Later on, many mathematicians like Aryabhatta, Bhaskaracharya, Shridhar, etc. were seen in the country. Of them Bhaskara-charya wrote Siddhanth Shiromani in 1150. This great book has four parts: (1) Leelavati, (2) Algebra, (3) Goladhyaya, and (4) Graha Ganit.

In his book Bhaskaracharya, Shri Gunakar Muley writes that Bhaskaracharya has acknowledged the basic eight works of mathematics:

1. Addition

2. Subtraction

3. Multiplication

4. Division

5. Square

6. Square Root

7. Cube

8. Cube root.

All these mathematical calculations were prevalent in India for thousands of years. However, Bhaskaracharya tells Leelavati a strange thing, ?At the root of all these calculations there are only two basic calculations?rise and fall or increase and decrease. Addition is increasing and subtraction is decreasing. The entire mathematics permeates from these two basic acts.?

These days, the computer solves the biggest and the most difficult calculations in a short time. All calculations are made with only two signs of addition and subtraction (+ and -). These are turned into electric signals i.e., the positive flow for addition and the reverse flow for subtraction. With this, calculations can be made at the speed of lightening. We understand increase, decrease, one and zero today but Bhaskaracharya had the basic knowledge at that time.

These days, mathematics is considered a dry subject. But Bhaskaracharya'sLeelavati is an example of how it can be taught with fun by intermixing it with entertainment, curiosity, etc. Let us see an example from Leelavati:

?From a bunch of pure lotuses? 1/3? 1/5? and 1/6? parts were used for the worship of Shiv, Vishnu and Durga respectively; 1/4 was used to worship Parvati and the 6 that were left, were used for the worship of the guru'sfeet. Now, Leelavati, quickly tell me how many lotus flowers were there in the bunch?? The answer is 120 flowers.

Explaining the square and the cube, Bhaskaracharya says, ?Leelavati, a square shape and its area are called the square. The multiplication of two equal numbers is also called a square. Similarly, the multiplication of three equal numbers is called cube and a solid with 12 compartments and equal arms is also called a cube.?

?Mool? or root in Sanskrit, meant the root of a tree or plant or, in a more expansive form, it means cause of something or origin also. Hence, in ancient mathematics, square root meant the reason for the square or origin that is one arm of a square. Likewise, we can understand the meaning of cube root in the same way. A number of ways were prevalent to find out the cube root and the square root.

In the same way, Bhaskaracharya mentions the trairashik. It had trinomial sums hence its name. For example, if one gets ?pr? (as in pramaan) in ?ph? (as in phal), then what will we find in ?i? (as in ichchha, that is desire)?

In trairaashik[trinomial] questions, the phal number should be multiplied by the ichchha number and the result should be divided by the pramaan number. What is thus acquired is the itccha phal (desired fruit). About 2000 years ago, the trairaashik [trinomial] rule was discovered in India. It reached the Arab countries in the 8th century AD. The Arab mathematicians called the trairaashik ?free raashikaat al hind?. Later, it spread to Europe where it was given the title of the Golden Rule.

The ancient mathematicians had knowledge not only of the trairaashik but also of the pancharaashik, saptaraashik and the navaraashik. (penta-or quinqnomial, hepta or septunomial and nonomial)

**Algebra**

India is the birthplace of algebra. It was called indistinct or cryptic mathematics. The Arab scholar Moosa-Al-Khawarizmi came to India in the 9th century to learn this and wrote a book called Alijeb Oyal Muquabila. Thence, this knowledge went to Europe.

In ancient times, mathematicians such as Aapastamba, Bodhaayan, Katyaayan and later, Brahmagupta and Bhaskaracharya worked on algebra.

Bhaskaracharya says that algebra means unexpressed mathematics but the initial reason is expressed. Hence, in Leelavati, arithmetic, which has expressed mathematics, was discussed at the start.

In algebra, Bhaskaracharya talks about zero and infinity.

Vadha au viyat khan khenadhaate, khaharo bhavet khen bhaktashch raashih

This means that if zero is divided by any number or multiplied by any number, the result is zero. If any number is divided by zero, the result is infinity.

Zero and infinity are the two precious jewels of mathematics. Life can continue without jewels but mathematics is nothing without zero and infinity.

Zero and infinity have no place or name in the physical world and are only the creations of the human mind. Yet, through the medium of mathematics and science, they clarify even the most difficult mysteries of the world.

Brahmagupta discovered various equations. He gave them the names of ek varna, anek varna, madhyamaharan and mapit. There is one unknown number in an ek varna equation and many unknown numbers in anek varna.

*(This book is available with Ocean Books (P) Ltd, 4/19 Asaf Ali Road, New Delhi-110 002.)*

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