Eye of Horus and unit fractions in ancient Egypt

 

The most interesting thing about Eye of Horus to me is its missing fragment. The legend says Horus' eye was broken into 6 pieces. Ancient Egyptians gave each of them a fraction as a unit of measurement. They are:  1/2,  1/4,  1/8,  1/16,  1/32,  1/64. In the legend, the eye of Horus was restored and made whole. So, the sum of these 6 unit friction should be equal to 1. However, we know the sum was short of 1 by 1/64. There was a fragment missing. 

There are some assumptions on this point. 

Some people believe that the missing part was withheld by Thoth's magic. 

Some people said it could mean that nothing is perfect. 

Some people explained that if we see 1/2,  1/4,  1/8,  1/16,  1/32,  1/64 as a well-known Calculus 'infinite sequence', then this infinite series can be added up to exactly 1. The missing 1/64 was supplied by Calculus. 

He Tu 河图 ( River Chart )--Chinese Cosmology

He Tu is also called Yellow River Map. "He" 河 means River, river of stars or galaxy. It is cosmological diagrams used in ancient China. It was used for the movement of the stars through the Nine Palaces.

The Nine Palaces were the groupings of stars that were identified

that traversed the heavens.  

It also used in geomancy. The table shows the meaning of numbers when it serves to explain the correlation of universe and human life. 

Reference:

https://thekongdanfoundation.com/lao-tzu/the-heru-luoshu-and-nine-palaces/

Constructing a Magic Square

please work on figuring out this 3X3 magic square, where each number from 1 to 9 is used once, and where all the rows, columns and diagonals add to 15!

 My solution:

1. The first thing I thought was to put 5 in the center because I found these pairs (1, 9), (2, 8), (3, 7), (4, 6) plus 5 is equal to 15.  

 2. 9 is the biggest number, start looking at this one. I can only find two combinations with 9 that sum to 15 : 1+5+9=15, 2+4+9=15. So 9 cannot be placed at corner. Then 1, 2 , and 4 can be placed in the square.

  3.  Then the rest of the numbers can be placed in the square by simply using addition.

Reading response----Was Pythagoras Chinese- Revisiting an Old Debate

Pythagorean theory is one of the most important theories in Euclidean geometry. When the Chinese high school Math teachers introduce the gou-gu theorem (another name of Pythagorean theory in China), they always emphasis that, "No, the gou-gu theorem is not invented by Gou Gu, no such a person called Gou Gu. Gou represents the long leg, and Gu represents the short leg, Xian represents the hypotenuse. Gou three, Gu four, and Xian five." 

Students always try to get some information more or less from the name of the theories or laws. For example, Newton's laws of universal gravitation, and Newton's laws of motion. Students could know that these laws was firstly formulated and published by Newton. The theories, such as Euclid-Euler theorem and Galileo's Principle, get named after their originators probably because people could easily reference them by  their names. For example, Galileo Principle was given by Galileo in his second book " Dialogues concerning Two New Sciences" published in 1638.

For Math teachers, I think it is unnecessary to intend to avoid acknowledging non-European sources of mathematics. If the source of mathematics is appropriate to support their teaching practice, they could use it to enrich students' knowledge base.  There are hundreds of proofs of the Pythagorean theorem. Different proofs use different mathematical theories. For example, Garfield's Proof used the knowledge of area of trapezoid.  Chinese Zhaoshuang's proof used the knowledge of squares of binomials. Teachers could choose those proofs to support their teaching if the proofs are relevant and appropriate.  

The method of 'false position' (estimate, check, adjust)

Problem:  Tom has a glass of water. He pours out three eighth of the water, then he has 20 gallons of water. How many gallons of water does he have at first?

The modern solution:

   x  -  3/8 x = 20        5/8 x = 20       x = 32

Egyptian " False Position" solution:

Try x = 8,    if originally Tom has 8 gallons of water

3/8 x = 3,   then he pours out 3 gallons of water

 8 - 3 = 5,     then he has 5 gallons of water. 

However, in the problem, Tom has 20 gallons of water left, which is 4 times as much  as 5 gallons of water. So this time, we could try 32 ( 4 x 8 = 32 ).

If x = 32,  3/8 x 32 = 12 , 32 - 12 = 20 

Assignment 1-- Extension about Egyptian Fraction

Reference:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

(I put the reference at the beginning is because this is a great webpage about Egyptian fraction. I highly recommend you to visit it if you are interested in the Egyptian fraction. Using copy and paste if the hyperlink doesn't work. )

From Egyptian Mathematical Papyrus, we found that ancient Egyptians could handle of fractions 4000 years ago. They only have notations for Unit Fractions, but they can represent more general rational numbers as a sum of distinct unit fraction.   

                                

Here are two algorithms for finding Egyptian Fraction (during my presentation, I only introduced one):

Method 1: Using Splitting Equation

Example:     2/6 

Decompose a fraction into the sum of unit fractions

               2/6 =  1/6  +  1/6  

(All the unit fractions should be different. That's because when ancient Egyptian repeated the process of dividing, the reminder gets smaller and smaller.)

Convert one of the repeated unit fractions into the sum of distinct unit fraction by using the splitting equation:

               2/6 = 1/6  + 1/7  + 1/42   

                         

Method 2: Fibonacci's Greedy Algorithm

Fibonacci proved and gave this method in his book Liber Abaci, the same book for Fibonacci Number.

Reflection (written on Nov 17th):

Due to some technical issue with my computer, my part of the presentation went not well on that day. After I shared my screen, the audience looked at a screen that was different from what I looked at. We realized this problem when I almost reached the last piece of slides. I think if I were not that nervous and talking slowly, we could notice that problem earlier. 

Fortunately, the two most important and interesting slides, "Egyptian solution" and "Fibonacci’s Greedy Algorithm" were presented without a problem. There are so many interesting topics under Egyptian Fractions, and I just presented the tip of the iceberg of it. There is so much fun to learn about the history of mathematic, so I think I will introduce it to my students in the future. 

Response on The History of the Word Problem Genre

Teaching and Learning word problems could help students better understand Mathematics concepts and real-life situations. However, not all word problems reflect reality, or not all of them have the practicality. Some word problems might be designed for further exploration, or just for practicing mathematical strategies, or just for fun. From this point of view, word problems have generality and abstraction. Not only symbolic manipulation could be abstract, word problems could also be abstract, especially for those artificial word problem. " Pure " mathematics and "applied" mathematics cannot be completely separated, since even the most abstract mathematics could have applications.

Above statements are true for word problems either of Babylonians or of modern world. Although ancient Babylonians lived in more than 4000 years ago and their representations are limited by language and notations, they have the same wisdom as the people living in modern world. Our familiarity with contemporary algebra could not change the nature of the mathematics.