Jackson's Math History Blog

My thought process in solving the Magic Square is as follows: 1) Since all rows, columns and diagonals must have values with a total of 15, then the number appearing in the center of the square must be used in the middle row, the middle column and the two diagonals, for a total of four different combinations involving this number. 2) Using the same observation as above, each of the four numbers that appear on the corners must be used three different combinations that total 15. 3) Lastly, the four remaining numbers (non-center, non-corner) are used in two different combinations. Therefore, I can solve this problem by writing down the possible combinations starting from 1: 1, 5, 9 1, 6, 8 2, 4, 9 2, 5, 8 2, 6, 7 4, 2, 9 4, 3, 8 4, 5, 6 5, 1, 9 5, 2, 8 5, 3, 7 5, 4, 6 .....etc. Once all combinations are listed (there's a pattern emerging that resembles how polynomials work), the number 5 is the only one that can be used in four combinations, so it must be the c...

Before I began this week's reading on the history of the word problem genre, I was of the view that "pure" and "applied" mathematics are somewhat interrelated, especially as many abstract concepts were borne out of practical uses of math, so it would not be appropriate to separate them into two fields. This perspective was somewhat supported by several scholars mentioned in this chapter, including Oxford's Eleanor Robson, who argued against this kind of dichotomy in mathematics as problems can function in two levels using essentially the same idea. Like many have pointed out, some problems to the Babylonians might seem like practical subsistence problems at first, but many of them are too unrealistic and artificial for real life application. Also, resources available to the Babylonians (relative to the Greeks) may have dictated their mathematical methods, so they had no choice but to use a "unified" mathematics, using practicality as a means t...

The use of unit factions is convenient for everyday applications where we need to divide a set of things among an unequal number (usually larger) of individuals. For instance, suppose a group of eight friends wants to share three pies in equal portions, how can they accomplish that? A straightforward division would not solve this problem because 3 divides 8 equals 0.375, and the presence of decimals complicates things even more. I think by representing a fraction like 3/8 as a sum of two unit fractions (ie. 1/4 + 1/8) makes this kind of sharing problem much simpler. We can now infer that each person will get a quarter of a pie and one half of another quarter of a pie. Since dividing by half (or into quarters, one-eighths, etc.) is practically easy to do, this method could be useful for many similar problems, including the inheritance problem posed on the blog. If the man's inheritance is to be shared in 1/2, 1/3 and 1/12 parts among his children, then adding these unit fra...

It was fascinating to learn that the Babylonians used methods that resembled modern algebra to solve equations. In particular, they adopted what the author referred to as the "rhetorical algebra" using imaginative numerical tables (for multiplication, reciprocates, squares etc.) to expedite their problem-solving process. The logic and principles behind their techniques - solving math problems step by step using an established numbering system and framework - are still existent today. Prior to modern algebraic notations, it was reasonable to use terms like "length" and "area" that have actual physical meanings in their daily lives to denote quantities. This makes understanding a mathematical problem much easier. From there, "hidden knowledge" that were obtained from previous problems and transcribed into records can be used as mechanisms to produce answers for each question. As for facing a mathematical world without algebra, it is personal...

1 45 2 22, 30 3 15 4 11, 15 5 9 6 7, 30 8 5, 37, 30 9 5 10 4, 30 12 3, 45 15 3 16 2, 48, 45 18 2, 30 20 2, 15 24 1, 52, 30 25 1, 48 30 1, 30 36 1, 15 40 1, 7, 30 45 1

In this week's reading on the non-European roots of mathematics, what fascinated me the most was how the certain parallels between Greek and Indian philosophies could be drawn based on historical evidence, and that Pythagoras even went as far as India to pursue knowledge. The author brought up examples of these "mathematical crusades" to point out the importance of diverse transmissions of mathematics across cultures. Although India served as a meeting point for civilizations to trade goods and share ideas, the magnitude of influence was greater than I had imagined. A further look into the early exchanges of mathematics allows us to trace many concepts that are still in use today back to India (such as trigonometric functions). Secondly, a generalization made by the author that caught my eye was how an "Eurocentric bias" still exists across many disciplines, not just in mathematics. This is especially true in social sciences, where many philosophies and v...

My first impression of using the number 60 as base rather than 10 was simple: I thought of the clock system we use today. There are 60 seconds within a minute, and 60 minutes within an hour, so it made sense. Perhaps 60 was the "original" base number, but the number 10 was used later in many other scenarios to make counting much easier (akin to how the Chinese language was simplified). The number 12, which is a factor of 60, also serves significance in the Chinese culture, where the 12 zodiacs play a major role in people's beliefs, values and even social behavior. It's also interesting that how the square of 60, which is 360, equates to a full circle in degrees. Since the Sun and Moon were divine symbols in most ancient cultures, the number 60 could hold also significance. This was prior to being introduced the historical background of the Babylonians and doing the in-class activity with my peers. Further research into its origins somewhat confirms that I was on th...

As someone who studied and taught economics, I feel I'm not a stranger when it comes to introducing the historical background of certain concepts into the classroom. I remember during the first month of my undergraduate journey, my economics professor brought a film for us to watch in class, a film starring Russell Crowe called A Beautiful Mind , which happened to win several awards including the Golden Globe. So why did we spend an entire week's worth of classroom time watching a movie that seemed boring to most students? The reason is simple: it has historical significance, and while it's not a form of entertainment by choice, it certainly was better than listening to the professor and taking notes, so as students most of us (I'd assume) felt our attention was worth it. Besides, it was Russell Crowe. Hence, having experienced a "history within lecture" episode myself, I am a strong advocate for integrating history within teaching, not just for mathematics,...