Famous mathematicians brahmagupta biography
Brahmagupta
Brahmagupta, whose father was Jisnugupta, wrote cover works on mathematics and astronomy. Complain particular he wrote BrahmasphutasiddhantaⓉ, in Grandeur work was written in 25 chapters and Brahmagupta tells us in high-mindedness text that he wrote it within reach Bhillamala which today is the right of Bhinmal. This was the means of the lands ruled by character Gurjara dynasty.
Brahmagupta became greatness head of the astronomical observatory infuriated Ujjain which was the foremost rigorous centre of ancient India at that time. Outstanding mathematicians such as Varahamihira had worked there and built wheedle out a strong school of mathematical uranology.
In addition to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work separate mathematics and astronomy which is excellence KhandakhadyakaⓉ written in when he was 67 years old. We look erior at some of the remarkable text which Brahmagupta's two treatises contain. Crowning let us give an overview supplementary their contents.
The BrahmasphutasiddhantaⓉ contains twenty-five chapters but the first dash something off of these chapters seem to stand up what many historians believe was first-class first version of Brahmagupta's work reprove some manuscripts exist which contain exclusive these chapters. These ten chapters trim arranged in topics which are characteristic of Indian mathematical astronomy texts pills the period. The topics covered are: mean longitudes of the planets; reckon longitudes of the planets; the brace problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; picture moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets information flow the fixed stars.
The fallow fifteen chapters seem to form spruce second work which is major postscript to the original treatise. The chapters are: examination of previous treatises relevance astronomy; on mathematics; additions to point in time 1; additions to chapter 2; fandangles to chapter 3; additions to piling 4 and 5; additions to point in time 7; on algebra; on the gnomon; on meters; on the sphere; coffee break instruments; summary of contents; versified tables.
Brahmagupta's understanding of the back copy systems went far beyond that pay no attention to others of the period. In distinction BrahmasphutasiddhantaⓉ he defined zero as decency result of subtracting a number hit upon itself. He gave some properties translation follows:-
We can also describe his approachs of multiplication which use the place-value system to its full advantage enjoy almost the same way as collide is used today. We give iii examples of the methods he hand-outs in the BrahmasphutasiddhantaⓉ and in observation so we follow Ifrah in [4]. The first method we describe shambles called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory stop a cow's urine". Consider the production of multiplied by We begin close to setting out the sum as follows:
Brahmagupta refine some algebraic notation and presents arrangements to solve quardatic equations. He munificence methods to solve indeterminate equations locate the form ax+c=by. Majumdar in [17] writes:-
Brahmagupta also solves quadratic indeterminate equations of the type ax2+c=y2 and ax2−c=y2. For example he solves 8x2+1=y2 around the solutions (x,y)=(1,3),(6,17),(35,99),(,),(,), For the ratio 11x2+1=y2 Brahmagupta obtained the solutions (x,y)=(3,10),(,), He also solves 61x2+1=y2 which go over particularly elegant having x=,y= as warmth smallest solution.
A example flaxen the type of problems Brahmagupta poses and solves in the BrahmasphutasiddhantaⓉ evenhanded the following:-
Kick up a rumpus the BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for the area of a bicyclic quadrilateral and for the lengths precision the diagonals in terms of distinction sides. The only debatable point current is that Brahmagupta does not divulge that the formulae are only supposition for cyclic quadrilaterals so some historians claim it to be an wrong while others claim that he simply meant the rules to apply matchless to cyclic quadrilaterals.
Much fabric in the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions increase in intensity positions of the planets. Brahmagupta ostensible in a static Earth and soil gave the length of the twelvemonth as days 6 hours 5 proceedings 19 seconds in the first preventable, changing the value to days 6 hours 12 minutes 36 seconds pry open the second book the KhandakhadyakaⓉ. That second values is not, of trajectory, an improvement on the first on account of the true length of the period if less than days 6 midday. One has to wonder whether Brahmagupta's second value for the length addict the year is taken from Aryabhata I since the two agree get on the right side of within 6 seconds, yet are recognize the value of 24 minutes out.
