Presentaion_Iks_Aman_Kr(28)_Krish_Bansal(67).pdf

1. Vedic Period (1500–500 BCE)

Mathematics in India is deeply rooted in the Vedas, especially in texts like the Śulba Sūtras (800–500 BCE), which were manuals for constructing altars.

The Śulba Sūtras contain some of the earliest references to geometry, algebraic identities, and approximations of √2.

They also describe principles similar to the Pythagorean theorem, long before Pythagoras (6th century BCE).

1. The Enduring Legacy of Indian
Mathematics
This presentation explores the profound contributions of ancient Indian
mathematicians, whose innovations laid foundational groundwork across various
fields. We will journey through their remarkable achievements, from the precise
calculation of mathematical constants to the development of sophisticated
algebraic and combinatorial methods.

2. Agenda:Unveiling India's Mathematical Heritage
01
Approximations of Pi (π)
Tracing the evolution of pi approximations from Vedic times to
Ramanujan.
02
The Genesis of Trigonometry
Understanding the origins of sine and cosine and methods for
computing sine tables.
03
Algebraic Innovations (Bīja-gaṇita)
Exploring the Indian contributions to algebra, including negative
numbers and symbolic representation.
04
Binary Mathematics and Combinatorics
Delving into Piṅgala's pioneering work on binary sequences and
combinatorial problems.
05
The Art of Magic Squares (Bhadragaṇita)
Discovering the intricate history and construction of pan-diagonal
magic squares.
06
Key Takeaways
Summarizing the unique aspects and enduring relevance of Indian
mathematics.

3. The Quest for Pi (π):A Journey Through Centuries
The pursuit of an accurate value for pi has spanned millennia in the Indian mathematical tradition. From early geometrical approximations in
Vedic texts to precise calculations using infinite series, Indian mathematicians consistently refined this fundamental constant.
Śulba-sūtras (around 800
BCE)
3.08888 1 Geometrical
Jaina texts (500 BCE) √10 = 3.1623 1 Geometrical
Āryabhaṭa (499 CE) 3.1416 4 Polygon doubling (1024 sides)
Bhāskarācārya (Līlāvatī) 3927/1250 = 3.14164 Polygon doubling
Mādhava (1375 CE) 3.14159265359 11 Infinite series with end corrections
Ramanujan (1914 CE) 17 million - Modular equation
Mādhava's calculation of pi to eleven decimal places using an infinite series was centuries ahead of its Western counterparts. Ramanujan
further pushed the boundaries in the 20th century with millions of decimal places.

4. Trigonometry:The Science of Chords (Jyotpatti)
Indian mathematics made pivotal advancements in trigonometry, known as
jyotpatti, or the science of computing chords. The term jyā(later simplified
from jyārdha, meaning "half a bow-string") referred to what we now call
the sine function. The accompanying koṭijyāor kojyārepresented the
cosine.
The transmission of these concepts to the Arab world, where jīvābecame
jība, and then to Latin (sinus), highlights a fascinating linguistic and
mathematical journey. This led directly to the modern terms"sine" and
"cosine."
Āryabhaṭa's Sine Tables
Āryabhaṭa (499 CE) developed two groundbreaking methods for deriving sine tables: a geometric approach and an analytical method that foreshadowed
modern differential processes. His formula for R-sine differences provided a recursive way to compute sine values, a feat unrecognized in the West until
much later.
"The first R sine divided by itself and then diminished by the quotient will give the second difference. For computing any otherdifference. The sum of
all the preceding differences is divided by the first R sine and the quotient is subtracted from the preceding difference. Thus,all the remaining
differences (can be calculated)."

5. Algebra (Bīja-gaṇita):The Science of Analysis
Ancient Indian mathematicians considered algebra, or bīja-gaṇita("the science of calculation of
elements"), a subject of great utility. Their contributions were fundamental, establishing algebra
as a core discipline and providing clarity and precision to mathematical thought.
•Symbolic Representation:Brahmagupta, building on hints from Āryabhaṭa, introduced
symbols for unknowns, a crucial step towards modern algebraic notation.
•Arithmetic of Negative Numbers:Brahmagupta (628 CE) clearly articulated rules for
operations with negative numbers, including addition and subtraction, centuries before their
full acceptance in Europe.
•Solving Equations:Linear and quadratic equations were systematically solved by Āryabhaṭa
and Brahmagupta, demonstrating an advanced understanding of algebraic principles.
"The sum of two positive numbers is positive, of two negative numbers is negative; of a positive and a negative number is their difference." —Brahmagupta (628
CE)
Indian mathematicians also established fundamental arithmetic operators like 'yu' for addition, 'kṣa' for subtraction, 'gu' for multiplication, and 'bhā' for division. They
even used colors to denote different unknown variables, classifying equations as samīkaraṇa.

