Early Chinese and Indian mathematics made important contributions. In China, the Nine Chapters on the Mathematical Art from around 179-150 BCE included solving systems of linear equations, using algorithms to extract roots, and proving the Pythagorean theorem. Indians developed the decimal number system and concepts like zero, negative numbers, and algebra. Mathematicians like Aryabhata and Brahmagupta advanced trigonometry in India. Pingala used binary numbers and described Pascal's triangle. Madhava of Sangamagrama derived formulas for pi and trigonometric functions, anticipating calculus.

1. EARLY CHINESE AND INDIAN
MATHEMATICS
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RICLYN K. GONZAGA

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3. EARLY CHINESE MATHEMATICS
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4. Oracle bone script date back to
the Shang Dynasty 1600-1050 BC .
One of the oldest surviving
mathematical works is the Yi Jing,
which greatly influenced written
literature during the Zhou 0-256 BC.
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5. ➢ In 263 AD, Liu Hiu a
Chinese mathematician and
writer who edited and
published a book with
solution to mathematical
problems in daily life and it is
known as The Nine Chapters
on the Mathematical Art
(Jiuzhang Suanshu).
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6. The Nine Chapters on the
Mathematical Art
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7. Chapter 1. Field Measurement
This chapter includes problems to find the areas of
rectangles, triangles, trapezoids, circles, and related regions.
Arithmetic techniques are developed to carry out computations
with fractional quantities.
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8. Chapter 2. Millet and Rice
This chapter includes problems of proportions and unit
prices. The Rule of Three similar to cross multiplication for
solving proportions is seen as an extension of the work with
fractions in the first chapter.
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9. Chapter 3. Distribution by Proportion
This chapter extends the problems solved via proportion in
the previous chapter.
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10. Chapter 4. Short Width
This chapter seeks the side or diameter of a region from
known areas and volumes. Arithmetic results developed include
algorithms to extract square roots and cube roots by hand.
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11. Chapter 5. Construction Consultations
This chapter seeks the volumes of a number of solids that
occur in construction problems. Formulas for the solution of
such problems are developed, often in several different ways.
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12. Chapter 6. Fair Levies
Further extensions of the proportion problems of
Chapters 2 and 3 are developed here. There is also a
consideration of the sums of arithmetic progressions. The kinds
of word problems that appear include work problems and
distance and rate problems.
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13. Chapter 7. Excess and Deficit
The chapter title refers to a technique to solve two linear
equations in two unknowns [double false position]. Again, a
variety of word problems are solved using this technique.
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14. Chapter 8. Rectangular Arrays
Problems of agricultural yields, the sale of animals, and a
variety of other problems, all leading to systems of linear
equations, are solved by a method that is known in the West as
Gaussian elimination. The problems lead to systems ranging
from two equations in two unknowns to six equations in six
unknowns, and one problem leads to an indeterminate system
of five equations in six unknowns.
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15. Chapter 9. Right-angled Triangles
Problems in surveying and in the lengths of various line
segments are solved using the Gougu Rule, the Chinese version
of the theorem known in the West as the Pythagorean
Theorem. The problems demonstrate familiarity with
Pythagorean triples.
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16. The Chinese versions of
Pythagorean Theorem Chou Pei
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17. 206-221 BCE. it is one of the oldest recorded proofs of the
Pythagorean Theorem. The following proof is included in
the Zhou Bi Suan Jing, one of the oldest Chinese mathematical
works known to scholars. Surviving copies of this text date
back to the Han Dynasty
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Given: ABCD is a square with sides of length a+b. Another square, square EFGH, is inscribed inside ABCD. Each side of
square EFGH has a length c. There are also four right triangles inside square ABCD.
Prove: a2+b2=c2
Proof:
1.The area of square ABCD=(a+b)2
2.The area of each right triangle is 12ab
3.The area of square EFGH is c2
4.The area of square ABCD is equal to the sum of the area of square EFGH and the areas of the four right triangles:
1.area of square ABCD= area of square EFGH+4(area of AHE)
2.(a+b)2=c2+4(12ab)
5.Simplify the expression:
1.Expand:
1.(a+b)(a+b)=c2+2ab
2.a2+ab+ab+b2=c2+2ab
2.Simplify completely by subtracting out the ab terms:a2+b2=c2
Conclusion: This proof proves that a2+b2=c2 when a and b are legs of a right triangle and c is the hypotenuse.

19. The method of Gaussian elimination also known as
row reduction appears in the Chinese mathematical text
Chapter Eight: Rectangular Arrays or The Nine
Mathematical Art. The first reference to the book by
this title is dated to 179 CE, but parts of it were written
as early as approximately 150 BCE. It was commented
on by Liu Hui.
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20. Indian Mathematics
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21. Indian mathematics emerged in the Indian
subcontinent from 1200 BC until the end of the 18th century.
In the classical period of Indian mathematics 400 AD to 1200
AD, important contributions were made by scholars
like Aryabhata, Brahmagupta, Bhakara, and Varahamihira.
The decimal number system in use today was first recorded in
Indian mathematics. Indian mathematicians made early
contributions to the study of the concept of zero as a
number, negative numbers, arithmetic, and algebra. In addition,
trigonometry was further advanced in India also the modern
definitions of sine and cosine were developed there. These
mathematical concepts were transmitted to the Middle East,
China, and Europe and led to further developments that now
form the foundations of many areas of mathematics.
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22. The later Sulba-sutras represent the 'traditional' material
along with further related elaboration of Vedic mathematics.
The Sulba-sutras have been dated from around 800-200 BC,
and further to the expansion of topics in the Vedangas,
contain a number of significant developments.
These include first 'use' of irrational numbers, quadratic
equations of the form a x2 = c and ax2 + bx = c, unarguable
evidence of the use of Pythagoras theorem and Pythagorean
triples, Pythagoras BC 572 - 497 BC.
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23. Pingala was an Indian musicologist and
mathematician who used binary system also found
Pascal triangle,, This article develops the basic ideas and
how it was further developed by other Indian
mathematicians. The work also includes much
development on Fibonacci sequence and its
applications and the subject of combinatorics which
was applied to music, medicine, architecture and other
fields.
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24. 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
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And (without the leading 0s) we have the first 16 binary numbers:
Binary: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
Decimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
This is useful! To remember the sequence of binary numbers just think:
•"0" and "1" {0,1}
•then repeat "0" and "1" again but with a "1" in front: {0,1,10,11}
•then repeat those with a "1" in front: {0,1,10,11,100,101,110,111}
•and so on!
At each stage we repeat everything we have so far, but with a 1 in front.

25. Pascal Triangle
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26. Madhava of Sangamagrama
• An Indian mathematician and astronomer .
• He is considered the founder of the Kerala School of
Astronomy and Mathematics.
• Use of infinite series of fractions to give an exact
formula for π, sine formula and other trigonometric
functions, important step towards development of
calculus.
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28. Source
MacTutor History of Mathematics Archives
YouTube
https://www-history.mcs.st-
andews.ac.uk/HistTopics/Chinese_overview.html
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