Zhu Shijie and “Pascal's” Triangle

Zhu Shijie and “Pascal’s” Triangle

Gerry Moerkerken

Figure 1: Image showing Pascal’s Triangle as a fractal (self-created)

Historical Background and Context Zhu Shijie ( 朱世杰 ; 1249–1314) was born near present day Beijing. He is known for unifying the mathematics of the Chinese northern and southern traditions. Two of his works were highly influential. The first, released in 1299 was Introduction to Mathematical Science, which played a role in a calculation method further develop by the Japanese. The second was The Precious Mirror and the Four Elements released in 1303.13 The book itself tells us that Zhu was a teacher for 20 years travelling “over seas and lakes” throughout China and “the number who came to be taught by him increased each day”.8 Zhu Shijie lived during a troubled time in Chinese history. For the first time the entire empire was ruled by a foreign power: Khublai Khan. The Yuan dynasty was declared by the Mongolian khan in 1271, but the conquest was not completed until 1279. Khublai was a promoter of Science, Maths, and Architecture. This can be seen by the capital building designed by the khan near present day Beijing. Despite the hatred of the Chinese towards the Mongols, the economy seemed to flourish during this time, as was written about by the Venetian, Marco Polo, and Korean, Ch’oe Pu.1 The history of mathematics in China is difficult to unravel. Straffin writes that mathematicians did not receive the same social status as humanities scholars, which may have attributed to the large gap between the work of Lui Hui in the 3rd century and the work produced by 13th

century mathematicians such as Zhu.10 Martzloff adds to this that little is known of Chinese mathematicians and it is often difficult to define who the mathematicians were.8 The Chinese at the time of Zhu Shijie used counting rods. It is not entirely clear when they first became a method of counting, but there is evidence to suggest the counting rods were used at least as early as 480 BC.7 The counting rod system is a base 10 system using bamboo with a length of 13.8 cm and diameter of 0.7 cm. A combination of rods could be used to make various numbers.8 The earliest known texts of Chinese mathematics include Zhou Dynasty Canon of Gnomonic Computations and Computational Prescriptions in the Nine Chapters, both anonymous and written around 200 BC. From these texts it is evident that the Chinese used a decimal system, had methods for basic mathematical operations, and also knew about Pythagoras’ theorem. During the first century, it appears the Chinese used nine textbooks, but the level of mathematics relative to other cultures was quite low. However, the Chinese remainder theorem and several other significant results appeared in these textbooks. Unfortunately, they have been lost.8 Of interest, these books contained a proof of Pythagoras’s Theorem using tangrams!2 From the period of early texts until the 13th century there is little evidence of development in Chinese mathematics. During the 13th century mathematics blossomed. As noted, this was during the Mongol occupation, and it is speculated, but not backed by hard evidence, that Islamic and Chinese mathematics may have diffused at this time.8 This is a very brief overview of the historical context in which we find Zhu Shijie. For a very interesting and informative read I recommend A History of Chinese Mathematics by Jean-Claude Martzloof. Precious Mirror and the Four Elements Hoe writes that the title of the book has been erroneously translated. The original title as directly translated is “Mirror […] jade […] four origins”8. Jade has been translated to precious, while its intention is more likely to indicate brightness. Furthermore, origins has been translated to unknowns or elements, but is meant to refer to the elements of the earth as per Chinese philosophers, specifically, heaven, earth, man and matter.8 Like most Chinese mathematics texts, Precious Mirrors and the Four Elements is based on problems and solutions. An example is the beautifully written problem below: I take along a bottle with some wine in it for an excursion in the spring. On reaching a tavern I double its contents and drink one and nine-tenths dŏu in the tavern. After passing four taverns the bottle is empty. Permit me to ask how much wine was there at the beginning? 7 I think this is an excellent problem to introduce some numeracy in the English curriculum, and also develops engagement with Asia! In total, there are 288 problems in the book. It also consists of seven prefaces, four figures, and four preliminary problems. The figures include Pascal’s triangle!8 Lay-Yong describes how Shijie used the triangle to find the roots of equations to the 8th degree.4 For example if we take the equation (a + b)8 and expand, the transformed equation will be a8 + 8ba7 + 28b2a6 + 56b3a5 + 70b4a4 + 56b5a3 + 28b6a2 + 8b7a + b8 .

