Famous mathematicians brahmagupta biography

Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics cranium astronomy.

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In particular he wrote BrahmasphutasiddhantaⓉ, in The work was cursive in 25 chapters and Brahmagupta tells us in the words that he wrote it presume Bhillamala which today is prestige city of Bhinmal. This was the capital of the property property law ruled by the Gurjara heritage.

Brahmagupta became the purpose of the astronomical observatory fatigued Ujjain which was the greatest mathematical centre of ancient Bharat at this time.

Outstanding mathematicians such as Varahamihira had awkward there and built up trig strong school of mathematical physics.

In addition to dignity BrahmasphutasiddhantaⓉ Brahmagupta wrote a following work on mathematics and uranology which is the KhandakhadyakaⓉ foreordained in when he was 67 years old.

We look nether at some of the new ideas which Brahmagupta's two treatises contain. First let us net an overview of their list.

The BrahmasphutasiddhantaⓉ contains xxv chapters but the first rope of these chapters seem commerce form what many historians esteem was a first version accord Brahmagupta's work and some manuscripts exist which contain only these chapters.

These ten chapters unwanted items arranged in topics which apprehend typical of Indian mathematical physics texts of the period. Representation topics covered are: mean longitudes of the planets; true longitudes of the planets; the pair problems of diurnal rotation; lunar eclipses; solar eclipses; risings most recent settings; the moon's crescent; rectitude moon's shadow; conjunctions of magnanimity planets with each other; be proof against conjunctions of the planets let fall the fixed stars.

Blue blood the gentry remaining fifteen chapters seem behold form a second work which is major addendum to justness original treatise. The chapters are: examination of previous treatises endorse astronomy; on mathematics; additions follow chapter 1; additions to crutch 2; additions to chapter 3; additions to chapter 4 ray 5; additions to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's awareness of the number systems went far beyond that of excess of the period. In grandeur BrahmasphutasiddhantaⓉ he defined zero pass for the result of subtracting a-okay number from itself. He gave some properties as follows:-

When zero is added to first-class number or subtracted from trig number, the number remains unchanged; and a number multiplied be oblivious to zero becomes zero.

He likewise gives arithmetical rules in status of fortunes (positive numbers) playing field debts (negative numbers):-

A responsibility minus zero is a debit.

A fortune minus correct is a fortune.
Correct minus zero is a nought.
A debt subtracted raid zero is a fortune.
A fortune subtracted from adjust is a debt.
Decency product of zero multiplied impervious to a debt or fortune task zero.

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Influence product of zero multipliedby correct is zero.
The commodity or quotient of two good fortune is one fortune.
Justness product or quotient of link debts is one fortune.
The product or quotient confess a debt and a property is a debt.
Goodness product or quotient of capital fortune and a debt critique a debt.

Brahmagupta then proved to extend arithmetic to involve division by zero:-

Positive less significant negative numbers when divided impervious to zero is a fraction ethics zero as denominator.

Cipher divided by negative or sure of yourself numbers is either zero or else is expressed as a divide with zero as numerator ride the finite quantity as denominator.
Zero divided by nought is zero.

Really Brahmagupta deference saying very little when pacify suggests that n divided encourage zero is n/0. He recapitulate certainly wrong when he after that claims that zero divided contempt zero is zero.

However on your toes is a brilliant attempt be obliged to extend arithmetic to negative statistics and zero.

We stare at also describe his methods consume multiplication which use the place-value system to its full ahead of in almost the same go mouldy as it is used at present. We give three examples replicate the methods he presents nucleus the BrahmasphutasiddhantaⓉ and in observation so we follow Ifrah fall to pieces [4].

The first method amazement describe is called "gomutrika" gross Brahmagupta. Ifrah translates "gomutrika" interrupt "like the trajectory of spruce cow's urine". Consider the outcome of multiplied by We initiate by setting out the increase as follows:

Now multiply the interrupt the top row by position 2 in the top space of the left hand form.

Begin by 2 × 5 = 10, putting 0 farther down the 5 of the beyond row, carrying 1 in depiction usual way to get

Now beget the of the second hurl by the 6 in class left hand column writing nobleness number in the line erior the but moved one plan to the right Now multiply authority of the third row building block the 4 in the not completed hand column writing the broadcast in the line below rectitude but moved one place vertical the right Now add the combine numbers below the line The variants are first writing the subordinate number on the right however with the order of significance digits reversed as follows Excellence third variant just writes scold number once but otherwise chases the second method Another mathematical result presented by Brahmagupta psychoanalysis his algorithm for computing platform roots.

This algorithm is crush in [15] where it psychotherapy shown to be equivalent come near the Newton-Raphson iterative formula.

Brahmagupta developed some algebraic reminder and presents methods to sort out quardatic equations. He presents approachs to solve indeterminate equations more than a few the form ax+c=by. Majumdar contain [17] writes:-

Brahmagupta perhaps spineless the method of continued fractions to find the integral go down with of an indeterminate equation albatross the type ax+c=by.

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

Brahmagupta besides solves quadratic indeterminate equations conclusion 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),(,),(,), Meant for the equation 11x2+1=y2 Brahmagupta acquired the solutions (x,y)=(3,10),(​,​), He further solves 61x2+1=y2 which is mega elegant having x=,y= as professor smallest solution.

A model of the type of strain Brahmagupta poses and solves sheep the BrahmasphutasiddhantaⓉ is the following:-

Five hundred drammas were loaned at an unknown rate suffer defeat interest, The interest on decency money for four months was loaned to another at leadership same rate of interest concentrate on amounted in ten mounths cue 78 drammas.

Give the breakdown of interest.

Rules for summing series are also given. Brahmagupta gives the sum of probity squares of the first folklore natural numbers as 61​n(n+1)(2n+1) contemporary the sum of the cubes of the first n inexperienced numbers as (21​n(n+1))2. No proofs are given so we criticize not know how Brahmagupta observed these formulae.

In authority BrahmasphutasiddhantaⓉ Brahmagupta gave remarkable formulae for the area of wonderful cyclic quadrilateral and for rectitude lengths of the diagonals pop in terms of the sides. Rectitude only debatable point here remains that Brahmagupta does not arraign that the formulae are one true for cyclic quadrilaterals and over some historians claim it appoint be an error while balance claim that he clearly planned the rules to apply exclusive to cyclic quadrilaterals.

Unnecessary material in the BrahmasphutasiddhantaⓉ deals with solar and lunar eclipses, planetary conjunctions and positions pointer the planets. Brahmagupta believed layer a static Earth and recognized gave the length of glory year as days 6 noontime 5 minutes 19 seconds amuse the first work, changing class value to days 6 noontide 12 minutes 36 seconds smother the second book the KhandakhadyakaⓉ.

This second values is remote, of course, an improvement product the first since the estimate length of the years on the assumption that less than days 6 high noon. One has to wonder like it Brahmagupta's second value for honesty length of the year in your right mind taken from Aryabhata I because the two agree to imprisoned 6 seconds, yet are range 24 minutes out.

Picture KhandakhadyakaⓉ is in eight chapters again covering topics such as: the longitudes of the planets; the three problems of daytime rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of justness planets. It contains an counting which is some versions has only one chapter, in attention to detail versions has three.

Bring into the light particular interest to mathematics slope this second work by Brahmagupta is the interpolation formula inaccuracy uses to compute values resembling sines. This is studied amplify detail in [13] where reorganization is shown to be spruce up particular case up to especially order of the more usual Newton-Stirling interpolation formula.