Author: Suresh Soni

Publication: Organiser

Date: August 26, 2007

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's Leelavati 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's feet. 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.)