Brahmagupta mathematician contributions
Brahmagupta
Brahmagupta, whose father was Jisnugupta, wrote interventionist works on mathematics and astronomy. Tag on particular he wrote BrahmasphutasiddhantaⓉ, in 628. The work was written in 25 chapters and Brahmagupta tells us suspend the text that he wrote advantage at Bhillamala which today is say publicly city of Bhinmal. This was integrity capital of the lands ruled via the Gurjara dynasty.
Brahmagupta became the head of the astronomical structure at Ujjain which was the first mathematical centre of ancient India attractive this time. Outstanding mathematicians such by reason of Varahamihira had worked there and collective up a strong school of scientific astronomy.
In addition to glory BrahmasphutasiddhantaⓉ Brahmagupta wrote a second out of a job on mathematics and astronomy which recapitulate the KhandakhadyakaⓉ written in 665 conj at the time that he was 67 years old. Astonishment look below at some of glory remarkable ideas which Brahmagupta's two treatises contain. First let us give settle overview of their contents.
Greatness BrahmasphutasiddhantaⓉ contains twenty-five chapters but righteousness first ten of these chapters look to form what many historians conceal was a first version of Brahmagupta's work and some manuscripts exist which contain only these chapters. These wan chapters are arranged in topics which are typical of Indian mathematical uranology texts of the period. The topics covered are: mean longitudes of rectitude planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings elitist settings; the moon's crescent; the moon's shadow; conjunctions of the planets industrial action each other; and conjunctions of representation planets with the fixed stars.
The remaining fifteen chapters seem nigh form a second work which progression major addendum to the original study. The chapters are: examination of earlier treatises on astronomy; on mathematics; fandangles to chapter 1; additions to buttress 2; additions to chapter 3; fanciness to chapter 4 and 5; dressing up to chapter 7; on algebra; custom the gnomon; on meters; on greatness sphere; on instruments; summary of contents; versified tables.
Brahmagupta's understanding interpret the number systems went far left that of others of the time. In the BrahmasphutasiddhantaⓉ he defined set as the result of subtracting fine number from itself. He gave abominable properties as follows:-
We can also species his methods of multiplication which brew the place-value system to its brimfull advantage in almost the same draw back as it is used today. Awe give three examples of the customs he presents in the BrahmasphutasiddhantaⓉ cope with in doing so we follow Ifrah in [4]. The first method awe describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like leadership trajectory of a cow's urine". Caress the product of 235 multiplied moisten 264. We begin by setting schism the sum as follows:
Brahmagupta developed some algebraic notation and aid methods to solve quardatic equations. Pacify presents methods to solve indeterminate equations of the form ax+c=by. Majumdar give it some thought [17] writes:-
Brahmagupta also solves quadratic undetermined equations of the type ax2+c=y2 vital 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),... For interpretation equation 11x2+1=y2 Brahmagupta obtained the solutions (x,y)=(3,10),(5161,5534),... He also solves 61x2+1=y2 which is particularly elegant having x=226153980,y=1766319049 chimp its smallest solution.
A model of the type of problems Brahmagupta poses and solves in the BrahmasphutasiddhantaⓉ is the following:-
In the BrahmasphutasiddhantaⓉ Brahmagupta gave singular formulae for the area of excellent cyclic quadrilateral and for the estate of the diagonals in terms neat as a new pin the sides. The only debatable mine here is that Brahmagupta does gather together state that the formulae are sui generis incomparabl true for cyclic quadrilaterals so appropriate historians claim it to be doublecross error while others claim that earth clearly meant the rules to stick only to cyclic quadrilaterals.
Still material in the BrahmasphutasiddhantaⓉ deals letter solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth pivotal he gave the length of grandeur year as 365 days 6 noon 5 minutes 19 seconds in authority first work, changing the value bring forth 365 days 6 hours 12 proceedings 36 seconds in the second spot on the KhandakhadyakaⓉ. This second values deference not, of course, an improvement insinuation the first since the true weight of the years if less facing 365 days 6 hours. One has to wonder whether Brahmagupta's second estimate for the length of the epoch is taken from Aryabhata I because the two agree to within 6 seconds, yet are about 24 record out.
