Indian Classical period starts around 400 AD and ends in 1200 AD. This period is often considered as the golden period for Indians in terms of science, technology, mathematics, astronomy, engineering, art, logic, philosophy, and religion. Restricting us to mathematics, people like were born in this period and had made immense contributions in the field of mathematics and astronomy. [wptabs style=”wpui-narrow” effect=”slide” mode=”horizontal”] [wptabtitle] Aryabhata[/wptabtitle] [wptabcontent] It was this period in which Aryabhata was the first to tell many astronomical facts like earth is round, sun is the center point of solar system and all the planets revolve around sun, earth spins around its own axis, cause of day and night, etc. He also made huge contributions in mathematics as well. Although ancient Indians had been following decimal numeral system from the Harappan civilization, still there was no place value system. Decimal place value system was seen in Aryabhata’s Aryabhatiya book and this is considered as the turning point of mathematics and biggest contribution in the field of mathematics **which gave rise to ‘0’** . Aryabhata didn’t invent ‘0’ symbol. Infact he did not use any of the symbols (1-9) though Brahmi numerals were widely used around same time but he did invent a place value system where ‘0’ can be fixed (place holder). Also he used words for the numbers and to denote the place value system. He used the word to denote nothing (0) and it was later used as **“ZERO”**.*(see more in Hindu Numerals goes West Section)*

So Aryabhata is considered as one resposible for the introduction of 0 and the place value system in mathematics. Aryabhata was born in Taregna, small town in Bihar. For study purpose he had gone to Kusumapura (town in Bihar) and stayed there for long time.He had many contributions in the field of mathematics like arithmetic, algebra, geometry, plane & spherical trigonometry, Fractions, Equations, etc. At the age of 23 he wrote book which is a treatises on mathematics and astronomy. Along with Aryabhatia, he wrote couple of books, but only Aryabhatia could survive. Aryabhatia is divided in 4 padas/chapters:

- : Large unit of time-kalpa, manvantra, and yuga Eg: The duration of the planetary revolutions during a mahayuga is given as 4.32 million years with 1st ever sine table(see below)
- :
*See below for more deails* - : Different unit of time and their division (days, months, years) according movement of celestial bodies.
- : Relationship between Earth and Cosmos: Shape of Earth, cause of day and night, etc

### Aryabhata’s numeral table:

It is a system of numerals based on Sanskrit phonemes and was mentioned in *Gitikapada* of *Aryabhatiya*. Aryabhata’s numeral table

### Sine Table:

Aryabhata mentioned 1st ever sine (jya) table for 25 values in just 1 stanza.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |

घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

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### Approximation of pi:

Aryabhata worked on approximation of pi and concluded it as irrational number.He mentions

*caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.* i.e.

“Add four to 100, multiply by eight, and then add 62,000.

By this rule the circumference of a circle with a diameter of 20,000 can be approached.”

This implies that the ratio of the circumference to the diameter is

((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416.

### Astronomy:

Aryabhata could calculate 1 day as 23:56:4.1 and the modern value is 23:56:4.091.

Also the length of year as 365 days, 6 hours, 12 minutes, and 30 seconds while modern value is 365 days, 6 hours, 9 minutes, and 10 seconds which are very close to each other.

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[wptabtitle] Varahamihira[/wptabtitle] [wptabcontent]

Varahamihira (505 – 587 AD) lived in Ujjain which was center of Hindu mathematics at that time.

Very less is known about Varahamihira in terms of mathematics but he and his son made huge contributions in terms of astrology, astronomy and mathematics. He is well known for 2 books *‘Pancha Siddhantika’* which had topics on mathematics and astronomy and *‘Brihat Samhita’* had information related to planetary movements, astrology, human interest like relations, architecture, precious stones, etc.

