Entry slip (Sep 28)----Babylonian "algebra"

 

I could not imagine how Babylonian solved those puzzles before the development of algebra. I wonder how did they figure out these procedures? 

For example, in example 4.6 in the article, the third step of Babylonians' solution is " Take the reciprocal of 0; 33,20". I do not understand how they knew to do so. What is their logic to take the reciprocal in this step? Without the unknown variable x, what do these independent numbers 0; 33, 20 and 1; 48 mean to them? 

Another example is about solving quadratics. I am surprised at Babylonian's solution of Example 4.7 shown in the article. Their approach was equivalent to our quadratic formula. I cannot imagine how they figured it out. Their first step is " Halve 7",  but it does not make sense to me. Why halve 7? Even though they got the right answer, how could they explain logically to others? 

I guess they made many tables to summarize the rules and laws about   algorithms. They might not have a general mathematical principle, so they need to study and record many cases as the samples. 

I think mathematics is mostly about abstraction and I hope it could be more about generalization. Once we deal with numbers and do calculations, we are entering the Math world, and it is abstract. For example, you have 5 apples, and I give you another 5 apples, if I ask you how many apples do you have, you will think about 5 plus 5 equals 10. When you doing this calculation, there is no apple in your mind. You are using some symbols and rules, and there is nothing to do with apples. When you get 10, you jump back from abstract math world to the real world, and tell me you have 10 apples. So if mathematics could be more about generalization, we could use and apply it more easily. 

Without algebra, I think sometimes it is not easy to state general or abstract relationships in any areas of mathematics. Since if it is an general or abstract relationship, we need to use some symbols to represent every thing in such category. Algebraic expressions is used to represent mathematical relationships. So I think using algebra is a good way to state relationships. 

Class notes:

Here is a good piece elaborating on the origins and uses of base 60 by Mesopotamian/Babylonian mathematicians, from the excellent University of St. Andrews math history database. 

https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/

A Babylonian word problem:

Entry slip (Sep 25.)---Babylonian Table of Multiplying to 45

 

Class notes:

What might these tables mean?
The first column multiplied by the second column equals 60. Or, 1/2 = 30/60, 1/3 = 20/60, ...1/10=6/60, this is a table of unit fraction. 
Note that, in this notation, commas separate place values (for both whole numbers and fractions).

Can you figure out the common theme here?

Why are certain numbers missing from the left hand column? For example, there is no 7, 11, 13, etc.
Because 60 divided by these numbers will get recurring decimal numbers. 
How do fractions in the Babylonian style connect with our fractions? (Keep this in mind as we learn about ancient Egyptian fractions later on...)

Entry (Sep23)---- The Crest of the Peacock

 

" The Crest of the Peacock", what a beautiful and miracle book. When I read it, I feel like entering another world. From Egypt and Mesopotamia to Europe, from the Dark Ages to Renaissance, plenty of pictures appear in my mind. Mathematics as the bridges connect these cultural areas and eras, and also connect me and you.  

The four-thousand-year-old Babylonian clay tablet with the value of n^3+n^2 struck me. This clay tablet implies that "Babylonians may have used these values in solving cubic equations ". I wonder how and why they invented it. Is it used for calculating the orbit of stars or trying to figure out the relationship between tracks of sun and moon?

When I read about the cross-cultural contact between India and China, I recalled a famous classical historic Chinese novel named "Journey to the West". This story is about a Tang dynasty Buddhist monk traveled to India to find Buddhist scriptures. This novel is so popular that most of Chinese have read it. It is also an evidence of a cross-cultural contact between India and China. In Yuan dynasty, a Chinese mathematician Zhu Shi Jie wrote a Math Book. Some every large numbers used in this book were from India's Buddhist scriptures. 

One of the features of mathematical activity through the ages makes me feel admiration for ancient mathematicians' spirit of pursuing knowledge. This feature is stated as "the relative ineffectiveness of cultural barriers(or 'filters') in inhibiting the transmission of mathematical knowledge." Around 2500 years ago, Greek Mathematicians such as Pythagoras and Eudoxus need to travel on foot to another country to learn knowledge. One can imagine their journey must be full of difficulties. Nowadays, people could work and learn online at home, and obtain many resources remotely. We should cherish what we have today, and remember what they have done for us yesterday.  

