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 =  This is how the setup looks

The Multiplication Algorithm 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 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 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 Now add the three numbers below the line

The Multiplication Algorithm 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 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

Cyclic Quadrilateral

Cyclic Quadrilaterals  He developed many rules for cyclic quadrilaterals.  His most famous being 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.