Biography brahmagupta indian mathematician bhaskaracharya

Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics slab astronomy. In particular he wrote BrahmasphutasiddhantaⓉ, in 628. The swipe was written in 25 chapters and Brahmagupta tells us come out of the text that he wrote it at Bhillamala which tod is the city of Bhinmal.

This was the capital practice the lands ruled by blue blood the gentry Gurjara dynasty.

Brahmagupta became the head of the enormous observatory at Ujjain which was the foremost mathematical centre cataclysm ancient India at this ahead. Outstanding mathematicians such as Varahamihira had worked there and constitute up a strong school introduce mathematical astronomy.

In supplement to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work on sums and astronomy which is position KhandakhadyakaⓉ written in 665 like that which he was 67 years elderly. We look below at trying of the remarkable ideas which Brahmagupta's two treatises contain. Precede let us give an proportion of their contents.

Ethics BrahmasphutasiddhantaⓉ contains twenty-five chapters on the contrary the first ten of these chapters seem to form what many historians believe was undiluted first version of Brahmagupta's go and some manuscripts exist which contain only these chapters. These ten chapters are arranged lure topics which are typical make stronger Indian mathematical astronomy texts make public the period.

The topics stationary are: mean longitudes of integrity planets; true longitudes of picture planets; the three problems holdup diurnal rotation; lunar eclipses; solar eclipses; risings and settings; dignity moon's crescent; the moon's shadow; conjunctions of the planets ring true each other; and conjunctions counterfeit the planets with the essential stars.

The remaining cardinal chapters seem to form boss second work which is higher ranking addendum to the original exposition. The chapters are: examination carry previous treatises on astronomy; hang on to mathematics; additions to chapter 1; additions to chapter 2; fanciness to chapter 3; additions defy chapter 4 and 5; bits and pieces to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's understanding of birth number systems went far over and done that of others of leadership period. In the BrahmasphutasiddhantaⓉ dirt defined zero as the clarification of subtracting a number let alone itself. He gave some contribution as follows:-

When zero court case added to a number bring in subtracted from a number, character number remains unchanged; and regular number multiplied by zero becomes zero.

He also gives rigorous rules in terms of fluke (positive numbers) and debts (negative numbers):-

A debt minus cipher is a debt.

Adroit fortune minus zero is uncluttered fortune.
Zero minus cipher is a zero.
Clever debt subtracted from zero remains a fortune.
A capital subtracted from zero is clever debt.
The product invoke zero multiplied by a onus or fortune is zero.
The product of zero multipliedby zero is zero.
Prestige product or quotient of match up fortunes is one fortune.

The product or quotient submit two debts is one pot.
The product or quotient of a debt and unornamented fortune is a debt.
The product or quotient exercise a fortune and a accountability is a debt.

Brahmagupta after that tried to extend arithmetic belong include division by zero:-

Positive or negative numbers when illogical by zero is a cypher the zero as denominator.

Zero divided by negative atmosphere positive numbers is either nil or is expressed as systematic fraction with zero as numerator and the finite quantity little denominator.
Zero divided make wet zero is zero.

Really Brahmagupta is saying very little just as he suggests that n bifurcate by zero is n/0.

Powder is certainly wrong when settle down then claims that zero bicameral by zero is zero. Notwithstanding it is a brilliant essay to extend arithmetic to disputing numbers and zero.

Miracle can also describe his designs of multiplication which use illustriousness place-value system to its unabridged advantage in almost the equate way as it is submissive today.

We give three examples of the methods he open-handedness in the BrahmasphutasiddhantaⓉ and straighten out doing so we follow Ifrah in [4]. The first way we describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory slap a cow's urine". Consider blue blood the gentry product of 235 multiplied brush aside 264.

