# Aryabhata’s Contributions in Mathematics Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.”

His contributions to mathematics are given below.

Index of Article (Click to Jump)

## 1. Approximation of π

Aryabhata approximated the value of π correct to three decimal places which was the best approximation made till his time. He didn’t reveal how he calculated the value, instead, in the second part of ‘Aryabhatia’ he mentioned,

Add four to 100, multiply by eight, and then add 62000. By this rule the circumference of a circle with a diameter of 20000 can be approached.”

This means a circle of diameter 20000 have a circumference of 62832, which implies π = 62832⁄20000 = 3.14136, which is correct up to three decimal places. He also told that π is an irrational number. This was a commendable discovery since π was proved to be irrational in the year 1761, by a Swiss mathematician, Johann Heinrich Lambert.

## 2. Concept of Zero and Place Value System

Aryabhata used a system of representing numbers in ‘Aryabhatia’. In this system, he gave values to 1, 2, 3,….25, 30, 40, 50, 60, 70, 80, 90, 100 using 33 consonants of the Indian alphabetical system. To denote the higher numbers like 10000, 100000 he used these consonants followed by a vowel. In fact, with the help of this system, numbers up to {10}^{18} can be represented with an alphabetical notation. French mathematician Georges Ifrah claimed that numeral system and place value system were also known to Aryabhata and to prove her claim she wrote,

It is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.”

## 3. Indeterminate or Diophantine’s Equations

From ancient times, several mathematicians tried to find the integer solution of Diophantine’s equation of form ax+by = c. Problems of this type include finding a number that leaves remainders 5, 4, 3, and 2 when divided by 6, 5, 4, and 3, respectively. Let N be the number. Then, we have N = 6x+5 = 5y+4 = 4z+3 = 3w+2. The solution to such problems is referred to as the Chinese remainder theorem. In 621 CE, Bhaskara explained Aryabhata’s method of solving such problems which is known as the Kuttaka method. This method involves breaking a problem into small pieces, to obtain a recursive algorithm of writing original factors into small numbers. Later on, this method became the standard method for solving first order Diophantine’s equation.

## 4. Trigonometry

In trigonometry, Aryabhata gave a table of sines by the name ardha-jya, which means ‘half chord.’ This sine table was the first table in the history of mathematics and was used as a standard table by ancient India. It is not a table with values of trigonometric sine functions, instead, it is a table of the first differences of the values of trigonometric sines expressed in arcminutes. With the help of this sine table, we can calculate the approximate values at intervals of 90º⁄24 = 3º45´. When Arabic writers translated the texts to Arabic, they replaced ‘ardha-jya’ with ‘jaib’. In the late 12th century, when Gherardo of Cremona translated these texts from Arabic to Latin,  he replaced the Arabic ‘jaib’ with its Latin word, sinus, which means “cove” or “bay”, after which we came to the word ‘sine’. He also proposed versine, (versine= 1-cosine) in trigonometry. ## 5. Cube roots and Square roots

Aryabhata proposed algorithms to find cube roots and square roots. To find cube roots he said, (Having subtracted the greatest possible cube from the last cube place and then having written down the cube root of the number subtracted in the line of the cube root), divide the second non-cube place (standing on the right of the last cube place) by thrice the square of the cube root (already obtained); (then) subtract form the first non cube place (standing on the right of the second non-cube place) the square of the quotient multiplied by thrice the previous (cube-root); and (then subtract) the cube (of the quotient) from the cube place (standing on the right of the first non-cube place) (andwrite down the quotient on the right of the previous cube root in the line of the cube root, and treat this as the new cube root. Repeat the process if there is still digits on the right).”

To find square roots, he proposed the following algorithm, Having subtracted the greatest possible square from the last odd place and then having written down the square root of the number subtracted in the line of the square root) always divide the even place (standing on the right) by twice the square root. Then, having subtracted the square (of the quotient) from the odd place (standing on the right), set down the quotient at the next place (i.e., on the right of the number already written in the line of the square root). This is the square root. (Repeat the process if there are still digits on the right).”

## 6. Aryabhata’s Identities

Aryabhata gave the identities for the sum of a 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 area of a triangle and wrote,

that translates to,

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