Differences Between Egyptian and Babylonian Numerals

The two number systems are that they both use place based notation to represent larger numbers. However, place is much more important in the Babylonian system because the symbols alternate between two simple shapes. In the Egyptian system, while they did seem to order their symbols in terms of place, the did not necessarily need to. Their symbols were all different, so if one was to shift the order around it would still be the same number. For example, shifting the Egyptian number system would only rephrase the problem to 50 + 100 + 1 + 1 + 10 000. This would not work in Babylonia, the number behind the symbols would be lost.

In Roman numerals, place only kind of matters. They overcomplicated their symbols by having the adjacent symbols impact one another. These additional rules caused basic addition and subtraction to be much more difficult than a place based system. 

Surveying in Ancient Egypt

In my engineering degree, I have learnt much about surveying. I have had jobs surveying, and have completed many courses on the subject. What surprised me was how little about this history I was taught. As I am seeing more often through the lens of this course, the majority of my STEM knowledge has been warped by western-centrism. I was only taught about how the Europeans used surveying, and the history of it was only taught back to the middle ages of British Kings. This article is amazing, and I feel robbed of this knowledge. I will ensure that my students will not suffer the same fate.

We have so much evidence of these different measurements, and the article mentions what they Egyptians knew about angles. I wonder how little they actually knew. Not every brilliant idea is recorded, so I can't help but think that there is something that the evidence is not telling us. We see that they have seqets for vertical angles, but that does not explain how they determined lateral angles. Perhaps there was a rule of thumb that surveyors of the time used, which was only passed on by word of mouth between generations.

The text mentions surveying techniques being used by priests and cultists. I wonder how much spiritual emphasis was placed on these mathematical concepts. We have heard the story of Pythagoras' math religion, but I know nothing about how math impacted Egyptian religion. Again, this is due to the western lens I have been taught through, but I wish to delve deeper to uncover the answer.

The Amazing Solution Behind the Russian Peasant Method

The Russian Peasant Method and the Egyptian Method are identical, except that the Russian Method uses a math trick to determine which factors of 2 must be used.

As a recap, to solve 9x13 using the Egyptian Method we find 1x9=9, 2x9=18, 4x9=36, 8x9=72. We then determine which combination of these equations gives us 13 9's. Finally, we sum up the right hand side of all of these equations. The end result is 13x9 = 1x9 + 4x9 + 8x9 = 9+36+72 = 117. 

The issue in the method is that we must determine which combinations of these equations gives us 13 9's.

When we first look at the Russian Method's algorithm, there are two steps that seem chaotic: Only paying attention to the odds on the left, and ignoring all the decimal remainders. BUT, these two steps actually compensate each other perfectly. The remainders are actually accumulated in the odd rows while we simplify the expression.

Let's take advantage of the fact that x*y = 1*x*y = 2/2 * x*y=x/2 * 2*y.

In equation 1, we have expanded 13 x 9 = 6 x 18 + 1 x 9. Note that we have converted this multiplication problem to a multiplication and addition problem, where we added a remainder. The most important note is that the remainder is EQUAL TO THE RIGHT NUMBER IN THE ROW ABOVE.

Equation 2 does not introduce a remainder, only halves and doubles the two numbers in the multiplication.

Equation 3 converts the multiplication aspect into two additions, one being a remainder and the final being the largest doubling we reach.

In the Russian Method, we add the Right numbers in the rows where the Left numbers are odd. But what we are actually doing is adding half of the Right numbers in rows that are not integers. It just happens that these two statements are equivalent.

Even rows will not create a remainder. Even rows with a remainder are just represented using the row above, and this row above must be odd because it created a remainder.

Therefore, the Russian Method is the same as the Egyptian Method but uses this neat remainder trick to determine which combinations of equations are needed!

Solving Division Using Egyptian Methods

 I decided to solve y=27/6 using the Egyptian method. Starting with doubling:

1x6=6
2x6=12
4x6=24
8x6=48

We stop here because 48 is larger than 27. The only numbers that we can combine and not exceed 27 is 24 alone. Even 24+6=30. Therefor, the whole number part of y is 4.

 27-24=3, so we are currently missing 3. Now onto the halving:

(1/2)x6=3

After only one iteration, we have found the missing 3, and we get it by adding (1/2) to y. Therefore,
y=4 +1/2 = 4.5

Babylonian Word Problems

 Word problems have the potential to deepen understanding in students, but it also has the ability to further the divide between a student and the subject matter. By needlessly adding a layer of complexity to a problem, students may disengage. These problems will only maximize their potential for learning when the problems reflect the real world authentically, not just needlessly shoehorning a random scenario that is not likely in the real world.