Restricting us to mathematics. Varahamihira is known for discovery of very basic trigonometric formulae’s like

sin^{2}x + cos^{2}x = 1

sinx = cos (Π/2 – x)

1-cos2x = 2sin^{2}x

Varahamihira improved the accuracy of the sine tables of Aryabhata I and calculated the binomial coefficients, known in the European civilization as Pascal’s triangle.

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[wptabtitle] Brahmagupta[/wptabtitle] [wptabcontent]

Brahmagupta (598 – 668 AD) was born in Bhinmal town of Rajasthan.

It is however said that he lived and worked later in Ujjain where Varahamihira worked previously. He had large contribution in the field of mathematics and astronomy and was best known for his work in *‘Brahmasputa Siddhanta’* in the year 628 AD. He along with **Bhaskara I were the 1st to use the ‘0’ symbol** in their respective Siddhantas.

Brahmagupta had contributions with the modern integers, method to calculate square roots, solving linear and quadratic equations, Series summation, Fibonacci identity, etc

Brahmagupta stated the rules to work with zero and negative numbers.

He dealt with zero, positive (property) and negative (debt) numbers are follows:

- When zero is added to a number or subtracted from a number, the number remains unchanged
- A number multiplied by zero becomes zero.
- A debt minus zero is a debt
- A fortune minus zero is a fortune.
- Zero minus zero is a zero.
- A debt subtracted from zero is a fortune.
- A fortune subtracted from zero is a debt.
- The product of zero multiplied by a debt or fortune is zero.
- The product of zero multiplied by zero is zero.
- The product or quotient of two fortunes is one fortune.
- The product or quotient of two debts is one fortune.
- The product or quotient of a debt and a fortune is a debt.
- The product or quotient of a fortune and a debt is a debt.
- But he was wrong when it came to concept of division by 0, he said “
“*Zero divided by zero is zero**(Later corrected by Bhaskara II)*.

### Algorithm for calculating squareroot:

*Self explainatary; Every 2 steps are repeated till 0 is obtained else continue to get answer in decimal form; Hence applicable to Perfect/Non-perfect square*

### Brahmagupta’s Formula:

Area of cyclic quadrilateral

Brahmagupta calculated the area of cyclic quadrilateral (quadrilateral inscribed in a circle) which is very similar to that Herons for triangle.

K = √[(s-a)(s-b)(s-c)(s-d)]

where s, the semiperimeter, is defined to be

s = ½ * (a + b + c+ d)

Or

K = ¼ * √[(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)]

Area of cyclic quadrilateral

K = √[(s-a)(s-b)(s-c)(s-d) – abcdcos2x]

Another Brahmagupta’s Formula for cyclic quadrilateral is

“If a cyclic quadrilateral has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite sides”.

### Summation of squares and cubes:

1^{2} + 2^{2} + 3^{2} + ….. + n^{2} = 1/6 * n (n+1) (2n+1)

1^{3} + 2^{3} + 3^{3} + ….. + n^{3} = (1+2+3+….+n)^{2}

**Gravitional Force**as “Bodies fall towards the Earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow”.It is believed that Arabs have learnt from Brahmagupta. Muhammad al-Fazari translated Brahmugupta’s work into Arabic and which gave rise to Arab astronomy and their calendar.

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[wptabtitle] Bhaskara I[/wptabtitle] [wptabcontent]

Bhaskara I (600 – 680 AD) was born in Parbani district of Maharashtra.

He wrote commentary on Aryabhatiya in 629 and named it as *Aryabhatiyabhashya* which had contents/comments on Aryabhata’s work of mathematics and astronomy. He also wrote 2 more books called as *Mahabhaskariya* and *Lahubhaskariya* which were again had topics of mathematics and astronomy.

Bhaskara along with Brahmagupta were the the 1st to make use of number ‘0’ in their mathematical books and like Aryabhata categorized pi as irrational number.