The coefficients correspond to the values in the eighth row of Pascal’s triangle. Another interesting mathematical concept that appears to be introduced by Zhu is the idea that the product of two numbers with the same sign (negative or positive) is always positive, whilst numbers with opposite signs will always yield a negative result.5 What is particularly interesting is that Pascal himself was not born till 1623 yet the triangle was named after him! I find these ideas fascinating because I did not know about the Pascal triangle rule to solve equations to the nth degree, and I really struggled to understand the idea that two negative numbers multiplied by each other would yield a positive. There is a wealth of information about Zhu Shijie and his contributions to Chinese mathematics, but much of it is at a level beyond my own understanding. However, I believe a gifted year 11 or 12 student might be very interested in exploring these concepts further as part of an extension activity.

Figure 2: Yanhgui triangle – Wikimedia Figure 3: Pascals Triangle - Wikimedia

Teaching Activity The following activity has been adapted from http://mathforum.org/workshops/usi/pascal/hs.color_pascal.html & https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-two- more-patterns 9 I think this is suitable for year 7 or 8. It could also be used as a remedial activity for students who struggle in year 9. Materials • Blank template of Pascal’s triangle (one per student): http://mathforum.org/workshops/usi/pascal/pascal_handouts.html • A set of colouring pencils (enough for all students) (at least 5 colours, and 6 of each colour) • Tape • Scissors Steps • Arrange the students in groups of 6 • Give each student a blank template of Pascal’s triangle • Ask the groups to fill in the triangle numbers using a pencil – you will likely need to demonstrate on the board • Give each group 6 colouring pencils – all the same colour per group, but each group can represent a different colour • Have the groups colour all the blocks with the number one it • The groups will now swap colouring pencils so they have a new colour (but ensure everyone in each group has the same colour) • Now have the groups colour in all blocks that divide by 5 without a remainder • Again have the groups swap colours and repeat step 7 except now colour cells that divide by 5 but have a remainder of 1 • Repeat till all the cells are coloured • Now have the groups cut out their triangles and combine them into hexagons – unique patterns will be clearly visible!

Rationale Several ideas were explored to develop for an activity related to Chinese Mathematics, but this one caught my eye because of the beauty of the designs that can be developed. At first I thought an activity like this might be a bit rudimentary for a high school class, but Lemon has tried a slight variation with great success.6 She reports a high level of enthusiasm amongst students, which I suspect is because the pattern is so clear and will really create an environment where students no longer feel intimidated by mathematics. In addition this activity could be further enhanced by dressing up as a Chinese mathematician of the time would have, incorporating some Chinese music, adding in some Chinese decorations and finally reading some of the beautiful Chinese poetry written about the same time. An example of a Chinese Mathematics problem was provided in a prior section, but there are many that could be explored. Australian Curriculum Links Integrating Chinese mathematics and specifically the work of Zhu Shijie and the Pascal triangle in teaching can be linked into the Australian curriculum in so many ways that I struggled to keep my topic concise, and picked a few codes that have very clear connections. Maths • Factorise algebraic expressions by identifying numerical factors (ACMNA191) • Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213) • Expand binomial products and factorise monic quadratic expressions using a variety of strategies (ACMNA233) • understand the concepts and techniques in combinatorics, geometry and vectors (Year 11, Mathematics Specialist Syllabus) • 1.1.16 identify the coefficients and the degree of a polynomial (Year 11, Mathematics Methods Syllabus) • 1.3.5 use Pascal’s triangle and its properties (Year 11, Mathematics Methods Syllabus) ICT • Binary is used to represent data in digital systems (ACTDIK024) • Design algorithms, represented diagrammatically and in structured English, and validate plans and programs through tracing (ACTDIP040) History • ACDSEH1(29 – 33) Depth study 2: Investigating one ancient society (Egypt, Greece, Rome, India, China) Languages • Chinese language studies are included in the Australian curriculum as an option, and for those classes a neat exercise might be to have a look at some of the original works of Zhu Shijie. Cross Curricular Priorities • Asia and Australia's engagement with Asia

