Transcript MA 330 Project Presentation

MA 330 Project Presentation
“ The Mathematical Genius of
Brahmagupta
Nick Moon
March 23, 2005
Outline
 A brief overview of Brahmagupta’s life
 Brahmagupta’s multiplication algorithm
 Other Areas of Study
 Conclusion
History of Brahmagupta
 Brahmagupta was born in 598 A.D. in
northwest India
 He likely lived most of his life in
Bhillamala(this is most likely modern
Bhinmal )
 He belonged to the Ujjain school
History of Brahmagupta
 Brahmagupta wrote his most famous work
Brahma Sphuta Siddhanta at ≈ age 30
 This is translated “The Opening of the Universe”
~disputed translation
 Brahmagupta wrote another work, Khand
Khadyak in ≈ 665
 He is said to have died at age 70 ≈ 668
The Multiplication Algorithm
 This is called gomutrika
 This is translated as “like the trajectory of a cow's
urine ”
 Let’s take 123*223 = 27429
 This is how the setup looks
1 223
2 223
3 223
----------
The Multiplication Algorithm
1 223
2 223
3 223
----------
• Now multiply the 223 of the top row by the 1
in the left column. Start the process by 1*2=2,
now put the 2 below the two of the last line. If
there were something to carry, this would be
done in the usual way.
The Multiplication Algorithm
1 223
2 223
3 223
---------223
• Now multiply the 223 of the second row by the 2
in the left column writing the number in the line
below the 223 but moved one place to the right.
The Multiplication Algorithm
1 223
2 223
3 223
---------223
446
• Now multiply the 223 of the third row by the 3 in
the left column writing the number in the line
below the 446 but moved one place to the right
The Multiplication Algorithm
1 223
2 223
3 223
---------223
446
669
• Now add the three numbers below the line
The Multiplication Algorithm
1 223
2 223
3 223
---------223
446
669
---------27429
• This is the exact number that we wanted to
calculate.
Other Areas of Interest
 He developed the algorithm we have seen
in class to solve x2 - Dy2 = ±1
 It is believed that he did work with
continued fractions to solve ax – by = c
 He also did a lot of work with astronomy
Cyclic Quadrilateral
Cyclic Quadrilaterals
 He developed many rules for cyclic
quadrilaterals.
 His most famous being
Area = (s - a)(s - b)(s - c)(s - d)
where 2s = a+b+c+d
Conclusions
 Brahmagupta’s strategies for solving equations
were astounding
 Brahmagupta greatly influenced the world of
Indian Mathematics
 Bhaskara II was greatly influenced by
Brahmagupta's work and gave Brahmagupta the
title Ganaka Chakra Chudamani, the gem of the
circle of mathematicians.