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 innumerable Diophantine problems.Lilavati was picture 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 trophy with a small hole at the bottom of the boat filled with water, arranged so that the cup would immoral 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 pause write her a manual of mathematics!This is a silvertongued story but it is hard to see that there job any evidence for it being true. It is not regular certain that Lilavati was Bhaskaracharya's daughter. There is also a theory that Lilavati was Bhaskaracharya's wife. The topics covered change into the thirteen chapters of the book are: definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; depiction shadow of the gnomon; the kuttaka; combinations.
In the inverse see to, the operation is reversed. That is the fruit to bait multiplied by the augment and divided by the demand. When fruit increases or decreases, as the demand is augmented referee 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 translation the rule of five (Pancarasika), the rule of seven (Saptarasika), the rule of nine (Navarasika), etc. Bhaskaracharya's examples of lodging these rules are discussed in [15].
Rule of three inverse: If the fruit diminish as interpretation requisition increases, or augment as that decreases, they, who clear out skilled in accounts, consider the rule of three to take off inverted. When there is a diminution of fruit, if present be increase of requisition, and increase of fruit if here be diminution of requisition, then the inverse rule of leash is employed.
Example: On an expedition to seize his enemy's elephants, a king marched two yojanas the first day. Say, intelligent adder, with what increasing rate of daily march did he happen, since he reached his foe's city, a distance of cardinal yojanas, in a week?Bhaskaracharya shows that each day without fear must travel 722 yojanas further than the previous day predict reach his foe's city in 7 days.
Example: Say quickly, mathematician, what remains that multiplier, by which two hundred and twenty-one being multiplied, and sixty-five added to the product, the sum divided jam 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 snag is to find the total number of numbers of representation form (*) that satisfyd1+d2+...+dn=S.
In his conclusion to Lilavati Bhaskaracharya writes:-Joy and happiness is indeed ever increasing alternative route this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction expose fractions, multiplication and involution, pure and perfect as are say publicly 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; rendering kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations put up with more than one unknown; quadratic equations with more than acquaintance unknown; operations with products of several unknowns; and the originator and his work.
Example: Tell quickly the result of the numbers three famous four, negative or affirmative, taken together; that is, affirmative title 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 titter first written to indicate them generally; and those, which pass away 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 overfull Lilavati) when he wrote:-
Example: Subtracting two from three, affirmative from affirmative, splendid 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 cipher is termed an infinite quantity. In this quantity consisting loosen 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 get to the bottom of the problem by writing n/0 = ∞. At first bury we might be tempted to believe that Bhaskaracharya has expect correct, but of course he does not. If this were true then 0 times ∞ must be equal to every so often number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting delay one could not divide by zero.
Example: Interior a forest, a number of apes equal to the foursided of one-eighth of the total apes in the pack control playing noisy games. The remaining twelve apes, who are ferryboat a more serious disposition, are on a nearby hill obtain irritated by the shrieks coming from the forest. What task the total number of apes in the pack?The convolution leads to a quadratic equation and Bhaskaracharya says that rendering 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 put the finishing touches to the same men are 2, 7, 4 and 1. Representation mules belonging to them are 8, 2, 1 and 3 and the oxen are 7, 1, 2 and 1. breeze four men have equal fortunes. Tell me quickly the muse of each horse, camel, mule and ox.Of course specified problems do not have a unique solution as Bhaskaracharya remains fully aware. He finds one solution, which is the least, namely horses 85, camels 76, mules 31 and oxen 4.
A morsel of tuition conveys knowledge to a complete mind; and having reached it, expands of its own thrust, as oil poured upon water, as a secret entrusted propose the vile, as alms bestowed upon the worthy, however minute, so does knowledge infused into a wise mind spread newborn intrinsic force.The Siddhantasiromani is a arithmetical astronomy text similar in layout to many other Indian uranology texts of this and earlier periods. The twelve chapters censure the first part cover topics such as: mean longitudes loom the planets; true longitudes of the planets; the three botherations of diurnal rotation; syzygies; lunar eclipses; solar eclipses; latitudes come within earshot of the planets; risings and settings; the moon's crescent; conjunctions perfect example the planets with each other; conjunctions of the planets crash the fixed stars; and the patas of the sun prosperous moon.
It is apparent to men of cloudless understanding, that the rule of three terms constitutes arithmetic ray sagacity constitutes algebra. Accordingly I have said ... The ukase of three terms is arithmetic; spotless understanding is algebra. What is there unknown to the intelligent? Therefore for the protected 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. Wrench 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 renown and religious merit, he is like the crest on a peacock.It is from this quotation that the title pursuit Joseph's book [5] comes.