### Algebraic formula for sine function

He gave rational approximation of sine function in book *Aryabhatiyabhashya* as

sin x = 16x (π – x)/[5 π^{2} – 4x (π – x)] for the angles between 0 and 90 degrees which reveals a relative error less than 1.9%.

He also had contribution in other mathematics topics like solving linear equations, quadratic equations, indeterminate equations, etc and in astronomical topics like determining longitudes of planets, association of planets with each other, Eclipses, Rising and setting phenomenon of sun, etc.

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[wptabtitle] Mahavira[/wptabtitle] [wptabcontent]

Mahavira (around 850 AD) was a jain mathematician from Gulbarg town of Karnataka.

He wrote a book on mathematics called *‘Ganita Sarasangraha’*. Because of his work he became very popular in south in those days.

### Square root:

He was the 1st to tell that square root of negative number does not exist.

### Volume of sphere:

He provided the volume of sphere as 9/2 * (d/2)^{3} which is very close to the actual volume 9/10 * 9/2 * (d/2)^{3}.

### Cyclic Quadrilateral:

He provided formulae to calculate area and perimeter of cyclic quadrilateral along with that of ellipses.

He also provided the ways of getting length of diagonals if all the sides of quadrilateral are known as

x = √[(ad+bc)(ac+bd) / (ab+cd)]

y = √[(ab+cd)(ac+bd) / (ad+bc)]

Then xy = ac + bd

where x & y are the diagonals and a, b, c & d are sides of cyclic quadrilateral.

### Combinations:

He provided the general formula for Combinations as

^{n}C_{r} = {n(n – 1)(n – 2)…(n – r + 1)}/1.2.3. … .r

### Quadratic Equations:

Mahavira knew that the quadratic equation have 2 roots.

x = ½ * [b/a ± √[(b/a – 4c)b/a]

### Geometric Progression:

He gave the formula as: a (r^{n} – 1) / (r-1)

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[wptabtitle] Bhaskara II[/wptabtitle] [wptabcontent]

Bhaskara II or Bhaskaracharya (Bhaskara, The Teacher) (1114 – 1185 AD) was born in Vijayawada, Karnataka.

He was the head of astronomical observatory in Ujjan, the leading mathematical center of ancient India. He was mathematician and an astronomer and is considered as one of the greatest mathematician along with Aryabhata and Madhava.

Bhaskara II dealt with arithmetic, algebra, solving equations, trigonometric and differential calculus. Bhaskara had shown very good amount of work related to calculus. Bhaskara used his vital mathematical on astronomy to calculate exact astronomical quantities.

He wrote *Siddhanta Shiromani* around in the year 1150 AD which had 4 sections:

- Lilavati (Arithmatic)
- Bijganita (Algebra)
- Goladhyaya (Spheres/Earth)
- Grahaganita (Mathematics of planets)

Lilavati and Bijaganita together had considerable amount of mathematics like algebra, geometry, basic arithmetical operations like multiplication, division, squaring , square roots, rules of division by 5,7 etc; solving linear, quadratic and cubic equations; plane and solid geometry; Arithmatic and geometric progressions, Estimation of pi, working with negative numbers and Zero; Surds; also solving the indeterminate equations of 1st (‘Kuttaka’) and 2nd order (‘Chakraval’).

- He also corrected Brahmagupta division by Zero concept by stating
*Any Number divided by 0 yields infinite* - He also worked with polygon having more than 96 sides and thus arriving at very good approximation of pi as
**3.141666**. - He was the first to provide the following trigonometric important formulae:
sin (a + b) = sin a cos b + cos a sin b

sin (a – b) = sin a cos b – cos a sin b. - Bhaskara II also worked on calculus almost 500 years before Newton. He is particularly known for the discovery of the principles of differential calculus and its application to astronomical problems and computations. He was the first to conceive the differential coefficient and differential calculus.

Bhaskara II died in year 1185 and this brought to the end of Golden Era (Classical Age) of Mathematics.

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Lata maganur says

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Lata maganur says

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