Download Indian Mathematics

Early Mathematical Contributions from India
Early Indian Mathematics
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

From region that is
currently Iran

Vedic Mathematics is
contained in
Sulbasutras

16 Sutras – Rules for
Arithmetic

Gained in Popularity in
1900’s and again in
1980’s
You try!
1.
Multiply 134 x 246
2.
Multiply 942 x 108
3.
Multipy 450 x 123
1. 2,10,6+12+8,18+16,24 = 2,10,26,34,24=32964
2. 9,4,74,32,16=101736
3. 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?”

2
27

 3 4 (3 x) 

  52  8
7




2
10
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


Unknown: writing ya


karana “irrational”
Yavattavat “so much as”
Known integers: Ru

Rupa “the absolute number”
8 x  10  7

ya 8 bha ka 10 ru 7
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.