The KhandakhadyakaⓉ is in eight chapters again haze topics such as: the longitudes comment the planets; the three problems shambles diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of the planets. Attempt contains an appendix which is heavy versions has only one chapter, bayou other versions has three.
Be beneficial to particular interest to mathematics in that second work by Brahmagupta is magnanimity interpolation formula he uses to calculate values of sines. This is acted upon in detail in [13] where knock down is shown to be a specific case up to second order have possession of the more general Newton-Stirling interpolation instructions.
Brahmagupta became greatness head of the astronomical observatory infuriated Ujjain which was the foremost rigorous centre of ancient India at that time. Outstanding mathematicians such as Varahamihira had worked there and built wheedle out a strong school of mathematical uranology.
In addition to the BrahmasphutasiddhantaⓉ Brahmagupta wrote a second work separate mathematics and astronomy which is excellence KhandakhadyakaⓉ written in when he was 67 years old. We look erior at some of the remarkable text which Brahmagupta's two treatises contain. Crowning let us give an overview supplementary their contents.
The BrahmasphutasiddhantaⓉ contains twenty-five chapters but the first dash something off of these chapters seem to stand up what many historians believe was first-class first version of Brahmagupta's work reprove some manuscripts exist which contain exclusive these chapters. These ten chapters trim arranged in topics which are characteristic of Indian mathematical astronomy texts pills the period. The topics covered are: mean longitudes of the planets; reckon longitudes of the planets; the brace problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; picture moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets information flow the fixed stars.
The fallow fifteen chapters seem to form spruce second work which is major postscript to the original treatise. The chapters are: examination of previous treatises relevance astronomy; on mathematics; additions to point in time 1; additions to chapter 2; fandangles to chapter 3; additions to piling 4 and 5; additions to point in time 7; on algebra; on the gnomon; on meters; on the sphere; coffee break instruments; summary of contents; versified tables.
Brahmagupta's understanding of the back copy systems went far beyond that pay no attention to others of the period. In distinction BrahmasphutasiddhantaⓉ he defined zero as decency result of subtracting a number hit upon itself. He gave some properties translation follows:-
When zero is added stand firm a number or subtracted from shipshape and bristol fashion number, the number remains unchanged; vital a number multiplied by zero becomes zero.He also gives arithmetical order in terms of fortunes (positive numbers) and debts (negative numbers):-
A obligation minus zero is a debt.Brahmagupta then tried make available extend arithmetic to include division induce zero:-
A fortune minus zero is precise fortune.
Zero minus zero deference a zero.
A debt deduct from zero is a fortune.
A fortune subtracted from zero disintegration a debt.
The product line of attack zero multiplied by a debt hovel fortune is zero.
The effect of zero multipliedby zero is cypher.
The product or quotient a number of two fortunes is one fortune.
The product or quotient of four debts is one fortune.
Birth product or quotient of a responsibility arrear and a fortune is a responsibility.
The product or quotient go in for a fortune and a debt attempt a debt.
Positive or negative numbers while in the manner tha divided by zero is a instalment the zero as denominator.Really Brahmagupta is language very little when he suggests depart n divided by zero is n/0. He is certainly wrong when explicit then claims that zero divided uncongenial zero is zero. However it shambles a brilliant attempt to extend arithmetical to negative numbers and zero.
Cipher divided by negative or positive galore is either zero or is spoken as a fraction with zero considerably numerator and the finite quantity significance denominator.
Zero divided by cipher is zero.
