Bhaskaracharya biography in telugu
Bhaskara
Mahesvara himself was famed as an astrologer. That happened frequently in Indian kinship with generations of a affinity being excellent mathematicians and ofttimes acting as teachers to block out family members.
Bhaskaracharya became head of the astronomical construction at Ujjain, the leading exact centre in India at focus time.
Outstanding mathematicians such trade in Varahamihira and Brahmagupta had mannered there and built up top-hole strong school of mathematical physics.
In many ways Bhaskaracharya represents the peak of precise knowledge in the 12th c He reached an understanding depict the number systems and key equations which was not disapprove of be achieved in Europe call upon several centuries.
Six activity by Bhaskaracharya are known however a seventh work, which evaluation claimed to be by him, is thought by many historians to be a late fake. The six works are: Lilavati(The Beautiful) which is on mathematics; Bijaganita(Seed Counting or Root Extraction) which is on algebra; righteousness Siddhantasiromani which is in bend in half parts, the first on accurate astronomy with the second means on the sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya's own commentary on the Siddhantasiromani ; the Karanakutuhala(Calculation of Great Wonders) or Brahmatulya which practical a simplified version of illustriousness Siddhantasiromani ; and the Vivarana which is a commentary disrupt the Shishyadhividdhidatantra of Lalla.
Security is the first three pressure these works which are greatness most interesting, certainly from decency point of view of math, and we will concentrate manner the contents of these.
Given that he was goods on the knowledge and upheaval of Brahmagupta it is surprising that Bhaskaracharya understood get there zero and negative numbers.
Quieten his understanding went further collected than that of Brahmagupta. Collect give some examples before surprise examine his work in a-one little more detail we banknote that he knew that x2=9 had two solutions. He besides gave the formula
a±b=2a+a2−b±2a−a2−b
Bhaskaracharya studied Pell's equation px2+1=y2 adoration p = 8, 11, 32, 61 and 67.When p=61 he found the solutions x=226153980,y=1776319049. When p=67 he found loftiness solutions x=5967,y=48842. He studied myriad Diophantine problems.
Let gruelling first examine the Lilavati.
First it is worth continuance the story told by Fyzi who translated this work smart Persian in 1587. We sift the story as given close to Joseph in [5]:-
Lilavati was the name of Bhaskaracharya's maid.Biography michaelThis is a charming tale but it is hard standing see that there is impractical evidence for it being work out.From toss her horoscope, he discovered mosey the auspicious time for cook wedding would be a enormously hour on a certain period. He placed a cup snatch a small hole at excellence bottom of the vessel unabridged with water, arranged so lose concentration the cup would sink mine the beginning of the well-timed hour. When everything was motive and the cup was to be found in the vessel, Lilavati a split second out of curiosity bent talk of the vessel and a treasure requency from her dress fell experience the cup and blocked grandeur hole in it.
The blessed hour passed without the tankard sinking. Bhaskaracharya believed that decency way to console his despondent daughter, who now would at no time get married, was to draw up her a manual of mathematics!
It is not even sure that Lilavati was Bhaskaracharya's girl. There is also a judgment that Lilavati was Bhaskaracharya's little woman. The topics covered in loftiness thirteen chapters of the publication are: definitions; arithmetical terms; interest; arithmetical and geometrical progressions; aeroplane geometry; solid geometry; the darkness of the gnomon; the kuttaka; combinations.
In dealing criticism numbers Bhaskaracharya, like Brahmagupta in the past him, handled efficiently arithmetic to negative numbers. He is timbre in addition, subtraction and be in the black involving zero but realised think about it there were problems with Brahmagupta's ideas of dividing by cypher. Madhukar Mallayya in [14] argues that the zero used tough Bhaskaracharya in his rule (a.0)/0=a, given in Lilavati, is meet to the modern concept imitation a non-zero "infinitesimal".
Although that claim is not without reinforcement, perhaps it is seeing meaning beyond what Bhaskaracharya intended.
Bhaskaracharya gave two methods loom multiplication in his Lilavati. Surprise follow Ifrah who explains these two methods due to Bhaskaracharya in [4]. To multiply 325 by 243 Bhaskaracharya writes glory numbers thus:
243 243 243 3 2 5 ------------------- Now working with the rightmost of the three sums blooper computed 5 times 3 next 5 times 2 missing brawn the 5 times 4 which he did last and wrote beneath the others one tighten to the left.
Note go off this avoids making the "carry" in ones head.
243 243 243 3 2 5 ------------------- 1015 20
------------------- Now add the 1015 present-day 20 so positioned and pen the answer under the next line below the sum get the gist to the left.
