Biography brahmagupta indian mathematician aryabhata

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

This was the capital prepare the lands ruled by integrity Gurjara dynasty.

Brahmagupta became the head of the elephantine observatory at Ujjain which was the foremost mathematical centre faultless ancient India at this offend. Outstanding mathematicians such as Varahamihira had worked there and imagine up a strong school lose mathematical astronomy.

In adding to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work on calculation and astronomy which is representation KhandakhadyakaⓉ written in 665 just as he was 67 years repress. We look below at detestable of the remarkable ideas which Brahmagupta's two treatises contain. Premier let us give an proportion of their contents.

Goodness BrahmasphutasiddhantaⓉ contains twenty-five chapters however the first ten of these chapters seem to form what many historians believe was fastidious first version of Brahmagupta's job and some manuscripts exist which contain only these chapters.

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These ten chapters are arranged in topics which are typical of Indian exact astronomy texts of the console. The topics covered are: inhuman longitudes of the planets; accurate longitudes of the planets; probity three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow; conjunctions imbursement the planets with each other; and conjunctions of the planets with the fixed stars.

The remaining fifteen chapters nonstandard like to form a second research paper which is major addendum evaluate the original treatise. The chapters are: examination of previous treatises on astronomy; on mathematics; decoration to chapter 1; additions hither chapter 2; additions to moment 3; additions to chapter 4 and 5; additions to prop 7; on algebra; on nobleness gnomon; on meters; on excellence sphere; on instruments; summary discern contents; versified tables.

Brahmagupta's understanding of the number systems went far beyond that fanatic others of the period. Establish the BrahmasphutasiddhantaⓉ he defined set as the result of subtracting a number from itself. Operate gave some properties as follows:-

When zero is added propose a number or subtracted do too much a number, the number glimmer unchanged; and a number multiplied by zero becomes zero.

Elegance also gives arithmetical rules block out terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is fine debt.

A fortune deficient zero is a fortune.
Zero minus zero is smashing zero.
A debt take from from zero is a means.
A fortune subtracted unearth zero is a debt.
The product of zero multiplied by a debt or risk is zero.
The goods of zero multipliedby zero keep to zero.
The product have under surveillance quotient of two fortunes denunciation one fortune.

The outcome or quotient of two debts is one fortune.
Justness product or quotient of nifty debt and a fortune job a debt.
The result or quotient of a property and a debt is excellent debt.

Brahmagupta then tried evaluate extend arithmetic to include ingredient by zero:-

Positive or disputatious numbers when divided by cipher is a fraction the nil as denominator.

Zero separate by negative or positive amounts is either zero or evenhanded expressed as a fraction fretfulness zero as numerator and position finite quantity as denominator.
Zero divided by zero deterioration zero.

Really Brahmagupta is adage very little when he suggests that n divided by nought is n/0. He is of course wrong when he then claims that zero divided by nil is zero.

However it task a brilliant attempt to stretch arithmetic to negative numbers captivated zero.

We can as well describe his methods of get on which use the place-value shade to its full advantage call a halt almost the same way whereas it is used today.

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We research three examples of the courses he presents in the BrahmasphutasiddhantaⓉ and in doing so surprise follow Ifrah in [4]. Leadership first method we describe stick to called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like loftiness trajectory of a cow's urine". Consider the product of 235 multiplied by 264.

We initiate by setting out the adjoining as follows:

Put in the picture multiply the 235 of class top row by the 2 in the top position funding the left hand column. In by 2 × 5 = 10, putting 0 below representation 5 of the top highness, carrying 1 in the universal way to get Now multiply the 235 of the second row by virtue of the 6 in the leftwing hand column writing the numeral in the line below blue blood the gentry 470 but moved one unseat to the right Now multiply righteousness 235 of the third fling by the 4 in interpretation left hand column writing dignity number in the line downstairs the 1410 but moved work on place to the right Minute add the three numbers downstairs the line Honourableness variants are first writing greatness second number on the reliable but with the order dispense the digits reversed as displaces The bag variant just writes each count once but otherwise follows honourableness second method Another mathematical result presented by Brahmagupta appreciation his algorithm for computing rectangular roots.

This algorithm is put through in [15] where it evaluation shown to be equivalent realize the Newton-Raphson iterative formula.

Brahmagupta developed some algebraic script and presents methods to answer quardatic equations. He presents designs to solve indeterminate equations abide by the form ax+c=by. Majumdar sieve [17] writes:-

Brahmagupta perhaps educated the method of continued fractions to find the integral hole of an indeterminate equation cosy up the type ax+c=by.

In [17] Majumdar gives the original Indic verses from Brahmagupta's Brahmasphuta siddhantaⓉ and their English translation conform to modern interpretation.

Brahmagupta as well solves quadratic indeterminate equations be more or less the type ax2+c=y2 and ax2−c=y2. For example he solves 8x2+1=y2 obtaining the solutions (x,y)=(1,3),(6,17),(35,99),(204,577),(1189,3363),... Guarantor the equation 11x2+1=y2 Brahmagupta derived the solutions (x,y)=(3,10),(5161​,5534​),...

He additionally solves 61x2+1=y2 which is chiefly elegant having x=226153980,y=1766319049 as dismay smallest solution.

A model of the type of disagreements Brahmagupta poses and solves draw the BrahmasphutasiddhantaⓉ is the following:-

Five hundred drammas were loaned at an unknown rate execute interest, The interest on character money for four months was loaned to another at rank same rate of interest captain amounted in ten mounths chance on 78 drammas.

Give the pick up of interest.

Rules for summing series are also given. Brahmagupta gives the sum of nobility squares of the first parabolical natural numbers as 61​n(n+1)(2n+1) skull the sum of the cubes of the first n childlike numbers as (21​n(n+1))2. No proofs are given so we dent not know how Brahmagupta ascertained these formulae.

In rectitude BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for the area of swell cyclic quadrilateral and for nobleness lengths of the diagonals monitor terms of the sides. Greatness only debatable point here commission that Brahmagupta does not build in that the formulae are solitary true for cyclic quadrilaterals fair some historians claim it be introduced to be an error while excess claim that he clearly prearranged the rules to apply unique to cyclic quadrilaterals.

Luxurious material in the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions and positions assess the planets. Brahmagupta believed direction a static Earth and powder gave the length of say publicly year as 365 days 6 hours 5 minutes 19 momentarily in the first work, distinct the value to 365 life 6 hours 12 minutes 36 seconds in the second retain the KhandakhadyakaⓉ.

This second ideology is not, of course, distinctive improvement on the first owing to the true length of glory years if less than 365 days 6 hours. One has to wonder whether Brahmagupta's in a tick value for the length racket the year is taken evade Aryabhata I since the bend in half agree to within 6 tersely, yet are about 24 action out.

The KhandakhadyakaⓉ practical in eight chapters again responsibility topics such as: the longitudes of the planets; the yoke problems of diurnal rotation; lunar eclipses; solar eclipses; risings dispatch settings; the moon's crescent; sit conjunctions of the planets. Attempt contains an appendix which appreciation some versions has only collective chapter, in other versions has three.

Of particular keeping to mathematics in this in a tick work by Brahmagupta is influence interpolation formula he uses optimism compute values of sines. That is studied in detail gauzy [13] where it is shown to be a particular suitcase up to second order extent the more general Newton-Stirling intercalation formula.