MATHS PPT

ACKNOWLEDGEMENTACKNOWLEDGEMENTI Kartik Gupta of class XI-C of Kulachi I Kartik Gupta of class XI-C of Kulachi Hansraj Model School would like to Hansraj Model School would like to report my sincere thanks to my report my sincere thanks to my mathematics teacher mathematics teacher MRS. AARTIMRS. AARTI who acted as my guide and mentor who acted as my guide and mentor and without whose help completion and without whose help completion of this project would have been of this project would have been impossible.impossible.

About His LifeAbout His Life

Aryabhatta (476-550 A.D.) was born Aryabhatta (476-550 A.D.) was born in Patliputra in Magadha, modern in Patliputra in Magadha, modern Patna in Bihar. Many are of the view Patna in Bihar. Many are of the view that he was born in the south of that he was born in the south of India especially Kerala and lived in India especially Kerala and lived in Magadha at the time of the Gupta Magadha at the time of the Gupta rulers; time which is known as the rulers; time which is known as the golden age of India. Aryabhata is the golden age of India. Aryabhata is the author of several treatises on author of several treatises on mathematics and astronomy, some mathematics and astronomy, some of which are lost.of which are lost.

1.)The surviving text is Aryabhata's masterpiece

the Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. Its mathematical section contains 33 verses giving 66

mathematical rules without proof. The Aryabhatiya contains an

introduction of 10 verses, followed by a section on mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of

50 verses being on the sphere and eclipses.

2.) A problem of great interest to Indian Mathematicians since ancient times has

been to find integer solutions to equations that have the form ax + by = c, a topic

that has come to be known as diophantine equations. This is an example

from bhaskara’s commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the

remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations,

such as this, can be notoriously difficult. They were discussed extensively in

ancient Vedic text Sulba sutra, whose more ancient parts might date to 800 BCE.

Aryabhata's method of solving such problems is called the kuttaka  method.

Kuttaka means "pulverizing" or Kuttaka means "pulverizing" or "breaking into small pieces", and the "breaking into small pieces", and the

method involves a recursive method involves a recursive algorithm for writing the original algorithm for writing the original

factors in smaller numbers. Today this factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in algorithm, elaborated by Bhaskara in 621 CE, is the standard method for 621 CE, is the standard method for

solving first-order diophantine solving first-order diophantine equations and is often referred to as equations and is often referred to as

the Aryabhata algorithm.The the Aryabhata algorithm.The diophantine equations are of interest diophantine equations are of interest

in cryptology, and the RSA in cryptology, and the RSA Conference, 2006, focused on Conference, 2006, focused on

the kuttaka method and earlier work the kuttaka method and earlier work in the Sulbasutras.in the Sulbasutras.

3.) Hindu - Arabic number system is 3.) Hindu - Arabic number system is consdered to be a universal human consdered to be a universal human

language, without which mathematics, language, without which mathematics, science and commerce would be science and commerce would be

almost impossible.  Russian almost impossible.  Russian mathematician Michel Ostrogradski mathematician Michel Ostrogradski

stated invention of the Hindu - Arabic stated invention of the Hindu - Arabic number system is the greatest number system is the greatest

discovery after writing.  discovery after writing.  Aryabhatta worked on different place Aryabhatta worked on different place value notations and finally developed value notations and finally developed the decimal place value notation and the decimal place value notation and the place holder. He calculated the the place holder. He calculated the

value of pi. This number system and value of pi. This number system and mathematics went to Europe and was mathematics went to Europe and was

known as "Modus Indorum" or "method known as "Modus Indorum" or "method of the Indians" during the middle ages. of the Indians" during the middle ages.

This method of the Indians is none This method of the Indians is none other than our arithmetic today.other than our arithmetic today.