243 243 243 3 2 5 ------------------- 1015 20 ------------------- 1215 Work out the middle appendix as the right-hand one, reread avoiding the "carry", and combine them writing the answer stygian the 1215 but displaced given place to the left.
243 243 243 3 2 5 ------------------- 4 6 1015 8 20 ------------------- 1215 486 Finally work out the heraldry sinister most sum in the different way and again place position resulting addition one place habitation the left under the 486.
243 243 243 3 2 5 ------------------- 6 9 4 6 1015 12 8 20 ------------------- 1215 486 729 ------------------- Finally add the several numbers below the second slope to obtain the answer 78975.
243 243 243 3 2 5 ------------------- 6 9 4 6 1015 12 8 20 ------------------- 1215 486 729 ------------------- 78975 Despite avoiding justness "carry" in the first rise, of course one is quiet faced with the "carry" be sure about this final addition.
Nobleness second of Bhaskaracharya's methods payoff as follows:
325 243 -------- Multiply the bottom publication by the top number play with the left-most digit weather proceeding towards the right.
Unsaddle depose each row one place single out for punishment start one place further modest than the previous line. Foremost step
325 243 -------- 729 Second step
325 243 -------- 729 486 Position step, then add
325 243 -------- 729 486 1215 -------- 78975 Bhaskaracharya, like profuse of the Indian mathematicians, ostensible squaring of numbers as festive cases of multiplication which payable special methods.
He gave unite such methods of squaring detect Lilavati.
Here is idea example of explanation of transposed proportion taken from Chapter 3 of the Lilavati. Bhaskaracharya writes:-
In the inverse method, greatness operation is reversed. That remains the fruit to be multiplied by the augment and bifid by the demand.As well as righteousness rule of three, Bhaskaracharya discusses examples to illustrate rules pay money for compound proportions, such as honesty rule of five (Pancarasika), rank rule of seven (Saptarasika), blue blood the gentry rule of nine (Navarasika), etc.When development increases or decreases, as high-mindedness demand is augmented or offend, the direct rule is encouraged. Else the inverse.
Critical of three inverse: If primacy fruit diminish as the summons appropriation increases, or augment as make certain decreases, they, who are fine in accounts, consider the aspire of three to be bottom up.When there is a cut of fruit, if there eke out an existence increase of requisition, and extend of fruit if there facsimile diminution of requisition, then rectitude inverse rule of three survey employed.
Bhaskaracharya's examples of using these rules are discussed in [15].
An example from Episode 5 on arithmetical and nonrepresentational progressions is the following:-
Example: On an expedition to clutch his enemy's elephants, a end marched two yojanas the be in first place day. Say, intelligent calculator, criticize what increasing rate of circadian march did he proceed, on account of he reached his foe's impediment, a distance of eighty yojanas, in a week?Bhaskaracharya shows that each day he mildew travel 722 yojanas further mystify the previous day to compass his foe's city in 7 days.
An example spread Chapter 12 on the kuttaka method of solving indeterminate equations is the following:-
Example: Remark quickly, mathematician, what is think about it multiplier, by which two tot up and twenty-one being multiplied, put forward sixty-five added to the invention, the sum divided by capital hundred and ninety-five becomes exhausted.Bhaskaracharya is finding integer indenture to 195x=221y+65.
He obtains interpretation solutions (x,y)=(6,5) or (23, 20) or (40, 35) and and on.
In the farewell chapter on combinations Bhaskaracharya considers the following problem. Let above all n-digit number be represented break through the usual decimal form in the same way
d1d2...dn(*)
where each digit satisfies 1≤dj≤9,j=1,2,...,n.Then Bhaskaracharya's problem give something the onceover to find the total integer of numbers of the epileptic fit (*) that satisfy
d1+d2+...+dn=S.
Brush his conclusion to Lilavati Bhaskaracharya writes:-Joy and happiness disintegration indeed ever increasing in that world for those who receive Lilavati clasped to their throats, decorated as the members sentinel with neat reduction of fractions, multiplication and involution, pure leading perfect as are the solutions, and tasteful as is description speech which is exemplified.Rectitude Bijaganita is a work cry twelve chapters.
The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations with advanced than one unknown; quadratic equations with more than one unknown; operations with products of various unknowns; and the author gleam his work.
Having explained how to do arithmetic twig negative numbers, Bhaskaracharya gives crushing to test the abilities have a high opinion of the reader on calculating cop negative and affirmative quantities:-
Example: Tell quickly the result elect the numbers three and cardinal, negative or affirmative, taken together; that is, affirmative and anti, or both negative or both affirmative, as separate instances; hypothesize thou know the addition castigate affirmative and negative quantities.Disputing numbers are denoted by evaluation a dot above them:-
The characters, denoting the quantities accustomed and unknown, should be greatest written to indicate them generally; and those, which become interdict should be then marked work stoppage a dot over them.In Bijaganita Bhaskaracharya attempted to improve on Brahmagupta's arrive at to divide by zero (and his own description in Lilavati) when he wrote:-
Example: Subtracting two from duo, affirmative from affirmative, and contradictory from negative, or the opposing, tell me quickly the elucidation ...
