Oct 4 Assignment 1 Solutions and Extension

To find positive integer solutions to Pythagorean triples, it is proposed that the ancient Babylonians used the reciprocal pairs method: Given s (some fraction greater than 1, which they wrote in sexagesimals) and 1/s we take their sum and difference, s+1/s and s-1/s. Next we take the difference of their squares: (s+1/s)^2 - (s-1/s)^2 which is always equal to 4 = 2^2. This was a great finding since the equation can be rearranged to (s-1/s)^2 + 2^2 = (s+1/s)^2. However this doesn't necessarily give postive integer solutions to x^2 + y^2 = z^2. To see how what could be done, since s is some fraction, write s as p/q where p>q>0. Then (s-1/s)^2 + 2^2 = (s+1/s)^2 becomes [(p^2-q^2)/pq]^2 + 2^2 = [(p^2+q^2)/pq]^2. Multiply both sides by (pq)^2 to get (p^2-q^2)^2 + (2pq)^2 = (p^2+q^2)^2. Given this formula now, positive integers p and q can be chosen such that x=p^2-q^2, y = 2pq, z = p^2-q^2. The problem is that this won't always generate a reduced Pythagorean triple. If the GCD of x,y,z is not 1, it'll have to be divided out from each of the three in order to get a reduced triple. Several observations/conditions were gleaned regarding the conditions for p and q to get a reduced triple, but no proof has been made on those claims. Modern solutions use more areas in known number theory to derive a proof on how to generate reduced Pythagorean triples. The conditions are that p and q are coprime, and exactly one of them is even and the other odd. This is a necessary and sufficient condition in order to have reduced triples. 

Reduced Pythagorean triples satisfy the Pythagorean equation x^2 + y^2 = z^2, where x,y denote the length of the two legs of a right triangle and z its hypotenuse length. The Pythagorean theorem, which applies to right triangles, is a special case of the cosine law, a^2+b^2-2abcos(C) = c^2, where the angle opposite to the hypotenuse, c, is 90°. Since cos(90°) = 0, it boils down to just a^2+b^2=c^2. The cosine law is useful for finding an angle measure when given all side lengths. It is also useful for finding a missing side when given the other sides and one angle measure. A great proof video on the derivation of the cosine law can be found on Khan Academy.

Oct 4 Comparing Ancient Egyptian and Ancient Babylonian Number Systems

The most notable difference between the two numeration systems is that the ancient Babylonians, as we have first studied, used a system in base 60, sexagesimals. 60 had many interesting properties and was helpful in concepts like time/representing fractions. The ancient Egyptians used the base 10 system which is what many are familiar with today. The symbols the Egyptians used to denote numbers are also more complex in looks as they are more pictorial. For the Roman numerals, it also uses a base 10 system. 1 is I, 2 is II, 3 is III. However there is a special symbol for 5, V. 4 is IV since it is one before 5, and 6 is VI since it is one after 5. 7 is VII, 8 is VIII for the same reason. But 10 is X, and 9 is IX, one before 10. Roman numerals have special symbols not only for powers of 10 but also for the powers of 10 times 5, i.e. 5 is V, 50 is L, 500 is D. Roman numerals are also more efficient in writing and carving onto blocks. In elementary, my social studies class went on a field trip. We learned that when European traders came to Canada, they etched Roman numerals on the trees since it was easy to carve lines instead of the curvyness of the Arabic numerals. A nice affordance for the Egyptian system is that it uses base 10 which we are familiar with. The Babylonian system uses base 60 which is helpful in representing fractions and time. A constraint for both systems is that it is slow since you have to write down literally 9 symbols to represent 9. The Egyptian system’s symbols are also complex which take more time to write out. Those pictorial symbols do have associations for the numbers they are representing though. It correlates with the quantity of/ level of importance associated with the symbol drawing.

Oct 4 Ancient Egyptian Land Surveying Response

The article mentions the symbolic connection of the surveying rope with the ram's head of the ancient Egyptian god Khnum, which may indicate the sanctity of measurements. I wonder what religious beliefs were attached to surveying in ancient Egypt. Were there rituals/ceremonies performed at the start or completion of important surveying tasks, especially given the importance of these measurements for agriculture and construction? The article also mentions the length of a remen which is half of √2 cubits long. I wonder if the ancient Egyptians had a way of calculating its decimal expansion to some place value. There is a trick to find square roots of numbers and giving an approximate decimal expansion, but I have not been taught about it at school. I only saw kids working on it while I was helping out at Kumon. I’m wondering if they knew about rational and irrational numbers as well. If they did, did they know that √2 was irrational, and what was their proof? One particular thing that surprised me was that the annual inundation of the Nile necessitated the need for consistent remeasurement and redistribution of arable land. This impacted not only on the practice and development of surveying but also on taxation and governance. The Nile was a source of life, but its yearly inundation would also erase boundary markers, making it vital for surveyors to redefine land divisions and boundaries. Such a consistent activity would eventually lead to advancements in surveying techniques.

