Knowledge Traditions And Practices Of India: Class 11 — Chapter 6: Mathematics in India

• Mohenjodaro (3000 BCE): Archaeological evidence reveals highly organized\pirban planning and advanced civilization along the Sindhu river.
• Brāhmaṇa Literature (2000 BCE): Post-Vedic texts containing ritualistic and philosophical treatises, showing continuous intellectual progress.
• Jainism: Highly valued mathematics, integrating gaṇita anuyoga (exposition of mathematical principles) and saṁkhyāna (science of numbers) as essential priestly accomplishments.
• Buddhism: Regarded arithmetic (gaṇanā saṁkhyāna) as the first and noblest of arts.

• Decimal Base: Number ten has historically formed the basis of numeration in India.
• High Numerals: Vedic works like the Yajurveda Saṁhitā express numerical denominations as large as $10^{12}$.
• Brahmi Numerals: Purely Indian invention dating back to King Asoka (300 BCE); early notations for 1, 2, and 3 were represented by horizontal lines stacked vertically.
• Discovery of Zero: Attributed to Āryabhaṭa I (born 496 CE) in connection with decimal place-value notation; represented as a circle (śūnya) to fill vacant places.

A highly productive era of mathematical systematization and advancement in India, featuring key scholars and treatises:

• Pāṭigaṇita: Compound word from pāṭi (board) and gaṇita (science of calculation), meaning mathematics requiring writing boards.
• Dhūli-karma: Literally "dust-work"; calculations performed by writing figures on a thin layer of dust spread over a board or ground, allowing easy erasure and correction.
• Conciseness: Indian scientific treatises favored highly compact, brief statements of formulae and results, often requiring expert commentary to interpret.

Classical Indian arithmetic recognizes eight core operations, which Bhāskara I simplified into two primary categories: increase (addition) and decrease (subtraction).

1. Addition (Saṁkalita / Saṁkalana)
2. Subtraction (Vyutkalita / Vyutkalana)
3. Multiplication (Guṇana)
4. Division (Bhāgahara)
5. Square (Varga)
6. Square-Root (Varga-mūla)
7. Cube (Ghana)
8. Cube-Root (Ghana-mūla)

• Addition (Saṁkalita): Defined by Aryabhata II as "making into one of several numbers." Processes include:
  • Direct Process: Adding digits from right to left (units place first, carrying over to tens, and so on).
  • Inverse Process: Adding digits from left to right (highest place value first), correcting previous digits as carries emerge.
• Subtraction (Vyutkalita): Defined as taking out a quantity from the total (sarvadhana or viyojya) to find the remainder (śeṣa or antara). The subtrahend is termed viyojaka. Performed in both direct and inverse orders.

• Multiplication (Guṇana): Historically defined as a process of repeated addition. Terminology: multiplicand (guṇya), multiplier (guṇaka), and product (guṇana-phala or pratyutpanna).
  • Methods: Brahmagupta lists gomūtrikā, khaṇḍa, bheda, and iṣṭa.
  • Kapaṭa-sandhi: "Door-junction" method described by Śrīpati, where the multiplier is placed above the multiplicand and shifted progressively left or right as digits are multiplied and rubbed out.
• Division (Bhāgahara): Regarded as the inverse of multiplication. Terminology: dividend (bhājya), divisor (bhājaka), and quotient (labdhi). Performed on the board by shifting the divisor to the right and rubbing out digits of the dividend to write the partial quotients.

• Vedic Era: The Ṛgveda mentions fractions like one-half (ardha) and three-fourths (tri-pāda). The Maitrāyaṇī Saṁhitā records one-sixteenth (kalā), one-twelfth (kuṣṭha), one-eighth (śapha), and one-fourth (pāda).
• Sulbasūtras: Systematically\pised fractions to state and solve geometric problems. Unit fractions were denoted by combining cardinal numbers with bhāga or aṁśa (e.g., pañca-daśa-bhāga for $1/15$).
• Composite Fractions: Expressed compound terms such as tri-aṣṭama ($3/8$) and dvi-saptama ($2/7$).

• Square (Varga / Kṛti): Defined by Āryabhaṭa I as the product of two equal quantities or the area of a four-sided equilateral figure. Calculated on the board by squaring digits and multiplying intermediate terms systematically.
• Square-Root (Varga-mūla): Derived from mūla or pāda (meaning root, base, or foundation). The Āryabhaṭīya provides a highly concise algorithm: divide the even place by twice the square-root of the preceding odd place, subtract the square of the quotient from the next odd place, and repeat.

• Sulbasūtras: Oldest known Indian mathematical texts (composed around 800 BCE by Baudhāyana, Āpastamba, and Kātyāyana) detailing geometry for constructing Vedic fire altars.
• Altar Designs: Required precise geometric shapes representing symbolic forms, such as a falcon in flight (śyena-cit), a chariot-wheel, or a tortoise.
• Triangles: Classified as equilateral (sama), isosceles (dvisama), and scalene (viṣamatribhūja).
• Pythagoras Theorem: Explicitly stated and applied in the Sulbasūtras for geometric constructions, such as transforming a rectangle into a square of equal area.
• Area Formulae: Āryabhaṭa I provided correct formulae for the area and perimeter of common geometric figures; Brahmagupta formulated the area of a cyclic quadrilateral given its sides.

• Terminology: Known as Bījagaṇita (science of calculation with elements/analysis) or avyakta-gaṇita (calculation with\pinknowns), in contrast to vyakta-gaṇita (arithmetic with knowns).
• Kuṭṭaka: Meaning "pulveriser"; a branch of algebra dealing with first-degree indeterminate equations. Brahmagupta referred to algebra as kuṭṭaka-gaṇita.
• Origins: Traced to geometric transformations in the Sulbasūtras, where converting a square to a rectangle of a given side is equivalent to solving a linear equation of the form $ax = c^2$.
• Binomial Expansion: Contributions by Āryabhaṭa I were\pitilized to evaluate powers, roots, and algebraic expansions.

• Functions: Indians invented the sine (jyā or jīvā) and cosine (ko-jyā) functions, defining them\pising a right triangle within a quadrant of a circle.
• Fundamental Identity: Expressed as: $$(R \text{ Sin } \theta)^2 + (R \text{ Cos } \theta)^2 = R^2$$ where $R$ is the radius of the circle.
• Sine Tables: Āryabhaṭa I calculated accurate values for $R \text{ Sin } \theta$ at intervals of $15^\text{o}$ and $3.75^\text{o}$.
• Interpolation Formulae: Brahmagupta and Govindaswami (880 CE) developed second-order interpolation methods to compute intermediate sine values, pre-dating Newton-Stirling and Newton-Gauss formulas.

Indian mathematicians calculated highly accurate values for the ratio of a circle's circumference to its diameter ($\text{Pi}$):

• Āryabhaṭa I (499 CE): Approximated $\text{Pi}$ as $3.1416$.
• Mādhava (1500 CE): Calculated $\text{Pi}$ to eleven decimal places as $3.14159265359$.
• Nīlakaṇṭha (1500 CE): Expressed the approximation as the fraction $355/113$.