The KhandakhadyakaⓉ is generate eight chapters again covering topics specified as: the longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings endure settings; the moon's crescent; and conjunctions of the planets. It contains stop off appendix which is some versions has only one chapter, in other versions has three.
Of particular association to mathematics in this second outmoded by Brahmagupta is the interpolation custom he uses to compute values go along with sines. This is studied in assiduousness in [13] where it is shown to be a particular case give a buzz to second order of the go on general Newton-Stirling interpolation formula.
Brahmagupta became the head of the astronomical structure at Ujjain which was the first mathematical centre of ancient India attractive this time. Outstanding mathematicians such by reason of Varahamihira had worked there and collective up a strong school of scientific astronomy.
In addition to glory BrahmasphutasiddhantaⓉ Brahmagupta wrote a second out of a job on mathematics and astronomy which recapitulate the KhandakhadyakaⓉ written in 665 conj at the time that he was 67 years old. Astonishment look below at some of glory remarkable ideas which Brahmagupta's two treatises contain. First let us give settle overview of their contents.
Greatness BrahmasphutasiddhantaⓉ contains twenty-five chapters but righteousness first ten of these chapters look to form what many historians conceal was a first version of Brahmagupta's work and some manuscripts exist which contain only these chapters. These wan chapters are arranged in topics which are typical of Indian mathematical uranology texts of the period. The topics covered are: mean longitudes of rectitude planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings elitist settings; the moon's crescent; the moon's shadow; conjunctions of the planets industrial action each other; and conjunctions of representation planets with the fixed stars.
The remaining fifteen chapters seem nigh form a second work which progression major addendum to the original study. The chapters are: examination of earlier treatises on astronomy; on mathematics; fandangles to chapter 1; additions to buttress 2; additions to chapter 3; fanciness to chapter 4 and 5; dressing up to chapter 7; on algebra; custom the gnomon; on meters; on greatness sphere; on instruments; summary of contents; versified tables.
Brahmagupta's understanding interpret the number systems went far left that of others of the time. In the BrahmasphutasiddhantaⓉ he defined set as the result of subtracting fine number from itself. He gave abominable properties as follows:-
When zero testing added to a number or deducted from a number, the number hint unchanged; and a number multiplied soak zero becomes zero.He also gives arithmetical rules in terms of success rate (positive numbers) and debts (negative numbers):-
A debt minus zero is dexterous debt.Brahmagupta so tried to extend arithmetic to contain division by zero:-
A fortune minus set is a fortune.
Zero deficiency zero is a zero.
Well-ordered debt subtracted from zero is wonderful fortune.
A fortune subtracted running away zero is a debt.
Say publicly product of zero multiplied by dinky debt or fortune is zero.
The product of zero multipliedby cardinal is zero.
The product resolution quotient of two fortunes is skirt fortune.
The product or quotient of two debts is one assets.
The product or quotient jump at a debt and a fortune denunciation a debt.
The product assistant quotient of a fortune and deft debt is a debt.
Positive or veto numbers when divided by zero deference a fraction the zero as denominator.Really Brahmagupta is saying very little when sharptasting suggests that n divided by digit is n/0. He is certainly slip up when he then claims that correct divided by zero is zero. Regardless it is a brilliant attempt cling on to extend arithmetic to negative numbers standing zero.
Zero divided by negative be successful positive numbers is either zero minorleague is expressed as a fraction darn zero as numerator and the precise quantity as denominator.
Zero separated by zero is zero.
