As per ancient Mathematical records, the first woman Mathematician is Hypatia who was also a Greek Philosopher .She was a famous teacher and a public speaker, who also taught her students to make astrolabes and hydrometers.
Her own accounts are recorded on Apollonius work conics (geometry) and Diophantus Arithmetica (number theory) are really amazing at that time
Now the first woman Indian Mathematician is Shakuntala Devi.
Also known as the ‘Human Calculator’ or ‘Human Computer’ , she demonstrated the multiplication of two 13 digit number which are 7,686,369,774,870 and 2,465,099,745,779 on an event held on 18th June 1980. These numbers were picked randomly by the computer department of Imperial Collage London where she answered the question in 28 seconds and that is 18,947,668,177,995,426,462,773,730.
She also earns a place in 1982 edition of the prestigious Guinness Book of World Records 2 year later based on the above mentioned event. Shakuntala Devi wrote a number of books which includes novels as well as texts about Mathematics, puzzles and astrology.
The 23rd root of a 201 digit number was given by her in 50 seconds which is 546,372,891. This answer was proven correct by the UNIVAC 1101 computer at US Bureau of Standards. They had to write a special program in order to incorporate such large calculation.