Eye of Horus and Unit Fractions

 The ancient Egyptian symbol “eye of Horus” (from the sky god Horus who was usually depicted as a falcon) was considered as a protective amulet to the Egyptian. It has been commonly painted on the bows of boats both protected the vessels and "saw" the way ahead. Moreover, the eye is constructed in six fractional parts, representing the shattering of the eye of Horus into six pieces. According to historical documents, these six parts also represent six senses of human. The inner corner of the eye indicates one half, the iris is one fourth, the eyebrow is one eighth, the outer corner of the eye is one sixteenth, and the decorations below the eye are one thirty-second and one sixty-fourth respectively. 

Note: The infinite geometric series with ratio less than one is:

The unit fraction numbers in the Eye of Horus are the first six terms of the geometric series:

Hence, it looks like Egyptian scribes would approximate one using the first six terms of this series because the sum: 1/2 + 1/4 +...+ 1/64 = 63/64.

In Milo Gardner paper (The Arithmetic used to Solve of an Ancient Horus-Eye Problem, 2006), explanations were given on how Egyptian scribes divide one (64/64) by number. In ancient Egypt hekat was a volume unit. One hekat can be considered 64/64 hekat. There are also other volume units such as hin, dja and ro where

1 hin = 1/10  hekat

1 dja = 1/64 hekat

1 ro = 1/320 hekat

Scribes would express the division of a hekat in terms of Horus Eye unit fractions and these smaller units of hekat. Here are some examples from Milo Gardner paper(2006).

In addition to Horus Eye unit fractions, Egyptians also used other unit fractions. All fractions can be represented as a sum of unit fractions (1/n, n is any natural number). According to (Katz, 2008, pg.5) the Egyptians expanded the fraction with non-unit numerators into the sum of unit fractions. Whenever a number of objects need to be equally divided into a number of receivers, this expansion method can be useful. For example, suppose we need to equally divide five loaves of bread to seven people. Then one can expand 5/7 into sum of unit fractions as follows: 

5/7  = 1/2 + 1/7 + 1/14 

This expansion also gives the plan of how to cut the bread. That is each person will get half of a bread loaf plus one seventh of a bread loaf plus one fourteenth of a bread loaf. So it is very interesting to know that some ancient peoples made good use of unit fractions in their daily business when they did not have the luxury of modern mathematics.

Many numbers have special properties, meaning or applications. Cultures are also connected to some numbers. In this Wichita State University website (http://www.math.wichita.edu/history/Topics/snumbers.html), there is some information on special numbers such as perfect numbers (a positive integers that is equal to sum of its divisors excluding itself such as 6 = 1+2+3). 

Amicable numbers are the pairs of numbers with the following properties:

1. Sum of divisors of first number (excluding the first number) equals the second number

2. Sum of divisors of second number (excluding the second number) equals the first number

An example is 220 and 284. Sum of divisors of 220 is:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

Sum of divisors of 284 is:

1 + 2 + 4 + 71 + 142 = 220.

There are also special numbers in cultures. Ancient Greeks are said to have believed in four elements (earth, water, air, and fire). Native American culture also talked about four directions (east, west, north, south) from where the wind came, and each of these winds had accompanying stories.

Constructing a magic square

 

Three by three magic square: Each square has a number from 1, 2, ..., 9 used once. Sum along any row, any column, and diagonal are all 15. What are the numbers in each square?

                                                                        1 + 2 + ... + 9 = 45

Write the square as 3 by 3 matrix.

The method of 'false position'

 The method of false position is suitable for solving an equation with one unknown. Usually the equation contains proportionality w.r.t. the unknown. Chinese dish problem discussed last week can be solved by this method.

Response on " Was Pythagoras Chinese?"

•  Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

Since I was born and raised in China before coming to Canada at age of 12, the foundation of my mathematical knowledge was learned in China. This includes arithmetic operations, integer operations, fraction operations, solving algebraic equations and system of equations. Because of the difference in mathematical curriculums, I was ahead of my class for several years since grade 7. In Canada, I learned pre-calculus in high school and more advanced mathematics in university. Hence, I have the experience learning math from two different cultures.