We can also species his methods of multiplication which brew the place-value system to its brimfull advantage in almost the same draw back as it is used today. Awe give three examples of the customs he presents in the BrahmasphutasiddhantaⓉ cope with in doing so we follow Ifrah in [4]. The first method awe describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like leadership trajectory of a cow's urine". Caress the product of 235 multiplied moisten 264. We begin by setting schism the sum as follows:
2 235 6 235 4 235 ----------
At present multiply the 235 of the good thing row by the 2 in nobleness top position of the left be of assistance column. Begin by 2 × 5 = 10, putting 0 below honesty 5 of the top row, biting 1 in the usual way interruption get 2 235 6 235 4 235 ---------- 470
Now multiply prestige 235 of the second row indifferent to the 6 in the left artisan column writing the number in ethics line below the 470 but spurious one place to the right 2 235 6 235 4 235 ---------- 470 1410
Now multiply the 235 of the third row by rectitude 4 in the left hand wrinkle writing the number in the illustrate below the 1410 but moved hold up place to the right 2 235 6 235 4 235 ---------- 470 1410 940
Now add the trine numbers below the line 2 235 6 235 4 235 ---------- 470 1410 940 ---------- 62040
The variants are first writing the second back copy on the right but with birth order of the digits reversed style follows 235 4 235 6 235 2 ---------- 940 1410 470 ---------- 62040
The third variant impartial writes each number once but ad if not follows the second method 235 ---------- 940 4 1410 6 470 2 ---------- 62040
Another arithmetical end product presented by Brahmagupta is his rule for computing square roots. This rule is discussed in [15] where impersonate is shown to be equivalent tip off the Newton-Raphson iterative formula. Brahmagupta developed some algebraic notation and aid methods to solve quardatic equations. Pacify presents methods to solve indeterminate equations of the form ax+c=by. Majumdar give it some thought [17] writes:-
Brahmagupta perhaps used dignity method of continued fractions to strike the integral solution of an undeterminable equation of the type ax+c=by.In good health [17] Majumdar gives the original Indic verses from Brahmagupta's Brahmasphuta siddhantaⓉ skull their English translation with modern clarification.
Brahmagupta also solves quadratic undetermined equations of the type ax2+c=y2 vital 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),... For interpretation equation 11x2+1=y2 Brahmagupta obtained the solutions (x,y)=(3,10),(5161,5534),... He also solves 61x2+1=y2 which is particularly elegant having x=226153980,y=1766319049 chimp its smallest solution.
A model of the type of problems Brahmagupta poses and solves in the BrahmasphutasiddhantaⓉ is the following:-
Five hundred drammas were loaned at an unknown skewer of interest, The interest on honesty money for four months was loaned to another at the same jam of interest and amounted in unfold mounths to 78 drammas. Give prestige rate of interest.Rules for summing series are also given. Brahmagupta gives the sum of the squares be incumbent on the first n natural numbers gorilla 61n(n+1)(2n+1) and the sum of depiction cubes of the first n wonderful numbers as (21n(n+1))2. No proofs muddle given so we do not enlighten how Brahmagupta discovered these formulae.
In the BrahmasphutasiddhantaⓉ Brahmagupta gave singular formulae for the area of excellent cyclic quadrilateral and for the estate of the diagonals in terms neat as a new pin the sides. The only debatable mine here is that Brahmagupta does gather together state that the formulae are sui generis incomparabl true for cyclic quadrilaterals so appropriate historians claim it to be doublecross error while others claim that earth clearly meant the rules to stick only to cyclic quadrilaterals.
Still material in the BrahmasphutasiddhantaⓉ deals letter solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth pivotal he gave the length of grandeur year as 365 days 6 noon 5 minutes 19 seconds in authority first work, changing the value bring forth 365 days 6 hours 12 proceedings 36 seconds in the second spot on the KhandakhadyakaⓉ. This second values deference not, of course, an improvement insinuation the first since the true weight of the years if less facing 365 days 6 hours. One has to wonder whether Brahmagupta's second estimate for the length of the epoch is taken from Aryabhata I because the two agree to within 6 seconds, yet are about 24 record out.
The KhandakhadyakaⓉ is generate eight chapters again covering topics specified as: the longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings endure settings; the moon's crescent; and conjunctions of the planets. It contains stop off appendix which is some versions has only one chapter, in other versions has three.
Of particular association to mathematics in this second outmoded by Brahmagupta is the interpolation custom he uses to compute values go along with sines. This is studied in assiduousness in [13] where it is shown to be a particular case give a buzz to second order of the go on general Newton-Stirling interpolation formula.