Eye of Horus and unit fractions in ancient Egypt

  I found that the different parts of the eye are represented by different fractions.

The right side of the eye = ​1⁄2

The pupil = ​1⁄4

The eyebrow = ​1⁄8

The left side of the eye = ​1⁄16

The curved tail = ​1⁄32

The teardrop = ​1⁄64

These fractions, all with powers of two in their denominators, were used to represent the fractions of hekat, the unit measure of capacity for grains. 

I think these unit fractions were significant during those times since they seem like important units of how they measure the amount of grain. I assume the ancient Egyptians would use them very often as they use grains for purposes like tax and trade so these fractions were commonly used in their society.

Coming from a Chinese background, the number 12 has been ubiquitous in my life. When we watch documentaries, TV shows and movies about ancient China, instead of 24 hours a day, there were only 12 divisions of time in a day. Each division is equivalent to 2 hours. Interestingly, the Chinese uses the 12 earthly branches to name the 12 divisions of time in a day. 

Also in the picture below, you can see the 12 zodiacs in the Chinese culture. One interesting thing I want to bring up is we all know that it is rude to ask people their age. So if you still want to know someone's approximate age and want to make it less awkward, Chinese likes to ask people which animal zodiac they belong to. Each animal cycle is 12 years. For example, the year 2020 is the year of rat. So if someone tells you he or she belongs to the rat zodiac, you know he or she was born in the one of the years of 2020, 2008, 1996, 1984, 1972, 1960.... etc... With this in mind, and by look at the person's physical appearance, you would have made a pretty good guess of how old that person is!

Magic 3 by 3 Box

 At the beginning I know that I have to put the 5 in the middle of the box since you need to have the number in the middle so it kind of balances the sum of each row, column and diagonal . With the 5 put down in the middle, I know that the other two entries of each row, column and diagonal need to add up to 10. That left us with 4 pairs of numbers to put in (1 and 9, 2 and 8, 3 and 7, 4 and 6, shown in step 1 in the picture). 

Next I just do trial and error. My first trial I put 1 and 9 on the diagonal (shown in second step), and I found out that 9 cannot be in the diagonal since it will come in 'contact' with too many other numbers and the problem with it is it is too big of a number. Since it is too big, the sum for certain row, column or diagonal may go over 15. So it is best that the 9 tug in middle column or the middle row where it comes into 'contact' with as few other numbers as possible. 

For my answer, I put 9 in the middle entry of the bottom row. So since it is a big number, the other 2 entries must add up to 6, and in this case you can only have 2 and 4. Once I have filled in the 2 and 4, the other numbers just fill in pretty smoothly. In the end, my answer is:

8 1 6

3 5 7

4 9 2

Was Pythagoras Chinese?

I think it makes a difference to our learning if we acknowledge the contributions of non-European sources of Mathematics. We may have to learn a totally new system of mathematics (like the sexagesimal system of Babylon) if we have to learn math in their context. It may be a good or bad thing. The good thing is that we teach students that mathematics is not Eurocentric and western. The world's oldest civilizations were responsible for developing a lot of the Math concepts and theorems of today. This is also a good chance for students to learn and appreciate other cultures. The bad thing is that the material is so rich that we may not even get to finish all the topics in time. Therefore, having history of mathematics as a separate course in the curriculum (which is what BC is doing) is more appropriate. This can be a more fun and interactive way of doing Math especially for students who see Math as something very boring.

I think in China, the Pythagorean Theorem is still called the gougu theorem. I also think that the naming of the theorems have to do with how modern textbooks were written. This has to go back to the the 18th or the 19th century when education was opened to public. During those times, countries like Egypt and India were colonies of the British empire. The Qing Dynasty of China was in a dire state due to conflicts like the Taiping Rebellion and the Sino-Japanese War. The country was also 'bullied' by the western powers. Therefore, a lot of the textbooks were written from the perspective of the Europeans and almost no acknowledgement was made to any of the oldest civilizations.  And the Europeans were the first ones to have a proper education system for the public so we can see a lot of mathematicians and scientists are white. But I think if we are learning a math theorem and its history, it is better to give credit to the creator even if the theorem itself is not named after him/her. When we are learning these mathematical concepts and theorems, we should learn the real history behind them. 

