Aryabhatta Math Homework Images For Kids


ARYABHATTA (476 - 550)




Born

476 AD (1707-04-15)
 

Died

550 AD

Residence

Kusumapura or Pataliputra (Patna)

Nationality

Indian

Fields

Mathematics, Astronomy

Institutions

Nalanda University


Aryabhatta was the first of great Hindu mathematician. He is also known as Aryabhatta I. He lived at Kusumapura or Pataliputra in ancient Magadhar or modern Patna.  He was born in 476 AD.

 At the age of 23 years Aryabhatt wrote two books on astronomy (1) Aryabhatiya (2) Arya-siddhanta. The aryabhatt deals with both mathematics and astronomy. It contains 121 stanzas in all. Aryabhatt is divided into 4 chapters called Pada (section)

Pada 1 – GitikaPada – 13 stanzas of basis definition of important astronomical parameters and tables.

Pada 2 – Ganita Pada – 33 stanzas deals with mathematics. The topics are geometrical figures with their properties and mensurations, series, linear and quadratic equations, methods for extracting the square roots, the cube roots etc.

Pada 3 – Kalakriya Pada – 25 stanzas deals with the true position of sun, moon and planets.

Pada 4 – Gola Pada – 50 stanzas deals with the motion of sun, moon and planets on the celestial sphere.


ARYABHATTA CONTRIBUTIONS TO MATHEMATICS

NUMBER NOTATION

Ø  Numerical values: he made a notation system in which digits are denoted with the help of alphabet numerals e.g., 1 = ka, 2 = Kha, etc.

Aryabhatta assigned numerical values to the 33 consonants of the Indian alphabet to represent 1,2,3…25,30,40,50,60,70,80,90,100.

Ø  Notation system: He invented a notation system consisting of alphabet numerals Digits were denoted by alphabet numerals. In this system devanagiri script contain varga letters (consonants) and avarga letters (vowels).1-25 are denoted by 1st 25 varga letters.

Ø  Place-value: Aryabhatta was familiar with the place-value system.

Ø  He knew numeral symbols and the sign for zero

Ø  Square root & cube root: His calculations on square root and cube root would not have been possible without the knowledge of place values system and zero. He has given methods of extracting square root cube root along with their explanation.

Ø  Interest: He formulated for the first time in India the formula for interest, time and other related ones, in the problems of interest.

ALGEBRA

Ø  Integer solutions: Aryabhatta was the first one to explore integer solutions to the equations of the form by =ax+c and by =ax-c, where a,b,c are integers. He used kuttuka method to solve problems.

Ø  Indeterminate equations:He gave general solutions to linear indeterminate equations ax+by+c= 0 by the method of continued fraction.

Ø  Identities: He had dealt with identities like (a+b)2=a2+2ab+b2and ab={(a+b)2-(a2-b2)}/2

Ø  He has given the following formula in aryabhatia

12+22+32+---------+n2=n(n+1)(2n+1)/6

13+23+33+---------+n3 = (1+2+3+------------+)2= {n2(n+1)2}/4

Ø  Algebraic quantities: He has given the method of addition, subtraction, multiplication of simple and compound algebraic quantities

Ø  Arithmetic series: He was given a formula for summing up of the arithmetic series after the Pth term The rule is S= n[a+{(n-1)/2+p} d]

S=(a+1) n/2

GEOMETRY

Ø  Discover the P Value : The credit for discovering the exact values P may be ascribed to the celebrated mathematician Aryabhatta.

Rule: Add 4 to 100, multiply by 8, add 62000. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.

This gives P =62832/20000=3.1416. Which is an accurate value of P. Aryabhatta discovered this value independently and also realized that P is an irrational number

Ø  Pythagorean Theorem: The Pythagorean theorem is stated as follows in his work “the square of the Bhuja (base) plus the square of the koti (perpendicular) is the square of the Karna”

(Buja and koti are the sides of a right-angled triangle. The Karna is the hypotenuse)

Ø  Circle Theorem: He has postulated a theorem relating to circle as follows “In a circle the product of two Saras is the square of the half chord of the two arcs” i.e. a*b=c2 where c is half the chord and the saras or arrows are the segments of a diameter which bisect any chord.

