Algorithms of division of
fractions in Chinese ancient
literature
Xuhua Sun xhsun@umac.mo
Faculty of Education, University
of Macau
自我介紹
• 1991.6 东北师范大学数学专业本科毕业，
获理学学士学位/硕士学位
• 1994. 9- 2003，广州师范学院 \广州大学副
教授
• 2003，1—2006，12 香港中文大学教育学
院 博士
• 2006-2007 香港教育学院 研究员
• 2008 –now 澳门大学 助理教授
Does calculation of fraction division
mean the procedure of “flip and
multiply”?
• Is historical research a powerful tool in
mathematics teaching.
Who could explain why the
algorithm of “flip and multiply” work
well ?
3 3 3 8
    2
4 8 4 3
• The six US textbooks provided the idea of
reciprocal and “flip-and-multiply”
algorithms for fraction division only , but
did not explain why the procedure works
(Li 2009).
• American pre-service and in-service
teachers experienced difficulty in making a
conceptual explanation for the underlying
principle and meaning of “flip and
multiply”(e.g., Ball, 1990; Ma, 1999; Li,
2008; Li & Kulm, 2008).
A typical mistake was: “flip both
dividend and divisor”.
Background
• My PHD thesis
• Division of fractions is often considered
the most mechanical and least understood
topic in schools.
• A range of studies have repeatedly shown
that both teachers and students have
difficulty in making a conceptual
explanation for the traditional algorithm of
“flip and multiply”.
Two ways
“flip and
multiply”
reducing
to same
denominat
or
3 3
3 8
    2
4 3
4 8
3 3 3 2 3 6 3
    63 2
 
4 8 4 2 8 8 8
Suan Shu Shu
• Suan Shu Shu 《算数书》was the oldest
book on mathematics.
• It was written from 202 B.C. to 186 B.C.,
or more than 300 years earlier than Jiu
Zhang Sun Shu《九章算术》.
Algorithm of Division of
Fractions in Suan Shu Shu
• 启从。广7分步之3，求田4分步之2,为从
（纵）几何？其从（纵）1步6分步之1。
• 求从（纵）术：广分子乘积分母为法，积
分子乘广分母为实，实如法一。
Procedure
2 3 2 7 14
1
   
1
4 7 4 3 12
6
• The another problem and procedure is as
follows:
1 1
23
(3   )  5  (18  3  2)  (5  6) 
2 3
30
.
The underlying rationale of “flip and
multiply” algorithm
• “Deleting denominator” possibly indicates
an underlying principle to solve problems
of fraction.
• The divisor and dividend become integers
after being multiplied by the same number
and the quotient is not changed
• This underlying principle should help to
understanding the reason .
Nine chapters
Algorithm of Division of
Fractions in Nine chapter
• 经分，又有三人，三分人之一，分六钱三
分钱之一，四分钱之三。问人得几何？
答曰：人得二钱、八分钱之一。经分术曰：
以人数为法，钱数为实，实如法而一。有
分者通之，重有分者同而通之.
Reducing to same denominator
1 3
1 19 3 10 85 40 85
1
(6  )  3  (  ) 



2
3 4
3
3 4
3 12 12 40
8
Besides, Liu Hui added a new
solution, namely, flip multiplication
(See Bai, 1983).
1 3
1 19 3 10 85 10 85 3
1
(6  )  3  (  ) 




2
3 4
3
3 4
3 12 3 12 10
8
Algorithms of Division of
Fraction in Shu Li Jing Yun (《数
理精蕴》)
• Shuli jingyun directly illustrated
• the algorithm of “numerator dividing
numerator and denominator dividing
denominator.
• .
2 1

9 3
=
.
=
2 1 2

93 3
• “Reductions of fractions to the same
denominator” was also regarded as that
the only algorithm illustrating the principle
of fraction division, namely, two
numerators are in proportion (the same
proportionate fraction unit is deleted).
“Reducing to same denominator”
• This algorithm makes up some weakness
of “flip and multiply”.
• Link to experiences of whole-number
division and that of fraction addition /
subtraction.
• Procedure of “flip and multiply” is based on
rod numerals system
Greece -- Geometry/ Euclid
Theorem-based tradition
Algebra
Analytic geometry Calculus
Two traditions of mathematics and Arithmetic
Euclid’s
Elements is
printed [1482]
1200
1300
1400
1500
1600
1700
1800
1900
2000
Arithmetic
1677 E. Cocker
Babylonian, Egyptian, Chinese, Indian…
Problem-based tradition
1738 Euler:
Introduction to the Art
of Arithmetic
[1202] Fibonacci’s Liber Abaci (The
Book of Calculation) first appears
• Thank you!