________________
[2.2] In the second section, we shall show that the Jaina school of Indian mathematics had
[a] an idea of evaluation of the nth power of a number and
[b] an exclusive idea of the act of lifting a number to its own power.
[2.2.1] The Method for evaluating a"
In context of finding the sum of a geometric progression, Mahavira (c. 850 A.D.) gives a method for evaluating a", when n is any positive integer, as follows:
[a] If n is even, it is divided by 2 and a zero is put is a separate column, and if n is odd, unity is subtracted from it and unity is put in the column. The process is repeated till zero is obtained in the end.
[b] In the column, the lowest term is always unity. It is multiplied by a and we get a. If there is zero above unity, this a is squared and if there is another unity, it is multiplied by a again. The process is continued till the highest term is disposed.
In this way, a" is evaluated.
If n
=
12, we shall have
Jain Education International
12 even
1226 even
6 ÷ 2 = 3 odd
312 even
2 2 1 odd
1-1=0
0
0
1
0
1
(a) = a12
12
(a3)" = a6
(a2). a = a3
(a)2 a2
=
the end.
This is what is known as the multiplication-square (gunana-vargaja) process. It was known to Pirigala (c. 200 B.C.) long before Mahavira (c. 850 A.D.) and Sridhara (c. 799 A.D.) and had been used by him in his Chandah-sutra (Rules of Meters) 5 for finding 2".
[2.2.2] Vargana-samavargana
1.a = a
The ancient philosopher Jainas have discussed cosmological system and
57
Arhat Vacana, 15 (4), 2003
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