Sept 27 Russian Peasant Multiplication and Ancient Egyptian Multiplication

It was interesting to see the video on Russian peasant multiplication. Numberphile videos never fail to amuse me. I had to scratch my head when trying to relate it to the ancient Egyptian multiplication method. The aforementioned method uses doubling, binary, and sums based on the multiplicands as shown in class. It's an efficient method that is more intuitive for students to understand how to multiply large numbers together. The Russian peasant multiplication method uses a different algorithm but is similar to the ancient Egyptians. For example:   

With the two examples above, the left side uses the Russian peasant method and the right side uses the ancient Egyptian method. I realized that the number of times I half each round for Russian is the same number of times I double each round for Egyptian. There was a consistency there. I also saw that writing [the left] multiplicand in binary, the rows with 1 for Egyptian corresponded with odd numbers for Russian, and the rows with 0 corresponded with even numbers. Therefore the rows with even numbers in the Russian peasant method are not used. The rows with 1/odd are highlighted in teal, and the rows with 0/even are highlighted in red. In essence, with a chosen multiplicand, one method is doubling starting from 1 and reaching the largest power of 2 that is less than or equal to that multiplicand, and the other method is starting from the multiplicand and halving (ignoring the remainder if odd) until 1 is reached. 

Sept 27 Ancient Egyptian Multiplication Example

Sept 20 Babylonian Word Problems

It is quite interesting to see that many ancient cultures, Babylonians, Egyptians, Greeks, Indians, and Chinese were able to think about and devise mathematical problems without any of the discoveries/technology we have today. Indeed their problems were devised based on practicality so it may prove useful to them. Other times it wasn't necessarily how realistic/practical it was, but using ideas that we can abstractly generate in our minds that could work in a different world. The problem is that if it doesn't make sense realistically (shooting an arrow off the moon) it gives people an impression that such knowledge is useless, so it is better to apply it using more appropriate examples instead (shooting an arrow in a space with no air friction). The point is to exercise our minds and train our brain to think in different ways. The same mathematical concepts were discovered by cultures all over the world that didn't have interactions with teachers back then, hence math is considered a universal notion. For me, pure math is proof based using rigorous axioms and theory build ups, whereas applied math is using what is learned and constructing them into problems we face daily or into problems that are imaginable. Conceptualization of these ideas doesn't require what we know about algebra per se. Even young kids can think about what situations math is needed without formal education in the subject. In essence we cannot look at the ancient peoples and say they weren't as smart as we were just because they didn't have the sufficient amount of resources. Instead they were the first ones to discover a lot of mathematical properties that we build off of today. The foundation is important and we should give credit to them for it.

Sept 20 Babylonian Algebra

The statement can be used using physical properties of spatial objects, eg. length, width, height, square, cube. One may want to know the dimensions of a particular field or construct a table of certain measurements. The ancient Babylonians were able to use word problems to describe different situations, which are now modeled more easily with the algebraic expressions we have today (for instance, buy and sell, property related questions). They also had a sophisticated number system which allowed them to perform arithmetic in meaningful ways with respect to time, distance, and volume. Mathematics isn't all about generalizations and abstraction. Linking it to physical properties is important. For calculus, it is used in finding volumes of shapes, used in engineering, and physics. Calculus also links to rates of change and accumulation of quantities. Graph theory requires not much algebra as it is simply a mathematical way of understanding vertex and edge relationships using some sets, but mostly diagrams. This connects with social relationships, such as for networking sites like LinkedIn. Graph theory also helps deal with transmission of virus linkages from a positive source, and there was a nice video with Eddie Woo and Poh Shen Loh talking about it.

Sept 18 Babylonian-style base 60 multiplication table for the number forty-five

Susan gave an example of 2 and 22,30 whose product is 45 in base 60

Below are five pairs of numbers whose product equals 45 in base 60:

- 4 and 11,15: 4 times 11 is 44, but 4 times 11 and a quarter (,15) is 44 + 1 = 45

- 6 and 7,30: 6 times 7 is 42, but 6 times 7 and a half (,30) is 42 + 3 = 45

- 8 and 5,37,30: 8 times 5 is 40, so to do 0.625 in sexagesimals, 0.625*60 = 37.5, so the tenths place is 37, and the hundredths place is the value that represents half of 60 which is 30

- 12 and 3,45: 12 times 3 is 36, but 12 times 3 and three quarters (3* ,15 = ,45) is 36 + 9 = 45

- 20 and 2,15: 20 times 2 is 40, but 20 times 2 and a quarter (,15) = 40 + 5 = 45