Link of the book:

http://www.ms.uky.edu/~sohum/ma330/files/Crest_of_the_peacock.pdf

Entry ( Sep 21) ----- Base 60

 

Thousands of years ago, the Babylonians used a number system based on 60. One of the reasons to use 60-based number system is because of its convenience. From 1 to 100, 60 is the only number has factors including 1, 2, 3, 4 and 5. That means if an ancient people has 60 fish, he could equally distribute these fish to 2, 3, 4, or 5 person, and if he has 2 times 60 fish, he still could evenly divide fish into 2, 3, 4, or 5 parts. With this property of 60, they divided many things including the time, the year, and the circle. 

Especially in the astronomy measurement, (yes, ancient people like to watch the sky, me too), people often need to divide angles. If they use Base-10, it would be difficult to divide an angle into 3 parts equally.  

In human history, there existed the base-8, base-12, base-16 and base-20 number systems. In China, people are still using the idiom "Half catty eight taels" to describe two things which have no difference to each other. It proves that Chinese people ever used base-16 number systems in the past. 

Nowadays, the base-10 number system is mostly used  in our daily life. Comparing with the base-60, the base-10 needs less symbols to represent digits, and can be easily learned. However, when we tell time and measure angles, the sexagesimal plays an irreplaceable role. Even today, in Chinese traditional calendar, they still use traditional way to  count year based on the sexagesimal. 

The base-60 number system embodies the ancient people's wisdom. Through writing this blog, I have an opportunity to learn and show my appreciation to those great endeavor.   

Below is the link of the reference article:" Why is a minute divided into 60 seconds, and hour into 60 minutes, yet there are only 24 hours in a day?"

https://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/#:~:text=The%20Babylonians%20made%20astronomical%20calculations,the%20first%20six%20counting%20numbers

Course Note:

Our detailed look at the history of mathematics will start in an area known as Mesopotamia (meso: 'between', potamia: 'rivers' -- between the Tigris and Euphrates Rivers), in what are now the countries of Iraq, Syria, parts of Turkey and Kuwait. The area is sometimes called the 'fertile crescent' because the river systems flowing down to the Persian Gulf made agriculture and cities viable.

In accounts of mathematics history, the mathematics of Mesopotamia circa 3100 BCE - 300 BCE is usually called Babylonian mathematics. But if you look more carefully at the history of Mesopotamia in this period, there were several different peoples and nations that governed this region, including the Sumerians, Akkadians, Assyrians and Babylonians. 

For our purposes, we will use the term 'Babylonian mathematics' to refer to the fairly unified mathematics traditions over this 3,000 year period (starting approximately 5,000 years ago). 

Babylonian writing was done with a wedge-shaped reed stylus in wet clay tablets the size of a person's palm, and then dried in the sun or in a kiln. Their ingenious writing system was known as cuneiform: 'wedge-shaped', and it was possible to write words, numbers and other symbols with just these wedge-shaped forms. Because these baked clay tablets are very durable, we have many of them in museum collections to this day -- and quite a few of these seem to be teaching tablets for scribes learning mathematics to take government jobs in the Mesopotamian cities!

Here is an example of one of the existing mathematical clay tablets from ancient Mesopotamia. Your job is to figure out what is written here, and how the writing system for this mathematical tablet works!

First reading (Sep 9)----Why teach Math history?

 

I firmly believe that Math teachers should mention some interesting historical Math stories when teaching a certain theory or concept. For example, when teaching the Gauss formula many Math teachers would tell a story of Carl Gauss about how he amazed his teacher with finding the sum of the integers from 1 to 100 when he was a young boy. Through telling stories, students will know the names of mathematicians, uncover how and why these concepts have been invented, and most importantly feel happy and relax. Students love stories. Teachers could tell the story of zero when explaining why zero cannot be the denominator, the story of the Pythagorean Theorem, the story of numbers when teaching place value, and so forth. Telling Historical Math stories is the only approach I knew about integrating history of mathematics in the classroom. Since the history is normally imparted chronologically, how to connect these pieces together to show students a whole picture of the history of Mathematics makes me feel I am not teaching history.

After reading the article of "Integrating history of Mathematics in the classroom: an analytic survey", I realized that "integrating the history of mathematics into the educational process" is so significant. The authors of the book elaborated Why from five aspects. Among them, "The appreciation of mathematics as a cultural endeavor" is the one that I strongly agree. Without Math,  technology, medical treatment, transportation and many other things wouldn't be possible. To show our sincere appreciation to Math, we need to know its history more or less.

Another part of the article inspired me is the Historical Problems. Problems with no solution or problems unsolved could show students what is the tenacious and perseverance. These are the spirits of Mathematics we want to pass on from generation to generation. Be courageous, do not fear the problem that looks hard. It is hard, and the only thing we need is taking a little bit more time on it.