Ø  Formula: Aryabhatta gives formulae for the areas of a triangle, square, rectangle, rhombus, circle etc.

TRIGONOMETRY

Ø  Sine Table: Aryabhatta gave a table of sines for calculating the approximate values at intervals of 90/24 = 3 45’. This was done using the formula for

sin (n+1)x -  sin nx in terms of sin nx and sin (n-1) x.

Ø  Versine: He introduced the versine (versin = 1-cosine) into trigonometry.

ASTRONOMY

Ø  Earth: Aryabhatta gave the circumference of the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas. Since1 yojana =5miles this gives the circumference as 24,835 miles, which is an excellent approximation to the currently accepted value of 24,902 miles.

Ø  He believes that the orbits of the planets are ellipses. He correctly explains the caused of eclipses of the Sun and the Moon.

Ø  Length of year: His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days and 6 hours.

Aryabhatta was one of those ancient scholars of India who is hardly surpassed by any one else of his time in his treatise on mathematics and astronomy. In appreciation of his great contributions to mathematics and astronomy, the government of India named the first satellite sent into space on 19-4-1975 as aryabhatta, after him.

Aryabhatta, also known as Aryabhatta I or Aryabhata (476-550?), was a famous Indian mathematician and astronomer, born in a place called Taregana, in Bihar (though some people do not agree with the evidence). Taregana (also spelled as Taragna) which literally means songs of stars in Bihari, is a small place situated nearly 30 km from Patna, which was then known as Kusumpura later Pataliputra, the capital of the Gupta Empire. This is the very empire that has been dubbed as the “golden period in Indian history”. The best introduction to the genius of past is seen in the words of Bhaskara I who said, “Aryabhatta 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”.

What was his name?

Varahamihira, the younger contemporary of Aryabhatta also mentions him as “Aryabhata”. In addition to this, Bhaskara I too mentions him as Aryabhata. It seems as if the correct name was Aryabhata and not Aryabhatta. This could mean that “Bhatta” was not his surname but as part of his first name. In fact, there is a lot of confusion about his name too. Perhaps he was called Arya and his surname was Bhat or Bhatta!

Where did Aryabhatta come from?

There is some disagreement about this birth place. Some are of the view that he was born in Patliputra while some are of the view that he was born in Kerala and moved to Patliputra and lived there. Those who say that he was in Bihar is because of this name. His name “Arya” and “Bhatta” indicates that he was from North India. His suffix “Bhatta” could have been either part of his name or his surname, till date it’s not known if this is correct or not. It is interesting to note that Aryabhatta himself have mentioned himself at only 3 places and as “Aryabhata” in his work Aryabhatiya.

The reason for not considering Kerala as his birthplace is that nowhere in his works he has mentioned Kerala. In addition, all works of Aryabhatta is in Sanskrit and Sanskrit was not used in Kerala. So to claim that Aryabhatiya was written in Kerala has no credibility. Furthermore, he has been identified by numerous mathematicians and in Arabic translations as someone who hailed from Kusumpura (modern Patna), the capital of Magadha. It therefore appears that Aryabhatta was born, lived, flourished and worked in Magadha. He has also been described as the head of the Nalanda University.

Influence of Aryabhatta on science and mathematics

Aryabhatta is considered to be one of the mathematicians who changed the course of mathematics and astronomy to a great extent. He is known to have considerable influence on Arabic science world too, where he is referred to as Arjehir. His notable contributions to the world of science and mathematics includes the theory that the earth rotates on its axis, explanations of the solar and lunar eclipses, solving of quadratic equations, place value system with zero, and approximation of pie (π).

Aryabhatta exerted influence on the Indian astronomical tradition to such an extent that his presence was felt in neighboring countries and cultures also. There have been various translations of his work among which the Arabic translation during the 820CE is very significant.

