined long before the globalization of our world, and it was rather independent of the political power of each nation at that time. Instead, reliable and understandable sources were more critical. Just as how Gustafson points out that "no English translation of the works has been made to date, despite the fact that [Jiu Zhang Suan Shu] is perhaps the most important compilation of ancient Chinese mathematics" (page 3), the barriers from documentation and translation of mathematics work from other cultures are likely the reason why they were not acknowledged by western societies. The naming of mathematical works possesses historical factors. Such debates is better to focus on the acknowledgement, rather than the naming itself.

Response to Babylonian Word Problems

Many word problems from either Babylonians, Greeks, or modern humans, all contains some degrees of practicality. This is because such problems are presented in a more dialogic form, which is closer to our daily life compared to some more abstract math languages. 

The ability to make mathematical language abstract, thus saving brain space for more efficient computations, did not appear until the Greeks. As a result, the Babylonians had to use a single form of discourse to communicate mathematical ideas. The concepts of "pure" and "applied" math were not distinguished during that time. 

Whether a word problem is abstract or practical does not rely on our familiarity with contemporary algebra because the purpose of such a problem is clear and self-explanatory. Jens Hoyrup has commented on Babylonian word problems: "as soon as you analyze the structure of known versus unknown quantities, the complete artificiality of the problem is revealed". Unpractical problems were designed to train students in mathematical methods, which then evolved into abstract problems with algebra. 

These ideas help us to study the social and pedagogic purposes of word problems at different times, which indicates the development of math discourses. 

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