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 x 2 - Dy 2 = ±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.