Brahmagupta’s contribution to Mathematics - Rules for "zero"

Against the popular perception, I would say that none including Aryabhat has invented ‘zero’ because the concept of ‘zero’ was always part of our life in different way academically or professionally since the origin of the universe.

Brahagupata’s major contribution to modern science and technology was - how to use zero in mathematical calculations that is used as base to calculate every field. He has formed the set of rules in the mathematics for positive and negative number in addition to forming the rules for using zeros in calculations and that has become the basis of mathematics and rest of the field including economics, finance, physics, chemistry, astronomy and so on…

Brahmagupta was born in 598 CE and died in c. 670. He lived in during king Vyagrahamukha in Bhillamala (modern Bhinmal) a capital of Gurjaradesa Chapa dynasty, one of the largesst kingdoms of Western India in parts of southern Rajasthan and Northern Gujarat.

He studied 5 traditional siddhantas in Brahmapaska in one of top schools for the astronomy during those days.

He also studied his predecessors’ (Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra) work to continue their research work. His rules of zero is in continuation of Aryabhata I ‘s work.

Brahmagupta gave the solution of the general linear in chapter eighteen of Brahmasphutasiddhanta,

bx + c = dx + e equivalent to x =< (e−c) / (b-d)


In chapter eighteen of Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,

18.30 (Additions)

· 3 + 4 = 7 (a positive number)

· -3 + (-4) = -7 (a negative number)

· -3 + 4 = 1 (positive number)

· 3 + (-4) = -1 (negative number)

· -3 + 3 = 0 (zero when both positive and negative numbers are same)

· -3 + 0 = -3 (negative number)

· 3 + 0 = 3 (positive number)

· 0 + 0 = 0 (no difference)

18.32 (subtractions)

· -3 – 0 = -3 (negative number)

· 3 – 0 = 3 (positive number)

· 0 - 0 = 0 (no difference)

· 3 - (-4) = -1 (negative number is added)

· -3 – 4 = - 7 (negative number is added)

18.33(Multiplication)

· 3 * - 4 = - 12 (one negative and one positive product becomes negative number)

· 3 * - 4 = 12 (two negative becomes positive)

· 3 * 0 = 0 (any number multiplied with zero is zero)

18.34 (Division)

·     3
     ----  = 1 (positive number)
      3

·    - 3
       ----  = 1 (positive number)
      -3

·    0
    ---  = 0 (zero)
     0

·     3
     --- = -1
     -3

·      -3
      ---  = -1
       3

·    √0   =  0

·    0 square   = 0

·    0
    ----
     0

·    3
    ---- = undefined
     0

Except the last definition of zero, we use all his rules in the modern mathematics.

Watch out this space for part II on his other contributions in calculating complex mathematical formulas.

References:

Brahmagupta (598–668)," Department of Mathematics, Simon Fraser University (British Columbia, Canada), http;//www.math.sfu.ca/histmath/India/7thCenturyAD/Brahmagupta.html (February 17, 2006).

"Brahmagupta (ca. 598–ca. 665)," Wolfram Research, http://www.scienceworld.wolfram.com/biography/Brahmagupta.html (February 17, 2006).

"Brahmagupta," School of Mathematics and Statistics, St. Andrews University, http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Brahmagupta.html (February 17, 2006).

"Brahmagupta," Vidyapatha, http://www.vidyapatha.com/scientists/brahmagupta.php (February 17, 2006).

http://www.encyclopedia.com/topic/Brahmagupta.aspx

Aryabhat and his work in Mathematics

Most of us know the invention of zero ‘0’ by Aryabhat in India; but very few know about him or his real work in detail. Aryabhat was a famous Indian mathematician and an astronomer who lived between 476 – 550 CE. Most of his work was done towards end of 4th century CE.

He received his early education in Kusumpura, which is now known as Patna in Bihar.

He had done enormous work in mathematics and astronomy that is complied into two major works and few notes. One of the major works is known as “Aryabhatiya” that has survived the passage of time. 

