Early Indian Mathematics Early Mathematical Contributions from India

Indus Valley 3000 BC Highly Ancient Indian Culture Harappan Civilization Archaeological Excavations at Mohenjo Daro Northeast of Karachin in Pakistan

Indus Valley Wide Streets Brick Dwellings Apartment Houses Tiled Bathrooms Covered City Drains Community Swimming Houses

Indus Valley Systems of Counting Writing Weights and Measures

Indus Valley 3000 BC Traded with Sumerians and Akkadians in Babylonia No Written Mathematical Documents from this Era

Aryan Settlement– 1800 B.C. Crossed over Himalayas into India Sanskrit word for “nobleman” or “owners of land” Some wandered into Europe The rest extended settlements throughout India Perfected written and spoken Sanskrit Introduced Caste System

Vedas – Sacred Texts 1500 B.C. Vedic People entered India Vedic Mathematics is contained in Sulba- sutras 16 Sutras – Rules for Arithmetic Gained in Popularity in 1900’s and again in 1980’s

You try! Multiply 134 x 246 Multiply 942 x 108 Multipy 450 x 123 2,10,6+12+8,18+16,24 = 2,10,26,34,24=32964 9,4,74,32,16=101736 4,13,22,15,0=55350

Vedic Mathematics Base 10 Invoked powers of 10 from 100 to 1 trillion Included Rules for Addition Subtraction Multiplication Division Fractions Squares Cubes Roots

Jaina Mathematics 600 BC – 1700 AD: Jainism religion and philosophy founded in India Replaced Vedic religion Surya Prajnapti and Jambidvipa Prajnapti – 400 B.C Texts Bhagabati Sutra – 300 B.C. mathematics text regarding combanitorics Sthananga Sutra – 200 B.C. mathematics text Number Theory Arithmetic Geometry Simple linear, cubic equations Combinatorics

To Infinity and Beyond Jainan religion concept of time and cosmology Was thought of as eternal and without form World was infinite – never created, always existed Space pervades everything – without form Were fascinated with large numbers

Large Numbers in Jaina Cosmology – time period 2588 Construction to stretch mind Start with Cylinder with radius = radius of earth Let h be the height Let n = number of mustard seeds that can be placed in this container Still the highest enumerable number has not been attained “Infinity is bigger than that” 5 different types of infinity Infinite in one direction Infinite in two directions Infinite in area Infinite everywhere Perpetually Infinite

Aryabhata – 476 AD – 550 AD Aryabhata 1 Wrote Aryabhatiya – mathematical and astronomical text 33 verses on mathematical rules without proof 25 verses on time and planetary models 50 verses on spheres and eclipses

Aryabhatiya - Mathematics Arithmetic Algebra Trigonometry on a plane Trigonometry on a sphere Continued Fractions Quadratic Equations Sums of Power Series Table of Sines

Aryabhatiya-Mathematical Contributions Calculations with zero Euclidean Algorithm Accurate approximation of pi = 3.141 Table of sine for each 3.45 degrees Introduced Cosine Sum of first n integers, first n squares and first n cubes Believed earth rotated on axis Believed Moon and Planets shine by reflected sunlight Correctly explained eclipses His value for a year = 365 days and 6 hours ( > actual value by minutes)

Brahmagupta –598 - 668 A.D Mathematician and Astronomer From Rajashtan – Northwest India Head of Astronomical Observatory at Ujjain in Central India Elliptic Verse Mathematics Poetic Ring

Brahmasphutasiddhanta “The revised system of Brahma” Mostly Astronomy 2 Chapters devoted to Math Algebraic Method of Inversion: “Beautiful Maiden with beaming eyes, tell me, as thou understands the right method of inversion, which is the number which multiplied by 3, then increased by ¾ of the product, then divided by 7, diminished by 1/3 of the quotient, multiplied by itself, diminished by 52, by the extraction of a square root, addition of 8, and division by 10 gives the number 2?”

Hindu Mathematical Writing Unlike Modern mathematics, addition was indicated by juxtaposition, rather than multiplication. Subtraction: dot over the subtrahend Multiplication: writing bha after factors bhavita “product” Division: writing divisor beneath the dividend Square Root: writing ka karana “irrational” Unknown: writing ya Yavattavat “so much as” Known integers: Ru Rupa “the absolute number”

Hindu Mathematical Insight Hindus included negatives and irrational numbers Recognized quadratics had two formal roots Solved quadratics by completing square Geometry was empirical Brahmagupta and Mahavira extended Heron’s Formula

Brahmagupta’s Formula

Find the area of the quadrilateral below. Identify if it’s cyclic first.