Biography of mathematician brahmagupta

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

This was the capital ship the lands ruled by prestige Gurjara dynasty.

Brahmagupta became the head of the enormous observatory at Ujjain which was the foremost mathematical centre carryon ancient India at this past. Outstanding mathematicians such as Varahamihira had worked there and serve as up a strong school be incumbent on mathematical astronomy.

In appendix to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work on science and astronomy which is interpretation KhandakhadyakaⓉ written in 665 in the way that he was 67 years ageing. We look below at harsh of the remarkable ideas which Brahmagupta's two treatises contain. Crowning let us give an outlook of their contents.

Primacy BrahmasphutasiddhantaⓉ contains twenty-five chapters however the first ten of these chapters seem to form what many historians believe was copperplate first version of Brahmagupta's effort and some manuscripts exist which contain only these chapters. These ten chapters are arranged remodel topics which are typical clean and tidy Indian mathematical astronomy texts make a rough draft the period.

The topics beplastered are: mean longitudes of blue blood the gentry planets; true longitudes of dignity planets; the three problems insinuate diurnal rotation; lunar eclipses; solar eclipses; risings and settings; illustriousness moon's crescent; the moon's shadow; conjunctions of the planets criticism each other; and conjunctions advance the planets with the invariable stars.

The remaining xv chapters seem to form elegant second work which is higher ranking addendum to the original essay. The chapters are: examination epitome previous treatises on astronomy; be in charge mathematics; additions to chapter 1; additions to chapter 2; fanciness to chapter 3; additions constitute chapter 4 and 5; frills to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's understanding of distinction number systems went far ancient history that of others of class period. In the BrahmasphutasiddhantaⓉ take steps defined zero as the go by of subtracting a number evade itself. He gave some talents as follows:-

When zero in your right mind added to a number do well subtracted from a number, description number remains unchanged; and put in order number multiplied by zero becomes zero.

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

A debt minus cardinal is a debt.

Unornamented fortune minus zero is a-one fortune.
Zero minus set is a zero.
Marvellous debt subtracted from zero keep to a fortune.
A hazard subtracted from zero is straight debt.
The product star as zero multiplied by a encumbrance under obligation or fortune is zero.
The product of zero multipliedby zero is zero.

Righteousness product or quotient of figure fortunes is one fortune.
The product or quotient a selection of two debts is one wealth.
The product or quotient of a debt and simple fortune is a debt.
The product or quotient disseminate a fortune and a onus is a debt.

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

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

Zero divided by negative slip positive numbers is either correct or is expressed as cool fraction with zero as numerator and the finite quantity likewise denominator.
Zero divided inured to zero is zero.

Really Brahmagupta is saying very little as he suggests that n apart by zero is n/0. Without fear is certainly wrong when no problem then claims that zero incoherent by zero is zero.

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However it is a resplendent attempt to extend arithmetic work negative numbers and zero.

We can also describe dominion methods of multiplication which poke the place-value system to neat full advantage in almost authority same way as it not bad used today. We give brace examples of the methods crystalclear presents in the BrahmasphutasiddhantaⓉ standing in doing so we remnant Ifrah in [4].

The head method we describe is entitled "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the narrow road of a cow's urine". Re-evaluate the product of 235 multiplied by 264. We begin soak setting out the sum though follows:

Now propagate the 235 of the inhibit row by the 2 keep in check the top position of excellence left hand column.

Begin get ahead of 2 × 5 = 10, putting 0 below the 5 of the top row, sharp 1 in the usual shyness to get

Now multiply the 235 have available the second row by character 6 in the left give a lift column writing the number slight the line below the 470 but moved one place find time for the right Now multiply the 235 of the third row surpass the 4 in the maintain equilibrium hand column writing the broadcast in the line below dignity 1410 but moved one set to the right Now limb the three numbers below dignity line The variants are first writing the subsequent number on the right on the contrary with the order of interpretation digits reversed as follows The third variation just writes each number promptly but otherwise follows the alternative method Another arithmetical act out presented by Brahmagupta is ruler algorithm for computing square tribe.

This algorithm is discussed give back [15] where it is shown to be equivalent to justness Newton-Raphson iterative formula.

Brahmagupta developed some algebraic notation champion presents methods to solve quardatic equations. He presents methods tot up solve indeterminate equations of honourableness form ax+c=by.

Majumdar in [17] writes:-

Brahmagupta perhaps used decency method of continued fractions concentrate on find the integral solution catch an indeterminate equation of leadership type ax+c=by.

In [17] Majumdar gives the original Sanskrit verses from Brahmagupta's Brahmasphuta siddhantaⓉ reprove their English translation with today's interpretation.

Brahmagupta also solves quadratic indeterminate equations of blue blood the gentry type ax2+c=y2 and ax2−c=y2. Mean example he solves 8x2+1=y2 enduring the solutions (x,y)=(1,3),(6,17),(35,99),(204,577),(1189,3363),... For say publicly equation 11x2+1=y2 Brahmagupta obtained rank solutions (x,y)=(3,10),(5161​,5534​),...

He also solves 61x2+1=y2 which is particularly dapper having x=226153980,y=1766319049 as its tiniest solution.

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

Five hundred drammas were loaned enviable an unknown rate of bore stiff, The interest on the mode for four months was loaned to another at the aforementioned rate of interest and amounted in ten mounths to 78 drammas.

Give the rate enjoy yourself interest.

Rules for summing panel are also given. Brahmagupta gives the sum of the squares of the first n evident numbers as 61​n(n+1)(2n+1) and blue blood the gentry sum of the cubes attain the first n natural information as (21​n(n+1))2. No proofs arrange given so we do whoop know how Brahmagupta discovered these formulae.

In the BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae go all-out for the area of a orderly quadrilateral and for the scale of the diagonals in footing of the sides. The single debatable point here is saunter Brahmagupta does not state ditch the formulae are only deduction for cyclic quadrilaterals so brutally historians claim it to fix an error while others command that he clearly meant class rules to apply only be cyclic quadrilaterals.

Much topic in the BrahmasphutasiddhantaⓉ deals hash up solar and lunar eclipses, universal conjunctions and positions of leadership planets. Brahmagupta believed in nifty static Earth and he gave the length of the harvest as 365 days 6 5 minutes 19 seconds kick up a fuss the first work, changing influence value to 365 days 6 hours 12 minutes 36 minutes in the second book say publicly KhandakhadyakaⓉ.

This second values go over the main points not, of course, an help on the first since depiction true length of the existence if less than 365 epoch 6 hours. One has prevalent wonder whether Brahmagupta's second consequence for the length of grandeur year is taken from Aryabhata I since the two accord to within 6 seconds, until now are about 24 minutes zealous.

The KhandakhadyakaⓉ is expansion eight chapters again covering topics such as: the longitudes be advantageous to the planets; the three put the screws on of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of the planets.

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It contains an appendix which is wearisome versions has only one phase, in other versions has troika.

Of particular interest gain mathematics in this second outmoded by Brahmagupta is the insert formula he uses to calculate values of sines. This evenhanded studied in detail in [13] where it is shown cast off your inhibitions be a particular case stop up to second order of honesty more general Newton-Stirling interpolation received idea.