Adyamadyena-Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)

Adyamadyena-Antyamantyena (आद्यमाद्येन अन्त्यमन्त्येन)

The Sutra “Adyamadyena-Antyamantyena” means “the first by the first and the last by the last”.

प्रथम को प्रथम के द्वारा तथा अन्तिम को अन्तिम के द्वारा।

Area of rectangle:

Example:

Find out the area of a rectangle whose length and breadth are respectively 5 ft.2 inches and 4 ft.5 inches.

Generally we continue the problem like this.

Area    = Length X Breadth

= 5’ 2" X 4’ 5"               (Since 1’ = 12")

= (5 X 12 + 2) (4 X 12 + 5) conversion in to single unit

= 62" X 53" = 3286 Sq. inches.

Since 1 sq. ft. =12 X 12 = 144 sq.inches

We have area

3286 /144 = Quotient is 22 and Remainder is 118.

Area of rectangle is 22 Sq. ft 118 Sq. inches.

Mental argumentation:

It is interesting to know the mental argumentation. It goes in his mind like this

    5’     2"

    4’     5"

First by first: 5’ X 4’ = 20 sq. ft.

Last by last: 2" X 5" = 10 sq. in.

Now cross wise 5 X 5 + 4 x 2 = 25 +8 = 33.

Adjust units  to left as  33 = 2 X 12 +9 , 2 twelve's as 2 square feet make the first 20+2 = 22 sq. ft ; 9 left becomes 9 x 12 square inches and go towards right  9 x 12 = 108 sq. in. gives 108+10= 118 sq.inch.

We got area in some sort of 22 sq ft and 128 sq. inches.

By Vedic principles "the first by first and the last by last"

5’ 2" can be treated as 5a + 2 and 4’ 5" as 4a + 5,

Where a= 1ft. = 12 inch and a² = 1 sq. ft = 144 sq. inch.

= (5a + 2) (4a + 5)

= 20a² + 25a + 8a + 10

= 20a² + 33a + 10

= 20a² + (24a+9a) + 10

= 20a²+ (2a+9) a + 10     writing 33 = 2X12 +9

= 22a²+ 9a + 10

= 22 sq. ft. + 9X12 sq. inch + 10 sq. inches

= 22 sq. ft. + 108 sq. inch + 10 sq. inches

= 22 sq. ft. + 118 sq. inch

Factorization of quadratics:

By Vedic process two sub-sutras are used to factorizing a quadratic.

(a) Anurupyena   (b) Adyamadyena-Antyamantyena

The usual procedure of factorizing a quadratic is as follows:

= 2 a² + 9a + 10

= 2 a² + 4a + 5a + 10

= 2a (a + 2) + 5 (a + 2)

= (a + 2) (2a + 5)

But by mental process, we can get the result immediately. The steps are as follows.

1.      Split the middle coefficient in to two such parts that the ratio of the first coefficient to the first part is the same as the ratio of the second part to the last coefficient. Thus we split the coefficient of middle term of 2 a² + 9a + 10 i.e. 9 in to two such parts 4 and 5 such that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2:4 and the ratio of the second pat to the last coefficient, i.e. 5: 10 are the same. It is clear that 2:4 = 5:10. Hence such split is valid. Now the ratio 2: 4 = 5: 10 = 1:2 give one factor (a+2). 

2.      Second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of the factor. i.e. the second factor is

2 a² + 9a + 10 = (a + 2) (2a + 5)