In China, elementary school math is taught with a blend of Chinese and Western sources. I remember learning how to use an abacus and we have our own version of Pascal’s triangle we call it “Yang Hui triangle”. Also, as mentioned in Gustafson, we learned right triangle theory as “gou-gu” theorem. Nonetheless, we also use Arabic numerals, and we call our variables x, y, z. Even though we have Chinese names for sine, cosine, and tangent, we write using the Western notations in problem solving. As a matter of fact, using various sources of mathematics have little impact on student’s learning. What made a huge difference between Canadian school and Chinese school is the way of teaching the knowledge. As mentioned in Gustafson’s work, memorization serves as a ground for knowledge building in Chinese education. Students are required to memorize definitions, theorems, as well as ways to solve problems. Moreover, no calculator is ever allowed in exams. In my opinion, this way serves well on building a good foundation but when students become more mature, I’d like being taught by Western approach more. When in advance level, students should use tools to aid achieving the purpose of understanding. Time should not be spent on doing arithmetic by hand, yet knowing how to do arithmetic by hand is also important. 

•  What are your thoughts about the naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle...check out its history.)

I would say, the way of naming mathematical theorems after the founder is sometimes bit misleading. Nowadays with the advancement in technologies, the development of knowledge are watched closely by academic society across the world. It would be easier to identify the source of a new idea. However, when communications cannot be done efficiently in the past, the validity on awarding one (or more, for co-founding) person for the contribution is questionable. Different people can independently come up with breakthroughs (e.g. Newton and Leibniz). In my opinion, the naming should be referring the knowledge instead of referring the person who come up with it (e.g. intermediate value theorem). Of course the acknowledging of the inventor is also important, but that can be done separately. Another benefit for not naming after a person is to remove the culture bias and barrier in knowledge. For example, when we look at the theorems in math courses, we get the Eurocentric feeling in mathematics because most of the theorems are named after European mathematicians. However, mathematics has been developed across different countries and cultures for quite a long time. Instead of fighting for who did it first, people should pay more attention on the evolvement of the knowledge. 

Babylonian word problems

 Word problems can be both practical and imaginary (pure math). For example, the first problem in our reading is “I have added seven times the side of my square to eleven times its area, and it is 6;15”. If we think from plane geometry point of view with assumption that the numbers ‘seven’ and ‘eleven’ do not have dimension (unit-less), then we would say there is no practical value.

7(side of square) + 11(Area of the square) = 6;15

It is because the equation representing the statement does not satisfy dimensional equality to be practical. The left hand side is the sum of two objects with the first object in dimension of [length] and the second object in the dimension of [area] or square of [length]. Some people would treat it as a puzzle. In fact, the term ‘square’ is not necessary. ‘Square’ and ‘side’ can be replaced by ‘circle’ and ‘radius’ or, more generally, ‘f(x)’ and ‘x’. Thus, if we use symbolic algebra, it can be written as:

7x + 11 f(x) = 6;15

In this statement the function f can be an abstraction such as a measure of an n-dimensional structure built from a component object of value x. We can further generalize it by replacing the constants with other objects (words, parameters) such as:

ax + b f(x) = c

On the other hand, we can assign the length dimension (7 meter, 7 feet, etc.) to the number seven; assign no dimension (unit-less) to the number 11. Then the problem becomes “the area of the rectangle made of seven meter side and the side of my square added to eleven times my square is 6;15”. 

The problem now has a practical value. Therefore, when we talk about a word problem or a set of word problems such as writings on clay tablets found in Babylon archaeological extraction from the so-called lens of ‘practicality’, ‘generality’, or ‘abstraction’ while sitting at our time, we would see that they are connected or we can build the connections. In other words, in the language of contemporary (symbolic) algebra in which we have a store of not only natural language but also mathematical symbols, we see the connection. We can see them separate, and we see them together because we can specialize, generalize, and replace some objects with abstract objects. Any applied problem can be generalized, then add abstraction to it. When we studied ‘vector space’ (abstract idea) in linear algebra, we could also transform in opposite direction.

 However, if we can transport ourselves to the time of Babylonia, what would be the case? We would not have symbolic algebra. We could start from ‘generality’. Generalization is possible using words. The above same example can be reworded into: “I have added seven times the size of a side of my object to the eleven times the size of my object. It is 6;15”. Here the word ‘object’ can be square, circle, cube, etc.

For abstraction without symbols, it is possible but it would be limited by the skills in using words and sentences, and experiences either real or imaginary. In above rewritten word problem, we could build the ‘object’ as an abstract object. In “Babylonian algebra (Crest of the Peacock)” we learned that Babylonians did power of four and eight even though they could only have power of two (area) and three (volume) in applications. Therefore, it is safe to assume that limited abstraction is possible without using symbolic algebra.

By this discussion, we find that by learning the history of mathematics (Babylonian and Egyptian), we are studying applied and pure mathematics of those days, and we could learn how generalization and abstraction can be possible and to what extent without using symbolic algebra.