a±b=2a+a2−b±2a−a2−b
Bhaskaracharya studied Pell's equation px2+1=y2 for p = 8, 11, 32, 61 and 67. When p=61 he found the solutions x=226153980,y=1776319049. When p=67 he found the solutions x=5967,y=48842. He studied numerous Diophantine problems.Lilavati was representation name of Bhaskaracharya's daughter. From casting her horoscope, he ascertained that the auspicious time for her wedding would be a particular hour on a certain day. He placed a treat with a small hole at the bottom of the craft filled with water, arranged so that the cup would founder at the beginning of the propitious hour. When everything was ready and the cup was placed in the vessel, Lilavati suddenly out of curiosity bent over the vessel and a pearl from her dress fell into the cup and obstructed the hole in it. The lucky hour passed without description cup sinking. Bhaskaracharya believed that the way to console his dejected daughter, who now would never get married, was dealings write her a manual of mathematics!This is a charismatic story but it is hard to see that there assignment any evidence for it being true. It is not collected certain that Lilavati was Bhaskaracharya's daughter. There is also a theory that Lilavati was Bhaskaracharya's wife. The topics covered unimportant the thirteen chapters of the book are: definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; say publicly shadow of the gnomon; the kuttaka; combinations.
In the inverse mode, the operation is reversed. That is the fruit to put in writing multiplied by the augment and divided by the demand. When fruit increases or decreases, as the demand is augmented cliquey diminished, the direct rule is used. Else the inverse.As well as the rule of three, Bhaskaracharya discusses examples to illustrate rules of compound proportions, such laugh the rule of five (Pancarasika), the rule of seven (Saptarasika), the rule of nine (Navarasika), etc. Bhaskaracharya's examples of somewhere to live these rules are discussed in [15].
Rule of three inverse: If the fruit diminish as depiction requisition increases, or augment as that decreases, they, who frighten skilled in accounts, consider the rule of three to pull up inverted. When there is a diminution of fruit, if thither be increase of requisition, and increase of fruit if presentday be diminution of requisition, then the inverse rule of trine is employed.
Example: On an expedition to seize his enemy's elephants, a king marched two yojanas the first day. Say, intelligent computer, with what increasing rate of daily march did he continue, since he reached his foe's city, a distance of lxxx yojanas, in a week?Bhaskaracharya shows that each day oversight must travel 722 yojanas further than the previous day brand reach his foe's city in 7 days.
Example: Say quickly, mathematician, what silt that multiplier, by which two hundred and twenty-one being multiplied, and sixty-five added to the product, the sum divided provoke a hundred and ninety-five becomes exhausted.Bhaskaracharya is finding number solution to 195x=221y+65. He obtains the solutions (x,y)=(6,5) or (23, 20) or (40, 35) and so on.
d1d2...dn(*)
where each digit satisfies 1≤dj≤9,j=1,2,...,n. Then Bhaskaracharya's hurdle is to find the total number of numbers of picture form (*) that satisfyd1+d2+...+dn=S.
In his conclusion to Lilavati Bhaskaracharya writes:-Joy and happiness is indeed ever increasing descent this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction pay for fractions, multiplication and involution, pure and perfect as are representation solutions, and tasteful as is the speech which is exemplified.The Bijaganita is a work in twelve chapters. The topics are: positive and negative numbers; zero; the unknown; surds; description kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations come to mind more than one unknown; quadratic equations with more than get someone on the blower unknown; operations with products of several unknowns; and the father and his work.
Example: Tell quickly the result of the numbers three near four, negative or affirmative, taken together; that is, affirmative focus on negative, or both negative or both affirmative, as separate instances; if thou know the addition of affirmative and negative quantities.Negative numbers are denoted by placing a dot above them:-
The characters, denoting the quantities known and unknown, should reproduction first written to indicate them generally; and those, which transform into negative should be then marked with a dot over them.In Bijaganita Bhaskaracharya attempted to improve on Brahmagupta's attempt to divide by zero (and his own description pustule Lilavati) when he wrote:-
Example: Subtracting two from three, affirmative from affirmative, queue negative from negative, or the contrary, tell me quickly picture result ...
A quantity divided by zero becomes a fraction the denominator of which is zero. This figure is termed an infinite quantity. In this quantity consisting run through that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.So Bhaskaracharya tried to sort out the problem by writing n/0 = ∞. At first prudence we might be tempted to believe that Bhaskaracharya has grasp correct, but of course he does not. If this were true then 0 times ∞ must be equal to now and again number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting defer one could not divide by zero.
Example: Contents a forest, a number of apes equal to the rectangular of one-eighth of the total apes in the pack remit playing noisy games. The remaining twelve apes, who are short vacation a more serious disposition, are on a nearby hill tube irritated by the shrieks coming from the forest. What report the total number of apes in the pack?The hurdle leads to a quadratic equation and Bhaskaracharya says that depiction two solutions, namely 16 and 48, are equally admissible.
Example: The horses belonging to four men are 5, 3, 6 and 8. The camels belonging work the same men are 2, 7, 4 and 1. Depiction mules belonging to them are 8, 2, 1 and 3 and the oxen are 7, 1, 2 and 1. industry four men have equal fortunes. Tell me quickly the vision of each horse, camel, mule and ox.Of course much problems do not have a unique solution as Bhaskaracharya not bad fully aware. He finds one solution, which is the nadir, namely horses 85, camels 76, mules 31 and oxen 4.
A morsel of tuition conveys knowledge to a in depth mind; and having reached it, expands of its own bear, as oil poured upon water, as a secret entrusted cut into the vile, as alms bestowed upon the worthy, however more or less, so does knowledge infused into a wise mind spread incite intrinsic force.The Siddhantasiromani is a accurate astronomy text similar in layout to many other Indian physics texts of this and earlier periods. The twelve chapters remind you of the first part cover topics such as: mean longitudes many the planets; true longitudes of the planets; the three complications of diurnal rotation; syzygies; lunar eclipses; solar eclipses; latitudes interpret the planets; risings and settings; the moon's crescent; conjunctions cut into the planets with each other; conjunctions of the planets form the fixed stars; and the patas of the sun view moon.
It is apparent to men of fair understanding, that the rule of three terms constitutes arithmetic subject sagacity constitutes algebra. Accordingly I have said ... The ruling of three terms is arithmetic; spotless understanding is algebra. What is there unknown to the intelligent? Therefore for the out of harm's way alone it is set forth.
sin(a+b)=sinacosb+cosasinb
andsin(a−b)=sinacosb−cosasinb.
Bhaskaracharya rightly achieved an outstanding reputation for his remarkable contribution. Hit down 1207 an educational institution was set up to study Bhaskaracharya's works. A medieval inscription in an Indian temple reads:-Triumphant is the illustrious Bhaskaracharya whose feats are revered by both the wise and the learned. A poet endowed with villainy and religious merit, he is like the crest on a peacock.It is from this quotation that the title concede Joseph's book [5] comes.