4.) Our sine and cosine functions 4.) Our sine and cosine functions come from Aryabhatta' Ardha come from Aryabhatta' Ardha Jya and Kotti Jya which were Jya and Kotti Jya which were

translated to Arabic as Jiba.  Arab translated to Arabic as Jiba.  Arab mathematicians added tangent mathematicians added tangent

function.  Jiba was translated to Latin function.  Jiba was translated to Latin as Sinus.  It is widely accepted that as Sinus.  It is widely accepted that Aryabhatta obtained Ptolemy's full Aryabhatta obtained Ptolemy's full

chord method and transformed it to chord method and transformed it to much easier half chord method. He much easier half chord method. He was the first mathematician to give was the first mathematician to give the 'table of the sines', which is in the 'table of the sines', which is in

the form of a single rhyming stanza, the form of a single rhyming stanza, where each syllable stands for where each syllable stands for increments at intervals of 225 increments at intervals of 225

minutes of arc or 3 degrees 45'.minutes of arc or 3 degrees 45'.

Alphabetic code has been used by him to define a set of increments. If we use Aryabhatta's table

and calculate the value of sin(30) which is

1719/3438 = 0.5; the value is correct. His alphabetic code is

commonly known as the Aryabhata cipher .

Sl. NoAngle ( A )(in degrees,arcminutes)

Value in Āryabhaṭa'snumerical notation

(in ISO1 5919 transliteration)

Value inArabic numerals

Āryabhaṭa'svalue ofjya (A)

Modern valueof jya (A)

(3438 × sin (A))

   1 03°   45′ makhi 225 225′ 224.8560

   2 07°   30′ bhakhi 224 449′ 448.7490

   3 11°   15′ phakhi 222 671′ 670.7205

   4 15°   00′ dhakhi 219 890′ 889.8199

   5 18°   45′ ṇakhi 215 1105′ 1105.1089

   6 22°   30′ ñakhi 210 1315′ 1315.6656

   7 26°   15′ ṅakhi 205 1520′ 1520.5885

   8 30°   00′ hasjha 199 1719′ 1719.0000

   9 33°   45′ skaki 191 1910′ 1910.0505

   10 37°   30′ kiṣga 183 2093′ 2092.9218

   11 41°   15′ śghaki 174 2267′ 2266.8309

   12 45°   00′ kighva 164 2431′ 2431.0331

   13 48°   45′ ghlaki 154 2585′ 2584.8253

Sl. NoAngle ( A )(in degrees,arcminutes)

Value in Āryabhaṭa'snumerical notation

(in ISO 15919 transliteration)

Value inArabic numerals

Āryabhaṭa'svalue ofjya (A)

Modern valueof jya (A)

(3438 × sin (A))

   14 52°   30′ kigra 143 2728′ 2727.5488

   15 56°   15′ hakya 131 2859′ 2858.5925

   16 60°   00′ dhaki 119 2978′ 2977.3953

   17 63°   45′ kica 106 3084′ 3083.4485

   18 67°   30′ sga 93 3177′ 3176.2978

   19 71°   15′ jhaśa 79 3256′ 3255.5458

   20 75°   00′ ṅva 65 3321′ 3320.8530

   21 78°   45′ kla 51 3372′ 3371.9398

   22 82°   30′ pta 37 3409′ 3408.5874

   23 86°   15′ pha 22 3431′ 3430.6390

   24 90°   00′ cha 7 3438′ 3438.0000

5.) In Aryabhatiya Aryabhatta provide elegant results for the submission of series of squares and cubes. FOR SQUARES:- 12+22+…..+n2 = n(n+1)(2n+1) 6 FOR CUBES:- 1 3+ 23 +….+ n3 = (1+2+…+n) 2

= (n(n+1) )2

4

6.) He already knew that the earth spins on its axis, the earth moves

round the sun and the moon rotates round the earth. He talks about the position of the planets in relation to its movement around the sun. He

refers to the light of the planets and the moon as reflection from the sun.

He goes as far as to explain the eclipse of the moon and the sun, day and night, the contours of the earth, the length of the year exactly as 365

days. He even computed the circumference of the earth as 24835 miles which is close to modern day

calculation of 24900 miles.

BIBLIOGRAPHYBIBLIOGRAPHYin.answers.yahoo.comin.answers.yahoo.comwikipedia.orgwikipedia.orgwww.mapsofindia.comwww.mapsofindia.comhubpages.com hubpages.com wiki.answers.com wiki.answers.com aryabhatta.net.comaryabhatta.net.comwww.trueknowledge.comwww.trueknowledge.com