Done properly, world problems can show a student the practicality of the knowledge they are learning. A student can see that this knowledge has a use in the real world. By implementing multiple word problems into a single unit, it can also show students just how versatile and general math concepts can be through a process of un-abstraction. Again, if this is done un-authentically, a student could see this as “See! This topic is so useless that my teacher could only come up with this random scenario.”

Educators have been making up inauthentic word problems that only test learner’s knowledge in a practicality vacuum for thousands of years. Even in Babylonian times, these problems focused more on exemplifying a pure math concept instead of focusing on how one would actually apply math in a real world problem. But just because this problem has existed for so long, doesn’t mean that it is unsolvable. I believe that if the proper time and energy is put into formulating these problems authentically, students will truly benefit from them.

Base 60 Examples

 These are the five examples I found of 2 base 60 numbers that multiply to equal 45:

45 = 6 * 7,30

45 = 30 * 1,30

45 = 2,24 * 18,45

45 = 8 * 5,37,30

45 = 40 * 1,7,30

Crest of the Peacock: Reflection

This text brought forward the importance of seeing math through a historical lens. I am used to seeing conflict through this historical lens and am therefore used to examining my bias. I know that the genocide of indigenous culture was a history written by the victors. I know that Africa had a rich culture before European subjugation, and my knowledge of African culture is written by the victors.

All my experience with historical bias is about devaluing cultures. I have not spent much time evaluating the bias in devaluing the impact of other cultures on an abstract institution such as math.

The most impactful part of this reading was seeing the traditional Eurocentric view of math’s trajectory vs a more global perspective on the subject. It was only after seeing the two diagrams that I realized how non-sensical the idea of a “dark age” was. It was almost like reading propaganda, “European maths are so fundamental that the entire global production of knowledge shut down.”

I am excited that this course takes down my pre-conceived notions that were instilled during my formative years. I am also excited to stop these biases from being transmitted to the next generation of learners during my career.

Why Base 60: Response

At first glance 60 is superior to 10 because of the large number of factors it has. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60; that is 12 factors! The number 10 only has 1, 2, 5, 10. Also notable is that the factors of 60 contain the low more commonly used numbers. Very rarely do I have more than 6 of a thing on my person. Just right now I have 4 pens, 1 helmet, 5 notebooks, and 2 waterproof clothes in my bag.

Having many factors could be important because it allows for simpler fractions. 60 divided by all its factors will produce nice non-decimal numbers. I could imagine an ancient tribe seeing a decimal number as their math “breaking.” So, through a trial-and-error method, a base 60 system caused their math to break the least.

We still use base 60s for time keeping. After some research, we also use it for measuring angles and China uses base 60 for many of their astronomical systems. The closest I can think of is that there are 12 inches in a foot, but that is not exactly base 60. There was a push for a metric time-keeping system, but it seemed that humans were much less willing to accept it. It seems like time is so intrinsic to human perception of our world that we are unwilling to change it. This reminds me of how there was a pushback to the ideas that time was a dimension of spacetime.

My research has shown that this idea was more or less the acceptable idea of the justification for the base 60 system. I have enjoyed learning about these Babylonian systems via my own private research in the past, and parts of this knowledge has luckily stuck in my brain. I am very happy that I am in this class, officially learning things that I have only dabbled in.

Response to Constantino's Integrating history of mathematics in the classroom: an analytic survey

 As I’ve been teaching math to students, I have always tried to sprinkle the history of math into my lessons. Yet, my current stance on the matter does not line up perfectly with Constantinos. Constantinos gives the impression that history of math should be its own class. That it should take up as much class time as the traditional math itself. This lends itself to criticism 07, that there is not enough time to teach both sides of math. History of math is a spice that enhances the traditional curriculum; a little goes a long way. When introducing the topic’s problem, framing it in a historical lens will foster further by-in from the students. After the introduction of a topic, I do not normally continue pushing the historical lens.

 

The number of examples and resources provided by Constantinos is amazing. As I was reading, I created a file with these alternative subjects. I am unsure if I will be able to fit them into the pre-existing BC curriculum. If I find myself with pedagogical freedom in my future, I will use these examples to further foster my student’s growth.