Mathematics
The technique of algebra and concept of zero originated in India.
The earliest book on Mathematics was written by Sulvasutra in 6th Century BC. There is a mention of Pi and concepts similar to Pythagorus theorem in the book.
Apastamba introduced the concept of right angles, acute angles and obtuse angles. This knowledge was needed in the construction of fire altars.
Aryabhatta wrote about his book Aryabhattiya in which methods of denoting big decimal numbers by alphabets was described. Aryabhatta also focused on astronomy. Aryabhatta also gave the formula for Pi (more accurately than Greeks), he formulated area of a triangle and discovered algebra.
Other contributions of Aryabhatta was that he proposed that earth is round and rotates around its axis. He also gave accurate reasons for occurrence of solar and lunar eclipse.
Brahmagupta in the 7th Century AD mentioned zero for the first time in his book. He also described negative and positive numbers.
Mahavircharya described the current methods for finding lowest common multiple.
Bhaskaracharya another important mathematician described the cyclic method for solving equations.
Atharva veda describes diseases and cures. However it mentions diseases being caused by demons that enter the body and cures are done by sue of magic spells.
However for rational and practical medicine, takshila and varanasi emerged as important centers around 600 BC.
Charak wrote charak samhita which deals with the use of plants and herbs for medicinal purposes. It deals with medicine as a science.
Susrut wrote Susrut samhita that dealt with practical problems of surgery and obsterics. Susruta studied anatomy with great detail with the use of dead body. Both these books were translated into arabic. Buddhist monks took the knowledge from India to China and Tibet too.
Unani system of medicine came to India from greece with the book Firdausu Hikmat written by Ali bin Rabban.
It is a system for organising days for social, religious, commercial or administrative purposes.
In India we have mainly three types of calendars across various regions :
Solar year : The solar year is 365 days 5 hours 48 mins and 46 seconds long. It has 12 months ad maintains closest correspondence between the year and the seasons.
Lunar year : This also has 12 months but each month is 29.6 days to 29.25 days long. This is because it is calculated as the time taken between two full moons or new moons. Thus it has 354 days. The shortage of 11 days every year is filled by an additional month every 2.5 years known as "Adhik masa".
Luni solar year : Here the year is calculated by the solar cycle and the month by the lunar division.
Vikram Samvat :
This era was started by King Vikramaditya and is 56 years before the christian era.
It is a lunar calendar.
There are 354 days and 12 months. the start of the new year is with the full moon in Chaitra month which falls in March - April.
An extra month is added every 3rd year and 5th year known as Adhik Masa. This counts for the deficiency in 11 days each year.
Saka era :
The zero year is 78 AD.
This calendar also has 12 months and 365 days. The months are named as Chaitya, vaishakh, jhestha, ashadh, shravan, bhadrapad, ashwin, karthik, margashirsha, pausha, magha and phalgun. The names are same as the names in the vikram samvat calendar.
Hijri calendar : There are 12 months with 354 days. It is a lunar calendar. The start of the year is 622 AD which is the year Prophet Muhammad went to Mecca from Medina. No adjustments are made for shortfall of 11 days every year and so hijri (islamic calendar) falls short of 1 year for every 30 years of the Gregorian calendar.
Gregorian calendar : The calendar is based on the birthday of the founder of Christianity Jesus Christ. It is a solar year with 365 days 5 hrs and 48 mins and 46 seconds. To compensate for the additional time every four years we have a leap year where the month of February has 29 days.
Saka calendar is the National calendar of India.
It was adopted on recommendations of the calendar Reforms committee in 1957.
30 different kinds of calendars were used in India during 1957 and the Saka calendar was adopted to synchronise the local calendars and ensure uniformity.