Enrichment Other Patterns in Pascal’s Triangle The unique, yet distinct patterns that can be found in Pascal’s triangle are amazing. Examples not already mentioned include: • Fibonacci Numbers – add up the numbers in the shallow diagonals to get the Fibonacci numbers.3 Figure 4 displays this pattern. • Pascal Petals – take any number in the pattern that has six surrounding numbers. If they are counted up in a circle (say 1 – 6) then the corresponding values of 1 * 3 * 5 will always equal the corresponding values of 2 * 4 * 6 Furthermore the product of all six numbers is a perfect square!12 • Combinatorics – If the triangle is split by horizontal lines and diagonal lines to form a grid it can be used to solve combinatorics problems. For example if there are 8 colours to pick from and 5 must be chosen we can determine the number of possible combinations by looking to Pascal’s triangle – the 9th horizontal row, and the 6th diagonal row : 56.3 Figure 6 & 7 below make this more evident. • Hockey Stick – if you add up all numbers starting at any of the ‘1’s in a given diagonal the sum will be equivalent to the number underneath in the opposite diagonal. • Parallelogram – take the 2nd and 3rd numbers in any row and the 2nd and 3rd numbers underneath and add them together. The sum will equal the square of the bottom left or right number - right or left depending on if the bottom numbers chosen were to the right or left of the top numbers. In addition the determinate will be equal to the bottom right number.11 Figure 5 helps identify this pattern. There are more, and the resources available on this topic are quite exhaustive. Getting a copy of the books or journal articles referenced in this paper will give you a good idea of the wealth of information available. It does make me wonder just how in depth the Chinese understood these concepts! Any one of these concepts could be explored and level of difficulty ranges from year 7 all the way to 12.

Figure 4: Fibonacci Numbers can be found adding up numbers in the displayed rows. (Golos, p. 10)

Figure 5: Various patterns can be seen here (Bell & Hyman, p. 69)

Figure 6: Diagonal Rows on Pascals Triangle (Golos, p. 8) Figure 7: Horizontal Rows on Pascals Triangle (Golos, p. 7)

ICT Activity ICT certainly has it place in various subject areas and the activity of colouring in Pascal’s triangle can be explored with ICT. The following website is an example of an interactive web version of the activity previously presented: http://www.shodor.org/interactivate/activities/ColoringRemainder/ In addition Pascal’s triangle can be used for enrichment in ICT programming lessons because there are some interesting problems that need to be solved when developing algorithmic solutions to the triangle.

References 1. Brook, T. (2010). Troubled Empire China in the Yuan and Ming Dynasties. Cambridge, MA: Harvard University Press.

2. Clarke, V. (2003). Effective strategies in the teaching of mathematics : A light from mathematics to technology. Lanham, MD: University Press of America.

3. Golos, E. (1981). Patterns in mathematics. Boston, MA: Prindle Weber & Schmidt.

4. Lay-Yong, L. (1980). The Chinese connection between the Pascal triangle and the solution of numerical equations of any degree. Historia Mathematica, 7(4), 407–424.

5. Lay-Yong, L., & Kangshen, S. (1989). Methods of solving linear equations in traditional China. Historia Mathematica, 16(2), 107–122.

6. Lemon, P. (1997). Pascal’s Triangle: Patterns, paths, and plinko. The Mathematics Teacher, 90(4), 270-273.

7. Lǐ, Y., & Dù, S. (1987). Chinese mathematics: A concise history. Oxford, UK: Clarendon.

8. Martzloff, J. (1997). A history of Chinese mathematics. Berlin: Springer.

9. Shannon, K., Bardzell, M., (2004). Patterns in Pascal's Triangle - with a Twist. Journal of Online Mathematics and its Applications, 3. Retrieved from https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a- twist-two-more-patterns

10. Straffin, P. (1998). Liu Hui and the first golden age of Chinese mathematics. Mathematics Magazine, 71(3), 163–181.

11. Topics in mathematics: Ideas for the secondary classroom. (1979). London: Bell & Hyman.

12. Usiskin, Z. (1973). Perfect square patterns in the Pascal triangle. Mathematics Magazine, 46(4), 203–208.

13. Zhu Shijie. (2014). In Encyclopaedia Britannica (Ed.), Britannica Concise Encyclopedia. Chicago, IL: Britannica Digital Learning. Retrieved from http://search.credoreference.com.ezproxy.library.uwa.edu.au/content/entry/ebconcise/zh u_shijie/0