A amount divided by zero becomes unmixed fraction the denominator of which is zero.Positive Bhaskaracharya tried to solve distinction problem by writing n/0 = ∞.This fraction equitable termed an infinite quantity. Drop this quantity consisting of ramble which has zero for hang over divisor, there is no adjustment, though many may be inserted or extracted; as no moderate takes place in the unlimited and immutable God when apples are created or destroyed, allowing numerous orders of beings net absorbed or put forth.
At first sight miracle might be tempted to deem that Bhaskaracharya has it right, but of course he does not. If this were genuine then 0 times ∞ be obliged be equal to every calculate n, so all numbers downright equal. The Indian mathematicians could not bring themselves to position point of admitting that lag could not divide by set.
Equations leading to make more complicated than one solution are obtain by Bhaskaracharya:-
Example: Inside out forest, a number of apes equal to the square holiday one-eighth of the total apes in the pack are playacting noisy games. The remaining dozen apes, who are of clever more serious disposition, are accord a nearby hill and burning by the shrieks coming cause the collapse of the forest.The problem leads to a quadratic equation gift Bhaskaracharya says that the fold up solutions, namely 16 and 48, are equally admissible.What is rank total number of apes efficient the pack?
Rectitude kuttaka method to solve tenuous equations is applied to equations with three unknowns.
The disconcert is to find integer solutions to an equation of description form ax+by+cz=d. An example operate gives is:-
Example: The roots belonging to four men watchdog 5, 3, 6 and 8. The camels belonging to interpretation same men are 2, 7, 4 and 1. The scuffs belonging to them are 8, 2, 1 and 3 stand for the oxen are 7, 1, 2 and 1.Of course such constraints do not have a solitary solution as Bhaskaracharya is magnificently aware. He finds one hole, which is the minimum, to wit horses 85, camels 76, scuff 31 and oxen 4.all match up men have equal fortunes. Emotion me quickly the price become aware of each horse, camel, mule subject ox.
Bhaskaracharya's conclusion to the Bijaganita is fascinating for the sympathy it gives us into grandeur mind of this great mathematician:-
A morsel of tuition conveys knowledge to a comprehensive mind; and having reached it, expands of its own impulse, similarly oil poured upon water, by the same token a secret entrusted to significance vile, as alms bestowed prep atop the worthy, however little, as follows does knowledge infused into graceful wise mind spread by basic force.Rank Siddhantasiromani is a mathematical physics text similar in layout concord many other Indian astronomy texts of this and earlier periods.
It is evident to men of clear mistake, that the rule of match up terms constitutes arithmetic and intelligence constitutes algebra. Accordingly I hold said ... The rule depart three terms is arithmetic; undefiled understanding is algebra. What equitable there unknown to the intelligent? Therefore for the dull get round it is set forth.
The twelve chapters of prestige first part cover topics much as: mean longitudes of decency planets; true longitudes of probity planets; the three problems forged diurnal rotation; syzygies; lunar eclipses; solar eclipses; latitudes of righteousness planets; risings and settings; say publicly moon's crescent; conjunctions of loftiness planets with each other; conjunctions of the planets with excellence fixed stars; and the catarrhine of the sun and communications satellit.
The second part contains thirteen chapters on the globule. It covers topics such as: praise of study of integrity sphere; nature of the sphere; cosmography and geography; planetary mode motion; eccentric epicyclic model fail the planets; the armillary sphere; spherical trigonometry; ellipse calculations; leading visibilities of the planets; shrewd the lunar crescent; astronomical instruments; the seasons; and problems make acquainted astronomical calculations.
There dash interesting results on trigonometry expect this work. In particular Bhaskaracharya seems more interested in trig for its own sake rather than his predecessors who saw blow only as a tool espousal calculation. Among the many compelling results given by Bhaskaracharya are:
sin(a+b)=sinacosb+cosasinb
andsin(a−b)=sinacosb−cosasinb.
Bhaskaracharya precisely achieved an outstanding reputation keep his remarkable contribution.In 1207 an educational institution was oversensitive up to study Bhaskaracharya's mill. A medieval inscription in characteristic Indian temple reads:-
Triumphant obey the illustrious Bhaskaracharya whose feats are revered by both dignity wise and the learned. Expert poet endowed with fame add-on religious merit, he is cherish the crest on a peacock.It is from this quote that the title of Joseph's book [5] comes.