Sept 13 Response to Crest of the Peacock

Many information/discoveries made by European expanders/explorers came to be over the past few centuries and that what we learned about these information/discoveries are from their observations. It’s as if they weren’t readily available before any encounters the Europeans had. We all know that isn’t the case. Many mathematical concepts and techniques often attributed to European scholars were known and used centuries earlier in other parts of the world. So for me, getting to observe how the ancients were able to do calculations and mathematical observations to such a fascinating degree where they could build the pyramids, roads with no pot holes, and the sewage system of the Forbidden Kingdom is truly remarkable (and no current European system can replicate such a feat). I never learned that a lot of these fathers of ancient mathematics from Greece (whose foundation became bedrock for modern Euro-mathematics cultural dependencies) actually interacted heavily with other ancient cultures from Egypt/Mesopotamia/India. That entails that indeed the honor and respect for these other cultural observations on math findings are important and somewhat embedded in math teachings today. I say somewhat due to the fact that there is so much more that could be extracted/implemented/used/taught from them, but also because the credit for them is hardly ever mentioned. So if a student (who learned their math history) came from one of these countries to North America and sees many concepts being credited to whites in relatively more recent times compared to the great mathematicians from their culture, would they not feel dejected and subverted? The numeracy skills and systems developed in those places back then were also observed to be more efficient, and in some places used in areas of our daily lives like in computer languages and time telling. So, to be flexible is important and to use and appreciate how vast this whole topic truly is will lead to a more inclusive and diverse field of expertise.

Sept 13 Why Base 60?

60 is a number with many factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Contrasting it with 10, which only has 4 factors. The large number of factors makes base 60 calculations more flexible in many situations, especially when it comes to divisions or fractions. For example, it's much easier to express a third in a base 60 system (as 20) than in base 10 (which results in an endless repeating decimal: 0.333...). So getting rid of the notion of repeated values for some rationals may allow people to think more clearly and focus more on the value of a third, rather than the repetitions. 

The factors of 60 are used in many places in time and geometry. The number of degrees in a circle is 360, which is very similar to the number of days in a year 365). 60 degrees is also the angle of a special right triangle. The number of days in a year is defined by the orientation of the star constellations so defining time with base 10 would be difficult. In many cultures around the world, there are 12 months in a year and are often represented with creatures real or mystical (like the Chinese and Greek zodiacs),  Time within the day is measured using hours, minutes, and seconds. There are 60 seconds in a minute and 60 minutes in an hour. There are two 12 hour half days in a day. For time, sometimes in English 7:15 will be stated as a quarter past 7, since 15 is a quarter of 60. 7:30 will be stated sometimes as half past 7, since 30 is a half of 60. 

After doing some research, I’ve found out that there are more uses for 60 in the [Babylonian] numeration system. For example the harmonic connotations. The number 60, being a multiple of 12, might have cultural or religious connotations in societies that revered the number 12. The Babylonians used base 60 for their calculations, and many of their mathematical/astronomical findings were foundational for subsequent civilizations, which lead to 60 being used instead of 10. The number 60 is also the lowest common multiple of all the numbers 1 through 6. This makes it easy to integrate or switch between different bases, such as base 2, 3, 4, 5, 6 within a base 60 framework. This property cannot be offered with base 10.

Sept 11 Integrating history of mathematics in the classroom: an analytic survey

    Math is a universal subject and its rich history of discovery across the world is worth mentioning. Humans are critical thinkers and math arouses topics that explain how things are around us. Different discoveries around the world lead to new findings, and applying how the people in the past figured things out without the use of technology is astounding. They understand that the math mysteries are deep within nature. A certain concept is called differently throughout history. The credit should be given to anyone who independently came up with the idea. I like to not only call it Pascal’s triangle, but also the Staircase of Mount Meru or Yang Hui’s triangle. How the information came to be in our school books doesn’t come in a blink. It is through eons of discovery and trials, and it gives the evidence of the first forms of applications/usage.

    Math is displayed as a learning subject that evolves throughout time. I see how math notations and instructional styles taught 300 years ago during Newton’s time are not the same as today. How students learn math around the world is also unique, so knowing the universal history of math allows a better context for the entirety of the subject’s exploration. It’s like many flowing streams branching out of a large basin. Why study and row in one stream when it's connected to all other streams? The deeper awareness of math in its intrinsic and extrinsic nature reminded me of the Philosophy of Mathematics course I earlier took. How to analyze math per se in its rigor and asking questions on the realistic or abstract meanings of mathematical concepts will allow students to think deeper and openly.

    The article allowed me to better know the benefits of having a deeper mathematical foundation through learning its rich history of discoveries. I know I will mostly bring up a lot of topics done differently in grade school compared to university (eg. the notations, change of definitions), so it is fascinating to see why that is. Finding and setting some time aside to implement some history in the classroom initially seems complicated, but bringing forth some historical problems which lead to what is being studied gives a background context for why certain topics are being studied in the classrooms.