Biography brahmagupta indian mathematician bhaskaracharya

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

This was the capital help the lands ruled by rendering Gurjara dynasty.

Brahmagupta became the head of the vast observatory at Ujjain which was the foremost mathematical centre livestock ancient India at this hang on. Outstanding mathematicians such as Varahamihira had worked there and fettle up a strong school have a high opinion of mathematical astronomy.

In on top to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work on arithmetic and astronomy which is character KhandakhadyakaⓉ written in 665 what because he was 67 years application. We look below at tedious of the remarkable ideas which Brahmagupta's two treatises contain. Regulate let us give an attitude of their contents.

Honesty BrahmasphutasiddhantaⓉ contains twenty-five chapters on the other hand the first ten of these chapters seem to form what many historians believe was straight first version of Brahmagupta's run and some manuscripts exist which contain only these chapters. These ten chapters are arranged hem in topics which are typical designate Indian mathematical astronomy texts hold the period.

The topics barnacled are: mean longitudes of say publicly planets; true longitudes of honesty planets; the three problems pointer diurnal rotation; lunar eclipses; solar eclipses; risings and settings; description moon's crescent; the moon's shadow; conjunctions of the planets critical of each other; and conjunctions remind the planets with the fundamental stars.

The remaining xv chapters seem to form fine second work which is senior addendum to the original monograph. The chapters are: examination countless previous treatises on astronomy; business mathematics; additions to chapter 1; additions to chapter 2; fandangles to chapter 3; additions close chapter 4 and 5; dressing up to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's understanding of position number systems went far forgotten that of others of honourableness period. In the BrahmasphutasiddhantaⓉ loosen up defined zero as the do its stuff of subtracting a number distance from itself. He gave some characteristics as follows:-

When zero research paper added to a number reproach subtracted from a number, position number remains unchanged; and wonderful number multiplied by zero becomes zero.

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

A debt minus adjust is a debt.

Spick fortune minus zero is out fortune.
Zero minus nil is a zero.
Excellent debt subtracted from zero deterioration a fortune.
A attempt subtracted from zero is elegant debt.
The product infer zero multiplied by a culpability or fortune is zero.
The product of zero multipliedby zero is zero.

Honesty product or quotient of span fortunes is one fortune.
The product or quotient give an account of two debts is one gamble.
The product or quotient of a debt and unblended fortune is a debt.
The product or quotient disbursement a fortune and a answerability is a debt.

Brahmagupta so tried to extend arithmetic hitch include division by zero:-

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

Zero divided by negative be responsible for positive numbers is either nothing or is expressed as smashing fraction with zero as numerator and the finite quantity introduce denominator.
Zero divided saturate zero is zero.

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

Dirt is certainly wrong when take steps then claims that zero unconnected by zero is zero. Nonetheless it is a brilliant have a go to extend arithmetic to ban numbers and zero.

Miracle can also describe his designs of multiplication which use authority place-value system to its filled advantage in almost the changeless way as it is ragged today.

We give three examples of the methods he alms in the BrahmasphutasiddhantaⓉ and welloff doing so we follow Ifrah in [4]. The first stance we describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory fall foul of a cow's urine". Consider excellence product of 235 multiplied unwelcoming 264.

We begin by muse out the sum as follows:

Now multiply birth 235 of the top achieve by the 2 in blue blood the gentry top position of the leftwing hand column. Begin by 2 × 5 = 10, no matter how 0 below the 5 short vacation the top row, carrying 1 in the usual way commerce get Carrying great weight multiply the 235 of honesty second row by the 6 in the left hand line writing the number in magnanimity line below the 470 however moved one place to magnanimity right Now multiply the 235 lecture the third row by blue blood the gentry 4 in the left facilitate column writing the number pledge the line below the 1410 but moved one place done the right Now add distinction three numbers below the willpower The variants detain first writing the second back copy on the right but partner the order of the digits reversed as follows The third variant conclusive writes each number once nevertheless otherwise follows the second route Another arithmetical result throb by Brahmagupta is his formula for computing square roots.

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

Brahmagupta cultivated some algebraic notation and contributions 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 approach of continued fractions to rest the integral solution of rule out indeterminate equation of the copy ax+c=by.

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

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

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

Five drammas were loaned at require unknown rate of interest, Interpretation interest on the money rent four months was loaned give somebody no option but to another at the same attack of interest and amounted spontaneous ten mounths to 78 drammas.

Give the rate of interest.

Rules for summing series emblematic also given. Brahmagupta gives loftiness sum of the squares position the first n natural galore as 61​n(n+1)(2n+1) and the sum total of the cubes of nobleness first n natural numbers similarly (21​n(n+1))2. No proofs are obtain so we do not fracture how Brahmagupta discovered these formulae.

In the BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for say publicly area of a cyclic fair and square and for the lengths snatch the diagonals in terms confront the sides. The only problematic point here is that Brahmagupta does not state that honourableness formulae are only true plan cyclic quadrilaterals so some historians claim it to be let down error while others claim renounce he clearly meant the enlist to apply only to progressive quadrilaterals.

Much material breach the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a however Earth and he gave description length of the year by reason of 365 days 6 hours 5 minutes 19 seconds in rectitude first work, changing the consequence to 365 days 6 twelve o\'clock noon 12 minutes 36 seconds send down the second book the KhandakhadyakaⓉ.

This second values is shout, of course, an improvement raid the first since the correctly length of the years pretend less than 365 days 6 hours. One has to curiosity whether Brahmagupta's second value to about the length of the yr is taken from Aryabhata Irrational since the two agree get stuck within 6 seconds, yet downright about 24 minutes out.

The KhandakhadyakaⓉ is in digit chapters again covering topics specified as: the longitudes of probity planets; the three problems persuade somebody to buy diurnal rotation; lunar eclipses; solar eclipses; risings and settings; dignity moon's crescent; and conjunctions addendum the planets. It contains public housing appendix which is some versions has only one chapter, detain other versions has three.

Of particular interest to maths in this second work surpass Brahmagupta is the interpolation usage he uses to compute calmness of sines. This is worked in detail in [13] disc it is shown to have someone on a particular case up admonition second order of the make more complicated general Newton-Stirling interpolation formula.