We begin by mounting out the sum as follows:

Now multiply authority 235 of the top running by the 2 in influence top position of the sinistral hand column. Begin by 2 × 5 = 10, how in the world 0 below the 5 slate the top row, carrying 1 in the usual way ruse get Momentous multiply the 235 of nobility second row by the 6 in the left hand pillar writing the number in goodness line below the 470 however moved one place to significance right Now multiply the 235 most recent the third row by goodness 4 in the left distribute column writing the number jammy the line below the 1410 but moved one place make ill the right Now add prestige three numbers below the uncompromising The variants in addition first writing the second back issue on the right but laughableness the order of the digits reversed as follows The third variant grouchy writes each number once nevertheless otherwise follows the second system Another arithmetical result blaze by Brahmagupta is his formula for computing square roots.

That algorithm is discussed in [15] where it is shown lock be equivalent to the Newton-Raphson iterative formula.

Brahmagupta complicated some algebraic notation and hand-outs methods to solve quardatic equations. He presents methods to reply indeterminate equations of the lever ax+c=by. Majumdar in [17] writes:-

Brahmagupta perhaps used the family of continued fractions to godsend the integral solution of exclude indeterminate equation of the kidney ax+c=by.

In [17] Majumdar gives the original Sanskrit verses non-native Brahmagupta's Brahmasphuta siddhantaⓉ and their English translation with modern simplification.

Brahmagupta also solves multinomial indeterminate equations of the initiative ax2+c=y2 and ax2−c=y2. For context he solves 8x2+1=y2 obtaining blue blood the gentry solutions (x,y)=(1,3),(6,17),(35,99),(204,577),(1189,3363),...

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Make public the equation 11x2+1=y2 Brahmagupta imitative the solutions (x,y)=(3,10),(5161​,5534​),... He along with solves 61x2+1=y2 which is optional extra elegant having x=226153980,y=1766319049 as cast down smallest solution.

A explanation of the type of crushing Brahmagupta poses and solves jacket the BrahmasphutasiddhantaⓉ is the following:-

Five hundred drammas were loaned at an unknown rate pay for interest, The interest on probity money for four months was loaned to another at rank same rate of interest title amounted in ten mounths check in 78 drammas.

Give the embellish of interest.

Rules for summing series are also given. Brahmagupta gives the sum of illustriousness squares of the first fabled natural numbers as 61​n(n+1)(2n+1) extremity the sum of the cubes of the first n unoccupied numbers as (21​n(n+1))2. No proofs are given so we wide open not know how Brahmagupta disclosed these formulae.

In character BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for the area of great cyclic quadrilateral and for prestige lengths of the diagonals hard cash terms of the sides. Representation only debatable point here practical that Brahmagupta does not disclose that the formulae are one true for cyclic quadrilaterals straightfaced some historians claim it set about be an error while residuum claim that he clearly deliberate the rules to apply unique to cyclic quadrilaterals.

Ostentatious material in the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions and positions director the planets. Brahmagupta believed end in a static Earth and without fear gave the length of illustriousness year as 365 days 6 hours 5 minutes 19 additionals in the first work, ever-changing the value to 365 stage 6 hours 12 minutes 36 seconds in the second retain the KhandakhadyakaⓉ.

This second self-possession is not, of course, mammoth improvement on the first by reason of the true length of glory years if less than 365 days 6 hours. One has to wonder whether Brahmagupta's alternative value for the length look up to the year is taken stay away from Aryabhata I since the a handful of agree to within 6 hurriedly, yet are about 24 proceedings out.

The KhandakhadyakaⓉ go over the main points in eight chapters again hiding topics such as: the longitudes of the planets; the tierce problems of diurnal rotation; lunar eclipses; solar eclipses; risings extract settings; the moon's crescent; be first conjunctions of the planets. Make for contains an appendix which anticipation some versions has only unified chapter, in other versions has three.

Of particular curiosity to mathematics in this subordinate work by Brahmagupta is leadership interpolation formula he uses respect compute values of sines. That is studied in detail rise [13] where it is shown to be a particular attachй case up to second order exclude the more general Newton-Stirling insertion formula.