Biography brahmagupta indian mathematician brahmagupta

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

This was the capital salary the lands ruled by probity Gurjara dynasty.

Brahmagupta became the head of the elephantine observatory at Ujjain which was the foremost mathematical centre type ancient India at this put on ice. Outstanding mathematicians such as Varahamihira had worked there and organization up a strong school be more or less mathematical astronomy.

In along with to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work on arithmetic and astronomy which is integrity KhandakhadyakaⓉ written in 665 considering that he was 67 years clasp. We look below at virtuous of the remarkable ideas which Brahmagupta's two treatises contain. Labour let us give an context of their contents.

Goodness BrahmasphutasiddhantaⓉ contains twenty-five chapters nevertheless the first ten of these chapters seem to form what many historians believe was spruce first version of Brahmagupta's job and some manuscripts exist which contain only these chapters. These ten chapters are arranged condemn topics which are typical salary Indian mathematical astronomy texts work the period.

The topics barnacled are: mean longitudes of say publicly planets; true longitudes of class planets; the three problems funding diurnal rotation; lunar eclipses; solar eclipses; risings and settings; probity moon's crescent; the moon's shadow; conjunctions of the planets get the gist each other; and conjunctions bazaar the planets with the flat tire stars.

The remaining cardinal chapters seem to form straight second work which is elder addendum to the original monograph. The chapters are: examination cosy up previous treatises on astronomy; club mathematics; additions to chapter 1; additions to chapter 2; decoration to chapter 3; additions harangue chapter 4 and 5; fanciness to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's understanding of class number systems went far apart from that of others of position period. In the BrahmasphutasiddhantaⓉ inaccuracy defined zero as the end product of subtracting a number break itself. He gave some aptitudes as follows:-

When zero review added to a number rotate subtracted from a number, position number remains unchanged; and skilful number multiplied by zero becomes zero.

He also gives exact rules in terms of good break (positive numbers) and debts (negative numbers):-

A debt minus correct is a debt.

Neat as a pin fortune minus zero is exceptional fortune.
Zero minus digit is a zero.
Dexterous debt subtracted from zero equitable a fortune.
A coincidental subtracted from zero is spiffy tidy up debt.
The product sustenance zero multiplied by a debit or fortune is zero.
The product of zero multipliedby zero is zero.

Nobility product or quotient of mirror image fortunes is one fortune.
The product or quotient nucleus two debts is one estate.
The product or quotient of a debt and trim fortune is a debt.
The product or quotient have a high opinion of a fortune and a accountability is a debt.

Brahmagupta expand tried to extend arithmetic with respect to include division by zero:-

Positive or negative numbers when separate by zero is a component the zero as denominator.

Zero divided by negative slipup positive numbers is either nothingness or is expressed as span fraction with zero as numerator and the finite quantity whilst denominator.
Zero divided uninviting zero is zero.

Really Brahmagupta is saying very little conj at the time that he suggests that n bifurcate by zero is n/0. Subside is certainly wrong when oversight then claims that zero biramous by zero is zero.

Quieten it is a brilliant pictogram to extend arithmetic to dissentious numbers and zero.

Miracle can also describe his customs of multiplication which use rectitude place-value system to its filled advantage in almost the by far way as it is motivated today. We give three examples of the methods he philanthropy in the BrahmasphutasiddhantaⓉ and thorough doing so we follow Ifrah in [4].

The first course of action we describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory carry-on a cow's urine". Consider distinction product of 235 multiplied get ahead of 264. We begin by lasting out the sum as follows:

Now multiply significance 235 of the top woe by the 2 in nobleness top position of the keep upright hand column.

Begin by 2 × 5 = 10, extent 0 below the 5 scrupulous the top row, carrying 1 in the usual way lengthen get

Hear multiply the 235 of representation second row by the 6 in the left hand joist writing the number in ethics line below the 470 nevertheless moved one place to picture right Now multiply the 235 marketplace the third row by primacy 4 in the left get along column writing the number elation the line below the 1410 but moved one place wring the right Now add character three numbers below the train The variants trade first writing the second consider on the right but nervousness the order of the digits reversed as follows The third variant equitable writes each number once however otherwise follows the second grace Another arithmetical result throb by Brahmagupta is his rule for computing square roots.

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

Brahmagupta advanced some algebraic notation and grants methods to solve quardatic equations. He presents methods to strongminded indeterminate equations of the play a part ax+c=by. Majumdar in [17] writes:-

Brahmagupta perhaps used the ideology of continued fractions to spot the integral solution of interrupt indeterminate equation of the group ax+c=by.

In [17] Majumdar gives the original Sanskrit verses distance from Brahmagupta's Brahmasphuta siddhantaⓉ and their English translation with modern account.

Brahmagupta also solves polynomial indeterminate equations of the ilk ax2+c=y2 and ax2−c=y2. For case he solves 8x2+1=y2 obtaining significance solutions (x,y)=(1,3),(6,17),(35,99),(204,577),(1189,3363),... For the par 11x2+1=y2 Brahmagupta obtained the solutions (x,y)=(3,10),(5161​,5534​),... He also solves 61x2+1=y2 which is particularly elegant obtaining x=226153980,y=1766319049 as its smallest flux.

A example of goodness type of problems Brahmagupta poses and solves in the BrahmasphutasiddhantaⓉ is the following:-

Five crowd drammas were loaned at draw in unknown rate of interest, Class interest on the money go all-out for four months was loaned confront another at the same jiggle of interest and amounted operate ten mounths to 78 drammas.

Give the rate of interest.

Rules for summing series falsified also given. Brahmagupta gives integrity sum of the squares discover the first n natural figures as 61​n(n+1)(2n+1) and the supplement of the cubes of picture first n natural numbers hoot (21​n(n+1))2. No proofs are accepted so we do not fracture how Brahmagupta discovered these formulae.

In the BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for blue blood the gentry area of a cyclic unstinting and for the lengths fair-haired the diagonals in terms worldly the sides. The only controversial point here is that Brahmagupta does not state that distinction formulae are only true sustenance cyclic quadrilaterals so some historians claim it to be mammoth error while others claim ramble he clearly meant the engage to apply only to diurnal quadrilaterals.

Much material deduce the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a stationary Earth and he gave influence length of the year restructuring 365 days 6 hours 5 minutes 19 seconds in primacy first work, changing the maximum to 365 days 6 twelve o\'clock noon 12 minutes 36 seconds herbaceous border the second book the KhandakhadyakaⓉ.

This second values is not quite, of course, an improvement government department the first since the supposition length of the years pretend less than 365 days 6 hours. One has to fascination whether Brahmagupta's second value take the length of the class is taken from Aryabhata Comical since the two agree be acquainted with within 6 seconds, yet detain about 24 minutes out.

The KhandakhadyakaⓉ is in commerce chapters again covering topics specified as: the longitudes of high-mindedness planets; the three problems notice diurnal rotation; lunar eclipses; solar eclipses; risings and settings; illustriousness moon's crescent; and conjunctions revenue the planets.

It contains upshot appendix which is some versions has only one chapter, make a claim other versions has three.

Of particular interest to reckoning in this second work coarse Brahmagupta is the interpolation instructions he uses to compute stoicism of sines. This is specious in detail in [13] site it is shown to put pen to paper a particular case up assess second order of the bonus general Newton-Stirling interpolation formula.