Name some famous Ancient Indian Mathematicians?What are their contributions to mathematics?
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The contributions of the Indian mathematicians start from the Indus valley civilization and the Vedas, up to the modern times. Indian mathematicians have made a number of important contributions to mathematics including placevalue arithmetical notation and the concept of zero.
During the Vedic Age, the noted Indian mathematicians include the names of Apastamba, Baudhayana, Katyayana, Manava, Panini and Yajnavalkya, who was credited with authorship of the Shatapatha Brahmana, which contains calculations related to altar construction. Baudhayana, the mathematician and also a priest was noted as the author of the earliest Sulba Sutra, appendices to the Vedas giving rules for the construction of altars  called the Baudhayana Sulbasutra. This book contains several important mathematical results. The Dharmasutra of Apastamba forms a part of the larger Kalpas?tra of ?pastamba, containing almost thirty prasnas, which literally means `questions` or books. The subjects of this Dharmasutra are well organized and conserved in good condition. These prasanas comprise of the ?rautasutra followed by Mantrapadha which is used in domestic rituals and is a collection of ritual formulas.
Aryabhatta is the first in the line of great Indian mathematiciansastronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya and AryaSiddhanta. Bhaskara was one of the 7th century Indian mathematicians, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and noteworthy rational approximation of the sine function in his interpretation on Aryabhata`s work. Jayadeva was a ninthcentury Indian mathematician, who further worked on the cyclic method (chakravala method) that was called by Hermann Hankel the premium success achieved in the theory of numbers before Lagrange (18th century). He also made momentous contributions to combinatorics.
Gopala, one of the noted Indian mathematicians studied the Fibonacci numbers in 1135, more than half a century before Fibonacci popularised these numbers in Europe. Pingala was an ancient Indian writer, famous for his work, the Chandas Shastra, a Sanskrit treatise on prosody considered one of the Vedanga and he developed advanced mathematical concepts for describing the patterns of prosody. Some of the famous Indian mathematicians of the later age are Srinivasa Ramanujan, A. A. Krishnaswami Ayyangar, Prasanta Chandra Mahalanobis, D. K. RayChaudhuri, HarishChandra, Calyampudi Radhakrishna Rao, Shreeram Shankar Abhyankar, Ramdas Lotu Bhirud, Jayant Narlikar and many others.
During the Vedic Age, the noted Indian mathematicians include the names of Apastamba, Baudhayana, Katyayana, Manava, Panini and Yajnavalkya, who was credited with authorship of the Shatapatha Brahmana, which contains calculations related to altar construction. Baudhayana, the mathematician and also a priest was noted as the author of the earliest Sulba Sutra, appendices to the Vedas giving rules for the construction of altars  called the Baudhayana Sulbasutra. This book contains several important mathematical results. The Dharmasutra of Apastamba forms a part of the larger Kalpas?tra of ?pastamba, containing almost thirty prasnas, which literally means `questions` or books. The subjects of this Dharmasutra are well organized and conserved in good condition. These prasanas comprise of the ?rautasutra followed by Mantrapadha which is used in domestic rituals and is a collection of ritual formulas.
Aryabhatta is the first in the line of great Indian mathematiciansastronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya and AryaSiddhanta. Bhaskara was one of the 7th century Indian mathematicians, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and noteworthy rational approximation of the sine function in his interpretation on Aryabhata`s work. Jayadeva was a ninthcentury Indian mathematician, who further worked on the cyclic method (chakravala method) that was called by Hermann Hankel the premium success achieved in the theory of numbers before Lagrange (18th century). He also made momentous contributions to combinatorics.
Gopala, one of the noted Indian mathematicians studied the Fibonacci numbers in 1135, more than half a century before Fibonacci popularised these numbers in Europe. Pingala was an ancient Indian writer, famous for his work, the Chandas Shastra, a Sanskrit treatise on prosody considered one of the Vedanga and he developed advanced mathematical concepts for describing the patterns of prosody. Some of the famous Indian mathematicians of the later age are Srinivasa Ramanujan, A. A. Krishnaswami Ayyangar, Prasanta Chandra Mahalanobis, D. K. RayChaudhuri, HarishChandra, Calyampudi Radhakrishna Rao, Shreeram Shankar Abhyankar, Ramdas Lotu Bhirud, Jayant Narlikar and many others.
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Other Answers (3)
Rated Highest
1. Bhaskara
2. Lilavati (Bhaskara's daughter)
3. Varaha mihira
4. Aryabhata
5. Brahma gupta
I use here the Sanskrit phonetic practice of ending names with a vowel and not the HindiUrdu (Hindustani) rule and that is why every name ends with a 'a'.
Many things in Mathematics with which we are familiar now are attributed to these (and some more). The format, notation and presentation of today are entirely different from those in ancient times. So, to say with certainity 'Yes! this is his invention', is difficult. Even though foundation for 'Calculus' was laid by both Newton and 'Leibnitz'; Newton's notation is cumbersome and we follow Leibnitz's. Almost all ancient Indian mathematicians were concerned with astronomy and solving astronomical problems. This is so because unlike today, those people used to look at the sky daily (often) and unwittingly keep track of happenings there (primary thing like moon's phases). Much of this knowledge is with us for calculations (algorithms) in Panchangam, the proof of which is the accuracy of predictions of eclipses. For instance Aryabhata devised a notation for Trigonometrical functions, not in the way we write like 'Sine', 'Cosine' but 'Versign' which simply is 'arc sine' of which the angle is.
Roughly their contributions are in Algebra (Beeja ganitham), Pythagous' proof of a rightangled triangle, solution of quadratic equation, continued fractions, number theory since 'zero' (0) was formulated and used by them much before Romans were grappling with Roman numerals and representation of numbers with them, elemnts of Trigonometry. I suggest you to go to 'Wikipedia' now, armed with the above names. 
aryabhatta

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