When mathematical students are confused with trigonometry even today, Aryabhatta had defined sine, cosine, versine and inverse sine back in his era, influencing the birth of trigonometry. The signs were originally known as jya, kojya, utkrama-jya and otkram jya. In Arabic they were translated as jiba and kojiba, which later when being translated into Latin was misunderstood to be ‘fold in a garment’ by Gerard of Cremona, who stated it as sinus, which meant fold in Latin. Aryabhatta was the first mathematician to detail both sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to 4 decimal places.

Aryabhatta’s astronomical calculations influenced the Arabians, who used the trigonometric tables to compute many astronomical tables. His calendared calculation has been in continuous use in India, on which the present day Panchangam is based. His studies are also base for the national calendars of Iran and Afghanistan today.

The Story of Numbers (0 and 1) Indian Numerals or Arabic?

Aryabhatiya

It is known that Aryabhatta has authored at least three astronomical books, in addition he also wrote some free stanzas. Among them “Aryabhatiya” is the only text that has survived to this day, whereas unfortunately his other works have been extinct. It is a small treatise written is 118 verses, which summarizes the Hindu mathematics of that time. This great mathematical masterpiece of the past starts with 10 verse introduction, which is then followed by mathematical section which is written in 33 verses that gives out 66 mathematical rules, but there is no proof to go with it. The mathematical part of the Aryabhatiya is about algebra, arithmetic, plane trigonometry and spherical trigonometry in addition to advanced mathematics on continued fractions, quadratic equations, sums of power series and a table of sines.

The next section consists of 25 verses which gives us glimpse into the planetary models. The final section of the book is dedicated to sphere and eclipses which runs into 50 verses. He states that the moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony where eclipses were believed to be caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by earth or those shadows that fall on earth. It is amazing how Aryabhatta could explain both lunar and solar eclipse so accurately.

There is some argument over the claim of Aryabhatta being the inventor of place value system that made use of zero. Georges Ifrah, in his work ‘Universal history of numbers: From prehistory to the invention of the computer (London, 1998)’ writes in work, “..it is extremely likely that Aryabhatta knew the sign for zero and the numerals of the place value system”. Georges Ifrah has studied the works of Aryabhatta and found that the counting and mathematical work carried out by him would have been not possible without zero or place value system.

Honouring Aryabhatta

The Indian ISRO (Indian Space Research Organization) named its first satellite after the genius mathematician and astronomer. A research establishment has been set up in Nainital, called the Aryabhatta Research Institute of Observational Sciences (ARIOS) to honor his contribution to the field of science. There is also a lunar crater and a species of bacteria discovered by ISRO named after Aryabhatta.

Some of the works of Aryabhatta include

  • Aryabhatta worked out the value of pi.
  • He worked out the area of a triangle. His exact words were, “ribhujasya phalashariram samadalakoti bhujardhasamvargah” which translates “for a triangle, the result of a perpendicular with the half side is the area”.
  • He discussed the idea of sin.
  • He worked on the summation of series of squares and cubes (square-root and cube-root).
  • He talks about the “rule of three” which is to find the value of x when three numbers a, b and c is given.
  • Aryabhatta calculates the volume of a sphere.
  • Aryabhatta described the model of the solar system, where the sun and moon are each carried by epicycles that in turn revolve around the Earth. He also talks about the number of rotations of the earth, describes that the earth rotating on its axis, the order of the planets in terms of distance from earth.
  • Aryabhatta describes the solar and lunar eclipses scientifically.
  • Aryabhatta describes that the moon and planets shine by light reflected from the sun.
  • Aryabhatta calculated the sidereal rotation which is the rotation of the earth with respect to the stars as 23 hours, 56 minutes and 4.1 seconds.
  • He calculated the length of the sidereal year as 365 days, 6 hours, 12 minutes and 30 seconds. The actual value shows that his calculations was an error of 3 minutes and 20 seconds over a year.

Although we know nothing about the personal history of Aryabhatta, he was the genius who continues to baffle mathematicians even to this day.

A new ebook (paperback coming soon) has been published called, “Life and Works of Aryabhata” which is available on Amazon.

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