6. Binary Mathematics and Combinatorics:Piṅgala's Insights
Piṅgala's *Chandaḥ-śāstra* (circa 300 BCE), a Sanskrit prosody treatise, pioneered the concept of binary
sequences. By representing syllables (laghu as '1' and guru as '0'), he effectively laid foundational
groundwork for binary mathematics, predating Western developments by centuries.
His "Prastāra" method for generating binary arrays employed a recursive process, mirroring modern
binary computation. This involved building sequences of increasing length by replicating existing arrays
and appending '0's and '1's.
Piṅgala also rigorously analyzed several key combinatorial problems related to these metrical patterns:
Prastāra
Generating all possible binary sequences for a
given length.
Saṃkhyā
Determining the total number of binary
sequences (equivalent to 2^n).
Naṣṭa
Identifying the binary sequence corresponding
to a specific row number.
Uddiṣṭa
Finding the row number for a given binary sequence.
Lagakriyā
Calculating the number of sequences with a
specific count of '1's or '0's (binomial
coefficients).

7. Magic Squares (Bhadragaṇita):Art and Utility in Numbers
In Indian mathematics, magic squares (Bhadragaṇita) were not just puzzles but subjects of serious mathematical inquiry and practical application.
These squares, where numbers in rows, columns, and main diagonals sum to a "magic" constant, demonstrate a blend of arithmetic and
combinatorial artistry.
Of particular interest were pan-diagonal magic squares (*sarvatobhadra*), where even
secondary diagonals also sum to the magic constant, often visualized as wrapping
around a torus.
Nāgārjuna (100 CE) provided mnemonic-based formulas for constructing 4x4 magic
squares, deriving cell values from a template for a desired sum.
•Ancient Origins:Attributed to Garga (100 BCE), with Kaksaputa by Nāgārjuna
describing 4x4 construction methods.
•Widespread Presence:Found in Jaina inscriptions, Gwalior Fort (1480 CE), and still
used today in various cultural contexts.

8. Building a Pan-Diagonal Magic Square:A Step-by-Step Guide
The construction of a 4x4 pan-diagonal magic square using numbers 1 to 16, with a magic sum of 34, exemplifies the ingenuity of Indian mathematicians.
The process relies on specific properties that reveal the underlying mathematical structure.
Step 1: Placement of 1 and 16
Place '1' in any cell. Place '16' two cells diagonally away from '1'.
Step 2: Neighbors of 16
Using Property 3 (elements 16 has neighbors 2, 3, 5, 9), place these numbers around 16 in any
order.
Step 3: Filling Remaining Cells
Utilize the properties to complete the square:
•Property 1:Entries of any 2x2 sub-square formed by consecutive rows and columns add up
to the magic sum (34).
•Property 2:The sum of an entry with another two squares away along a diagonal is half the
magic sum (17).
This systematic approach, formalized by mathematicians like Nārāyaṇa, demonstrates a deep theoretical understanding and practical application of
combinatorial principles.

9. Conclusion:The Enduring Contributions of Indian Mathematics
The rich history of Indian mathematics reveals a vibrant tradition of inquiry driven by both practical needs and intellectualcuriosity. From the earliest Vedic
period to the time of Ramanujan, Indian mathematicians developed sophisticated concepts that continue to resonate in modern mathematics.
Holistic Development
Mathematical concepts evolved across the
Indian subcontinent, from Gāndhāra to
Kerala, reflecting a widespread intellectual
pursuit.
Pioneering Concepts
Contributions include precise
approximations of Pi, foundational work in
trigonometry (sine tables), and the
development of algebra (negative numbers,
symbolic representation).
Algorithmic Thinking
Ancient texts like Āryabhaṭīya showcase a
well-developed algorithmic approach to
problem-solving and recursive methods.
Foresight in Computation
Piṅgala's work on binary sequences laid
critical groundwork for modern computing
and combinatorial theory.
Artistic and Utilitarian Integration
The study of magic squares (Bhadragaṇita)
illustrates the blend of mathematical rigor
with aesthetic and practical applications,
found even in contemporary architecture
and astrology.
The legacy of Indian mathematics is a testament to human ingenuity and its continuous pursuit of understanding the quantitative aspects of the universe.