Ancient Egyptian 'algebra': The method of 'false position' (estimate, check, adjust)

During the reign of HongWu, the first emperor of the Ming Dynasty of China, corruption of government officials was rampant and the emperor was very determined to wipe it out. Officials who embezzled more than the equivalent of 60 liang (one liang was around 30 grams) of silver were to be beheaded and then flayed, the skin publicly exhibited. The problem we have is inspired by this.

During a severe drought, relief funds were passed down from the central government to the provincial/municipal governments to aid affected people in the region. It has been a 'tradition' for these officials to embezzle a portion of these relief funds. When the funds were at the provincial government, the officials took half of it and passed on to the municipal government. Then the municipal government took half of the remaining funds before the funds were passed onto the people in need. In the end, the central government found out that a total of 90000 liang was missing. How much was the original amount of relief funds issued by the central government?

False Position:

We let x be the original amount of relief funds.

x/2 + x/4 = 90000

We try x = 60000 

60000/2 + 60000/4 = 45000

So we know 45000 is half of 90000, so we can just multiply the x by 2: 60000 * 2 = 120000.

120000/2 + 120000/4 = 90000

So a total of 120000 liang was issued by the central government.

Modern method: 

x/2 + x/4 = 90000

2x/4 + x/4  = 90000

3x/4 = 90000

x = 120000

Homework on the history of Babylonian word problems

 After reading the article, I have an impression that the Babylonians were stuck between the ideas of ‘pure’ and ‘applied’ mathematics. On one hand, we see that the Math problems the Babylonians had were all in the context of their daily lives back then. However, on the other hand, we also see that the numbers in these word problems were very unrealistic. We can also see that the problems themselves did not really reflect the reality as well.

These actually remind me of word problems we see in our textbooks and classes. A lot of word problems do not reflect the what is happening in the real world. We tend to take out a lot of factors like friction and air resistance and put in some ‘nice’ numbers so the problem seems easier to solve. I think we do this to focus on the topic on hand so that teachers or students do not digress. It seems like the Babylonians were doing exactly this as well to train their students.

Another reason I can think of is that due to the absence of modern algebra, the math problem seemed too complicated to be comprehended by the Babylonians. And all they could do was to use real life examples to represent the math problem itself. Metaphors and analogies had to be used to present a more suitable way for their students to learn ‘pure’ math. I still remember how my dad explained the concept of substitution to solve systems of linear equations (I was in grade 7 at that time, so I have not learnt system of equations yet). He used real life examples which employed ridiculous numbers (an orange costed $5 and an apple costed $10) so I could understand the problem without having to worry too much about the numbers since they were all ‘nice’ numbers.

These ideas showed how the ancient civilizations dealt with mathematics without modern algebra. To educate their students on more complex concepts, they had to resort to using real world problems with ‘nice’ numbers. I can see that mathematics in these civilizations originated from having to deal with real life problems. Then these problems developed into more concrete math concepts like geometry and trigonometry. However, without algebra, they had to go back to their real life to represent the problems and teach the next generations these concepts.

Assignment 1 Presentation

 Here is the link to our presentation for assignment 1:

Link

Reflection:

I worked on the Sagitta project with Karishma and Tyler. It has led to some interesting findings like the Ishtar Gate built during the times of King Nebuchadnezzar II. We were also amazed by the advance knowledge the Babylonians had with circle geometry. This allowed them to build some of the greatest architectures in the ancient world. We also found that sagitta means arrow in Latin. This inspired us to do a bow and arrow question for our extension of our problem.

I enjoyed working with both Karishma and Tyler. We divided the presentation equally among the 3 of us. Karishma did the introduction including the Ishtar Gate example. I did the proving of the equations and Tyler did the extension with the bow and arrow. And I do feel that our presentation was a success in terms of introducing the class to this interesting topic and getting them engaged.

This also leads to my thought of introducing this kind of group projects/presentations in our high school Math classrooms. It would be interesting for students to work on projects that talk about the history of any topic they learn in school. With such diverse cultures we have here in Canada, students can even research on the history of a certain math topic in their own cultures. This would give them different perspectives of the topics and they will appreciate the contributions made by different cultures to Mathematics (exactly what we are doing in this course).