1. MA 330 Project Presentation“ The Mathematical Genius of Brahmagupta Nick Moon March 23, 2005

2. Outline • A brief overview of Brahmagupta’s life • Brahmagupta’s multiplication algorithm • Other Areas of Study • Conclusion

3. 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

5. 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

6. 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 ----------

7. 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.

8. 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.

9. 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

10. The Multiplication Algorithm 1 223 2 223 3 223 ---------- 223 446 669 • Now add the three numbers below the line

11. The Multiplication Algorithm 1 223 2 223 3 223 ---------- 223 446 669 ---------- 27429 • This is the exact number that we wanted to calculate.

12. 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

13. Cyclic Quadrilateral

14. Cyclic Quadrilaterals • He developed many rules for cyclic quadrilaterals. • His most famous being where 2s = a+b+c+d

15. 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.