However, his other work in astronomical computation that is known, as ‘Arya-siddhanta’ could not survive.

Aryabhatiya mainly focuses on mathematics and astronomy and it consists of 4 parts:

1. The Dasagitika Sutra: It has 33 verses that defines the basics of astronomical tables

2. Ganita – Mathematics: This section has 33 verses that give 66 mathematical rules.

3. Kalakriya – The measure of time: This section has 25 verses that describe time, its measurement and planetary models.

4. Gola – The spherics: This section has 50 verses those details on sphere and eclipses.

Here is the extract of some of his work for us to gauge the depth of knowledge that India lost over a period of time.

i. Most basic & famous example of binomial:

(a + b) ² = a² + 2ab + b²

Or

a X b = (a+b)2  - (a2 – b2)

------------------------

2

ii. Value of Pi:

abhi svav Rushtim madey ays yudhyato raghveeriva pravaNe samsrutaya: | indro yad vajree dhRuSHamANo andhasA behind valasya paridheemriva trita: ||

paridhi m. circumference Paridhih +iva+ tritah. Paridhih means circumference, iva means like or for example, tritah means by 3. Thus, like circumference by three.

haturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"

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

= 3.14159265 i.e. the value of π is correct to 8 places

iii. The value of pie upto

गोपीभाग्य मधुव्रातः श्रुंगशोदधि संधिगः |

खलजीवितखाताव गलहाला रसंधरः ||

gopeebhaagya maDhuvraathaH shruMgashodhaDhi saMDhigaH

khalajeevithakhaathaava galahaalaa rasaMDharaH

ga-3, pa-1, bha-4, ya -1, ma-5, Dhu-9, ra-2, tha-6, shru-5, ga-3, sho-5, dha-8, Dhi -9, sa-7, Dha- 9, ga-3, kha-2, la-3, jee-8, vi-4, tha-6, kha-2, tha-6, va-4, ga-3, la-3, ha-8, la-3, ra-2, sa-7, Dha-9, ra-2

π  = 3.1415926535897932384626433832792...

iv. Measurement of earth:

About the circumference of the earth, Aryabhat measured as 4 967 yojanas (1 yojana = 5 miles) and its diameter as 1 5811/24 yojanas.

5 x 1 5811/24

= 24835

And, today it is measured as 24 902 miles (using all the advanced equipment, technologies)

Sine differences: The rules says each sine-difference reduced by quotients of all the previous differences and itself by the first difference.

d2 (sin x)

--------------   = - Sin x

dx2

For example, check the numbers - 51, 37, 22, 7

v. Earth and planetary light:

Aryabhat has mentioned that the Moon and planets shine by reflected sunlight.

vi. Rotation of earth:

In his golapada stanza 9, he wrote:

He also mentioned the unconventional discovery about the rotation of the heavens was due to the axial rotation of the Earth on daily basis; that apparently was accepted much later by the West.

Acknowledging Aryabhat’s contribution in mathematics and astronomy, India honored him by naming its first satellite as ‘Aryabhat.’

If the Indian government takes interest in teaching ancient languages and investing into ancient scientific scriptures, it can build stronger scientific base in future. Apparently, it was believed that Germans learned Sanskrit to rediscover Indian scientific and technological advancement in aeronautical field and they succeeded in building an advanced plane. That knowledge was eventually taken by NASA scientists after World War II and used in their research.

Image Credits: Wikimedia Commons

References:

J. J. O'Connor and E. F. Robertson, Aryabhata the Elder, MacTutor History of Mathematics archive

http://www.ms.uky.edu/~sohum/AAK/pdf%20files/aryabhatta.pdf_

Understanding (Third Edition). New York: W.H. Freeman and Company. p. 70. ISBN 0-7167-4361-2

https://en.wikipedia.org/wiki/Aryabhata

http://www-history.mcs.st-and.ac.uk/Biographies/Aryabhata_I.html