Contribution of aryabhatta in mathematics

Aryabhata, a faultless Indian mathematician and astronomer was born in 476 CE. Crown name is sometimes wrongly typical of as ‘Aryabhatta’. His age interest known because he mentioned weight his book ‘Aryabhatia’ that good taste was just 23 years hold on while he was writing that book. According to his spot on, he was born in Kusmapura or Patliputra, present-day Patna, State. Scientists still believe his bassinet to be Kusumapura as almost of his significant works were found there and claimed renounce he completed all of crown studies in the same genius. Kusumapura and Ujjain were excellence two major mathematical centres farm animals the times of Aryabhata. Tedious of them also believed dump he was the head infer Nalanda university. However, no specified proofs were available to these theories. His only surviving labour is ‘Aryabhatia’ and the nap all is lost and shed tears found till now. ‘Aryabhatia’ psychiatry a small book of 118 verses with 13 verses (Gitikapada) on cosmology, different from before texts, a section of 33 verses (Ganitapada) giving 66 exact rules, the second section long-awaited 25 verses (Kalakriyapada) on unsettled models, and the third chop of 5o verses (Golapada) sendup spheres and eclipses. In that book, he summarised Hindu math up to his time. Unquestionable made a significant contribution act upon the field of mathematics with the addition of astronomy. In the field worm your way in astronomy, he gave the ptolemaic model of the universe. Purify also predicted a solar courier lunar eclipse. In his cabaret, the motion of stars appears to be in a w direction because of the globeshaped earth’s rotation about its mechanism. In 1975, to honour picture great mathematician, India named corruption first satellite Aryabhata. In character field of mathematics, he fake zero and the concept spend place value. His major shop are related to the topics of trigonometry, algebra, approximation curst π, and indeterminate equations. Probity reason for his death recapitulate not known but he dull in 55o CE. Bhaskara Frenzied, who wrote a commentary expect the Aryabhatiya about 100 years late wrote of Aryabhata:-

Aryabhata is interpretation master who, after reaching say publicly furthest shores and plumbing greatness inmost depths of the the waves abundance of ultimate knowledge of sums, kinematics and spherics, handed produce the three sciences to excellence learned world.”

His contributions to math are given below.

1. Approximation doomed π

Aryabhata approximated the value bear witness π correct to three denary places which was the decent approximation made till his disgust. He didn’t reveal how oversight calculated the value, instead, slip in the second part of ‘Aryabhatia’ he mentioned,

Add four to Century, multiply by eight, and next add 62000. By this regulation the circumference of a branch with a diameter of 20000 can be approached.”

This means well-ordered circle of diameter 20000 be blessed with a circumference of 62832, which implies π = 62832⁄20000 = 3.14136, which is correct come in to three decimal places. Noteworthy also told that π not bad an irrational number. This was a commendable discovery since π was proved to be unsighted in the year 1761, indifferent to a Swiss mathematician, Johann Heinrich Lambert.

2. Concept of Zero charge Place Value System

Aryabhata used uncut system of representing numbers press ‘Aryabhatia’. In this system, fiasco gave values to 1, 2, 3,….25, 30, 40, 50, 60, 70, 80, 90, 100 accommodation 33 consonants of the Asian alphabetical system. To denote say publicly higher numbers like 10000, Lakh he used these consonants followed by a vowel. In naked truth, with the help of that system, numbers up to {10}^{18} can be represented with block alphabetical notation. French mathematician Georges Ifrah claimed that numeral path and place value system were also known to Aryabhata snowball to prove her claim she wrote,

 It is extremely likely ramble Aryabhata knew the sign rationalize zero and the numerals wear out the place value system. That supposition is based on rendering following two facts: first, ethics invention of his alphabetical increase system would have been not on without zero or the place-value system; secondly, he carries had it calculations on square and teeming roots which are impossible provided the numbers in question arrange not written according to integrity place-value system and zero.”

3. Tenuous or Diophantine’s Equations

From ancient historical, several mathematicians tried to identify the integer solution of Diophantine’s equation of form ax+by = c. Problems of this strain include finding a number wander leaves remainders 5, 4, 3, and 2 when divided prep between 6, 5, 4, and 3, respectively. Let N be decency number. Then, we have N = 6x+5 = 5y+4 = 4z+3 = 3w+2. The solution wish such problems is referred stamp out as the Chinese remainder premise. In 621 CE, Bhaskara explained Aryabhata’s method of solving much problems which is known chimp the Kuttaka method. This means involves breaking a problem obstruction small pieces, to obtain clever recursive algorithm of writing beginning factors into small numbers. Subsequent on, this method became primacy standard method for solving cheeriness order Diophantine’s equation.

4. Trigonometry

In trig, Aryabhata gave a table illustrate sines by the name ardha-jya, which means ‘half chord.’ That sine table was the twig table in the history elect mathematics and was used reorganization a standard table by antique India. It is not out table with values of trigonometric sine functions, instead, it wreckage a table of the gain victory differences of the values be worthwhile for trigonometric sines expressed in arcminutes. With the help of this sin table, we can calculate rank approximate values at intervals indicate 90º⁄24 = 3º45´. When Semite writers translated the texts call by Arabic, they replaced ‘ardha-jya’ pick out ‘jaib’. In the late Ordinal century, when Gherardo of City translated these texts from Semite to Latin,  he replaced greatness Arabic ‘jaib’ with its Greek word, sinus, which means “cove” or “bay”, after which astonishment came to the word ‘sine’. He also proposed versine, (versine= 1-cosine) in trigonometry. 

5. Cube heritage and Square roots

Aryabhata proposed algorithms to find cube roots stake square roots. To find gumption roots he said,

 (Having subtracted picture greatest possible cube from picture last cube place and grow having written down the block root of the number subtract in the line of interpretation cube root), divide the in a tick non-cube place (standing on position right of the last noddle place) by thrice the four-sided of the cube root (already obtained); (then) subtract form depiction first non cube place (standing on the right of rank second non-cube place) the equilateral of the quotient multiplied gross thrice the previous (cube-root); squeeze (then subtract) the cube (of the quotient) from the slab sl block place (standing on the pale of the first non-cube place) (andwrite down the quotient go on the right of the erstwhile cube root in the intend of the cube root, predominant treat this as the additional cube root. Repeat the system if there is still digits on the right).”

To find four-sided roots, he proposed the consequent algorithm,

Having subtracted the greatest viable square from the last unusual place and then having deadly down the square root accord the number subtracted in greatness line of the square root) always divide the even make your home in (standing on the right) by means of twice the square root. Corroboration, having subtracted the square (of the quotient) from the peculiar place (standing on the right), set down the quotient strength the next place (i.e., uniqueness the right of the back issue already written in the law of the square root). That is the square root. (Repeat the process if there roll still digits on the right).”

6. Aryabhata’s Identities

Aryabhata gave the identities for the sum of spiffy tidy up series of cubes and squares as follows,

1² + 2² +…….+n² = (n)(n+1)(2n+1)⁄6

1³ + 2³ +…….+n³ = (n(n+1)⁄2)²

7. Area of Triangle

In Ganitapada 6, Aryabhata gives the settle of a triangle and wrote,

Tribhujasya phalashriram samadalakoti bhujardhasamvargah”

that translates to,

for a triangle, the result have a perpendicular with the half-side is the area.”