Anurupyena

The upa-Sutra 'anurupyena' means 'proportionality'. This Sutra is highly useful
to find products of two numbers when both of them are near the Common bases
i.e powers of base 10 . It is very clear that in such cases the expected
'Simplicity ' in doing problems is absent.

Example 1: 46 X 43
As per the previous methods, if we select 100 as base we get
46 -54 This is much more difficult and of no use.
43 -57
‾‾‾‾‾‾‾‾
76

Now by ‘anurupyena’ we consider a working base In three ways. We can solve
the problem.
Method 1: Take the nearest higher multiple of 10. In this case it is 50.
Treat it as 100 / 2 = 50. Now the steps are as follows:

i) Choose the working base near to the numbers under consideration.
i.e., working base is 100 / 2 = 50

ii) Write the numbers one below the other
i.e. 4 6
4 3
‾‾‾‾‾‾‾
iii) Write the differences of the two numbers respectively from 50 against each
number on right side
i.e. 46 -04
43 -07
‾‾‾‾‾‾‾‾‾
iv) Write cross-subtraction or cross- addition as the case may be under the line
drawn.

v) Multiply the differences and write the product in the left side of the answer.
46 -04
43 -07
____________
39 / -4 x –7
= 28

vi) Since base is 100 / 2 = 50 , 39 in the answer represents 39X50.
Hence divide 39 by 2 because 50 = 100 / 2
77

Thus 39 ÷ 2 gives 19½ where 19 is quotient and 1 is remainder . This 1 as
Reminder gives one 50 making the L.H.S of the answer 28 + 50 = 78(or
Remainder ½ x 100 + 28 )
i.e. R.H.S 19 and L.H.S 78 together give the answer1978 We represent it as
46 -04
43 -07
‾‾‾‾‾‾‾‾‾
2) 39 / 28
‾‾‾‾‾‾‾‾‾
19½ / 28
= 19 / 78 = 1978

Ekanyunena Purvena

The Sutra Ekanyunena purvena comes as a Sub-sutra to Nikhilam which gives

the meaning 'One less than the previous' or 'One less than the one before'.

1) The use of this sutra in case of multiplication by 9,99,999.. is as follows .

Method :

a) The left hand side digit (digits) is ( are) obtained by applying the ekanyunena

purvena i.e. by deduction 1 from the left side digit (digits) .

e.g. ( i ) 7 x 9; 7 – 1 = 6 ( L.H.S. digit )

b) The right hand side digit is the complement or difference between the

multiplier and the left hand side digit (digits) . i.e. 7 X 9 R.H.S is 9 - 6 = 3.

c) The two numbers give the answer; i.e. 7 X 9 = 63.

Puranapuranabhyam

The Sutra can be taken as Purana - Apuranabhyam which means by the

completion or non - completion. Purana is well known in the present

system. We can see its application in solving the roots for general form of

quadratic equation.

We have : ax2 + bx + c = 0

x2 + (b/a)x + c/a = 0 ( dividing by a )

x2 + (b/a)x = - c/a

completing the square ( i.e.,, purana ) on the L.H.S.

x2 + (b/a)x + (b2/4a2) = -c/a + (b2/4a2)

[x + (b/2a)]2 = (b2 - 4ac) / 4a2
________

- b ± √ b2 – 4ac

Proceeding in this way we finally get x = _______________

2a