Babylonian algebra response

I found this article very interesting. I had assumed that Babylonians would not have had a similar method to solve problems like algebra. I was especially impressed by the way which they were able to deal with the fractions and roots which showed up in example problems. I’m still trying to wrap my head around how they figured all this out without easily available writing materials. I suppose that a lot of what is being discussed can be drawn out in the dirt and done graphically. The method of finding an “exact” value for a square root could be understood using geometric relations. I suppose, like the Greeks, Babylonians understood the relationship between numbers and space (maybe very literally).

 Me reading how babylonians did square roots: 

https://blogs.sas.com/content/iml/2016/05/16/babylonian-square-roots.html

EDIT: In terms of abstractions, these algebraic differences between the modern and the Babylonian only differ in the terms used to represent the mathematics (and some other language). I still find it very impressive that the algebra done by the Babylonians was so sophisticated without the use of what we consider to be algebra. In the example 4.7, the problem is stated as a length of a rectangle exceeds its width by 7 and has a totals area of 1,0 ( 60). In modern terms this might have been written a rectangle has a length which is 7 units larger than its width and has an area of 60 units. I would have done width = x then length = x + 7. Then x(x+7) = 60 and solved this quadratic. But this is a very abstract method of doing this problem. I would like to see how this was done graphically, which is probably how Babylonians would have solved these problems.

Word problems through time

It does appear that we have continued to use word problems throughout all of our math history, from the Babylonians to our modern classrooms. I agree that the ancient people of Babylon probably used word problems as a way of practicing problem types. I think that it’s unlikely that the ancient Babylonians had a good system of creating problems the way we have, and so putting a problem into words was the best way to convey the problem. Even in more recent times a problem would be given in words. The following is a recreation of a math problem and its solutions. I think that this helps to highlight that these problems (which would not have been expressed algebraically) are ways for earlier mathematicians to work on complex problems. https://old.maa.org/press/periodicals/convergence/recreational-problems-in-medieval-mathematics-men-buying-a-horse 

EDIT:

To expand on the idea that I was trying to get at previously, in modern day we can express many problems in abstraction through algebra. A problem like A^2 + B^2 = C^2 is abstracted from the graphical format which it originated with algebra. If we wanted to consider the same problem in the ancient world, we might have to begin by explaining the rules of our problem and drawing it. We have also reduced the need for explanations through a long history of education, a quadratic does not need a context for it to have a solution or root. Given a problem like x^2 -2 =0 we know how to solve it without context or explanation, but in the ancient world people would not readily know the type of problem simply from numbers.  To that end they needed context and diagrams to better understand the problems which appeared in their lives, or math that solely existed for fun.

45 multiplication base 60

 here is my multiplication table for x*y=45 for a few items!

Base 60 & 12 response

    I really liked these two articles (I think they are the articles I have read the most closely in this program). I found them both very thought provoking and engaging. I knew about the divisions of angles into minutes and seconds, but I didn’t really understand their context. I find that the two arguments for why base 60 or base 12 are quite compelling. I personally think that there reason that a society would agree on a specific number system is because it is convenient for a problem which is prevalent in their society. If a group of people are focusing on ideas of astronomy it would make sense to work in either base 12 or base 60. I also find that the idea of counting joints on one hand and using your other hand to count groups of 12 up to 5 groups of 12 also makes very logical sense (but why wouldn’t it extend to base 144?). I attached a method which the second resource was describing for counting to 60 on your hands, and could be extended to 144.

The units are being counted on my right hand and groups of 12 are being counted on my left hand. In the case being described in the second reading we would have 1 group of 12 and 8 more, but in the 144 system we would have 4 groups of 12 and 8 more. I found this interesting enough to include in my blog.

I think I didn’t really have any preconceptions on how we came to measure time, so it was all interesting and new to me. I liked the connection to the angles on the sundial with minutes and seconds, and was very interested to hear that minutes and seconds were not considered until the 14th and 16th centuries respectively. I was even more interested in how a standard hour wasn’t adopted until clocks came about.

I still wouldn’t be able to decide which of the reasons being discussed in these articles are the reason for the base 60 number system in ancient Babylon. Both feel intuitive enough to be widely adopted, and have very practical applications. 

Peacock response

 I think that with modern mathematics, we have come to think of math and science as a western invention. I believe that this is a gross reduction of the impressive mathematics of other lands that were far ahead of European mathematics. Especially in Asia where the Middle East, India and China were massively ahead of Europe for a long time. The maps in Peacock provided a great visual representation for the web of knowledge of mathematics in our “early” history. 

I’m looking forward to engaging in the diverse mathematics and many proofs which come together to form our current mathematical systems. I hope we have a chance to go through some of the mathematics which were used for cultures like the Mayans, and China. Both have impressive mathematical understanding, but in the case of the Mayans we see their application through the understanding of astrology. This is a very impressive feat of mathematics!

EDIT: 

Surprises: 

1. The fluidity of mathematics throughout the Levant and Mediterranean Sea. Aka that the Greek mathematics were kept alive by the arabs throughout the dark ages of Europe.
2. Figure 1.3 shows china directly working with Baghdad. This is surprising because they are really far apart, how did they get there? Boat?
3. The same figure shows a lot of back and forth in the east but I’m surprised that there seems to be very little back and forth from China. Basically just mathematical trade between China and India, otherwise China is only exporting these ideas.

why base 60

 I think that base 60 was used for astronomy. It is a good system to work with lunar systems and roughly works for the solar year. It would be convenient to calculate any of these using a base 60 system since 30 days would be 0.5*60 and 360 is 6*60. I cant think of a good reason to calculate 3600 other than decades.

The only thing I use base 60 for is time, it's very useful.

Looks like we currently use base 60 for time, angles and navigation. These are convenient because were on a globe and moving in a fairly circular pattern. 

pic from: 24hr Analogue Clock | Jadco Time

EDIT: 

Babylonians used this base 60 system to do a verity of complicated mathematics. We have seen that they could do complicated multiplications and divisions using this system. The big issue they ran into when using this was the concept of irrational numbers. I found that number like 7/60 would be impractical for them to use and they are excluded in their multiplication and division tables (tablets). However this is actually rather less of a weakness because 60 is more divisible than our base 10. Examples include 1/3 or 1/15. In decimal these are 3.33…/10 or 0.66…/10 , but in base 60 this is 20/60 or 4/60.

History of mathematics included in math class or not? Integrating history of mathematics in the classroom dialog

    Before reading this article I believed that a math teacher should include some of the mathematical history, because I believe that there are many ways for a student to connect with the subject material. Some students may find it difficult to “store” material which doesn’t have a story associated with it. If it helps a student relate to a concept through the struggles and story of a mathematician, it’s worth including. I also believe that it can offer an easy introduction into a topic which camouflages as not math.

    After reading, I feel that these ideas have been reinforced and I have considered how I might discuss the topics of math history in my classroom. I found that the first place I stopped and thought was in the objections, however I found the “practical” objections sounded like excuses to why a teacher might not put the effort into their class. I found another interesting topic to consider was the idea of providing famous false proofs. I really enjoyed the idea that a group of students might work together to find the flaw in a famous mathematician’s proof, or confirm a true one.

    I think this article didn’t exactly change my opinion, however it did provide an opportunity for me to engage in a way of approaching subject material I hadn’t considered previously.