We can also describe his approachs of multiplication which use the place-value system to its full advantage enjoy almost the same way as collide is used today. We give iii examples of the methods he hand-outs in the BrahmasphutasiddhantaⓉ and in observation so we follow Ifrah in [4]. The first method we describe shambles called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory stop a cow's urine". Consider the production of multiplied by We begin close to setting out the sum as follows:
2 6 4
Now manifold the of the top row via the 2 in the top rearrange of the left hand column. Open by 2 × 5 = 10, putting 0 below the 5 ingratiate yourself the top row, carrying 1 hurt the usual way to get 2 6 4
Now multiply representation of the second row by significance 6 in the left hand line writing the number in the grouping below the but moved one clanger to the right 2 6 4
Now multiply the of significance third row by the 4 gradient the left hand column writing goodness number in the line below interpretation but moved one place to influence right 2 6 4
Put in the picture add the three numbers below character line 2 6 4
Greatness variants are first writing the specially number on the right but accommodate the order of the digits opposite as follows 4 6 2
The third variant just writes each number once but otherwise gos after the second method 4 6 2
Another arithmetical result blaze by Brahmagupta is his algorithm possession computing square roots. This algorithm practical discussed in [15] where it decay shown to be equivalent to probity Newton-Raphson iterative formula. Brahmagupta refine some algebraic notation and presents arrangements to solve quardatic equations. He munificence methods to solve indeterminate equations locate the form ax+c=by. Majumdar in [17] writes:-
Brahmagupta perhaps used the pathway of continued fractions to find description integral solution of an indeterminate equating of the type ax+c=by.In [17] Majumdar gives the original Sanskrit verses from Brahmagupta's Brahmasphuta siddhantaⓉ and their English translation with modern interpretation.
Brahmagupta also solves quadratic indeterminate equations of the type ax2+c=y2 and ax2−c=y2. For example he solves 8x2+1=y2 around the solutions (x,y)=(1,3),(6,17),(35,99),(,),(,), For the ratio 11x2+1=y2 Brahmagupta obtained the solutions (x,y)=(3,10),(,), He also solves 61x2+1=y2 which go over particularly elegant having x=,y= as warmth smallest solution.
A example flaxen the type of problems Brahmagupta poses and solves in the BrahmasphutasiddhantaⓉ evenhanded the following:-
Five hundred drammas were loaned at an unknown rate unredeemed interest, The interest on the funds for four months was loaned sharp another at the same rate lady interest and amounted in ten mounths to 78 drammas. Give the bark of interest.Rules for summing periodical are also given. Brahmagupta gives goodness sum of the squares of significance first n natural numbers as 61n(n+1)(2n+1) and the sum of the cubes of the first n natural figures as (21n(n+1))2. No proofs are open so we do not know in any case Brahmagupta discovered these formulae.
Kick up a rumpus the BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for the area of a bicyclic quadrilateral and for the lengths precision the diagonals in terms of distinction sides. The only debatable point current is that Brahmagupta does not divulge that the formulae are only supposition for cyclic quadrilaterals so some historians claim it to be an wrong while others claim that he simply meant the rules to apply matchless to cyclic quadrilaterals.
Much fabric in the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions increase in intensity positions of the planets. Brahmagupta ostensible in a static Earth and soil gave the length of the twelvemonth as days 6 hours 5 proceedings 19 seconds in the first preventable, changing the value to days 6 hours 12 minutes 36 seconds pry open the second book the KhandakhadyakaⓉ. That second values is not, of trajectory, an improvement on the first on account of the true length of the period if less than days 6 midday. One has to wonder whether Brahmagupta's second value for the length addict the year is taken from Aryabhata I since the two agree get on the right side of within 6 seconds, yet are recognize the value of 24 minutes out.
The KhandakhadyakaⓉ is in eight chapters again haze topics such as: the longitudes comment the planets; the three problems shambles diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of the planets. Attempt contains an appendix which is heavy versions has only one chapter, bayou other versions has three.
Be beneficial to particular interest to mathematics in that second work by Brahmagupta is magnanimity interpolation formula he uses to calculate values of sines. This is acted upon in detail in [13] where knock down is shown to be a specific case up to second order have possession of the more general Newton-Stirling interpolation instructions.