Squared triangular number

Sridhar Acharya (c. 870, India – c. 930 India) was an Indian mathematician, Sanskrit pundit and philosopher. He was born in Bhurishresti (Bhurisristi or Bhurshut) village in South Radha (at present Hughli) in the 10th Century AD. His father's name was Baladev Acharya and mother's name was Acchoka. His father was also a Sanskrit pundit.

Works

He was known for two treatises: Trisatika (sometimes called the Patiganitasara) and the Patiganita. His major work Patiganitasara was named Trisatika because it was written in three hundred slokas. The book discusses counting of numbers, measures, natural number, multiplication, division, zero, squares, cubes, fraction, rule of three, interest-calculation, joint business or partnership and mensuration.

• He gave an exposition on zero. He has written, "If 0(zero) is added to any number,the sum is the same number; If 0(zero) is subtracted from any number,the number remains unchanged; If 0(zero) is multiplied by any number,the product is 0(zero)". He has said nothing about division of any number by 0(zero).
• In the case of dividing a fraction he has found out the method of multiplying the fraction by the reciprocal of the divisor.
• He wrote on practical applications of algebra separated algebra from arithmetic
• He was one of the first to give a formula for solving quadratic equations.
• He found the formula :-

$ax^{2}+bx+c=0$
$4a^{2}x^{2}+4abx+4ac=0$ (Multiply by 4a)

Biography

Sridhara is now believed to have lived in the ninth and tenth centuries. However, there has been much dispute over his date and in different works the dates of the life of Sridhara have been placed from the seventh century to the eleventh century. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of mathematics he was familiar with and also seeing which later mathematicians were familiar with his work. Some historians give Bengal as the place of his birth while other historians believe that Sridhara was born in southern India.

Sridhara is known as the author of two mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Information about these books was given the works of Bhaskara II (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493).

K.S. Shukla examined Sridhara's method for finding rational solutions of $Nx^{2}\pm \ 1=y^{2}$, $1-Nx^{2}=y^{2}$, $Nx^{2}\pm \ C=y^{2}$, $C-Nx^{2}=y^{2}$which Sridhara gives in the Patiganita. Shukla states that the rules given there are different from those given by other Hindu mathematicians.

Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Unfortunately, as indicated above, the original is lost and we have to rely on a quotation of Sridhara's rule from Bhaskara II:-

Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.

Proof of the Sridhar Acharya Formula,

let us consider,

$ax^{2}+bx+c=0$

• Multipling both sides by 4a,

$4a^{2}x^{2}+4abx+4ac=0$

• Substracting $4ac$ from both sides,

$4a^{2}x^{2}+4abx=-4ac$

• Then adding $b^{2}$ to both sides,

$4a^{2}x^{2}+4abx+b^{2}=-4ac+b^{2}$

• We know that,

$(a+b)^{2}=a^{2}+2ab+b^{2}$

• Using it in the equation,

$(2ax+b)^{2}=b^{2}-4ac$

• Taking square roots,

$2ax+b=\pm {\sqrt {b^{2}-4ac}}$

$2ax=-b\pm {\sqrt {b^{2}-4ac}}$

• Hence, dividing by $2a$ get

$x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$

In this way, he found the proof of 2 roots.

References

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