BRAHM-Ai • Vedic Math Lab

Futuristic Math — Aryabhata × Brahmagupta

Ancient wisdom, modern computation. Trigonometry, Kuttaka, cyclic quadrilaterals, and composition identities—reimagined.

π ≈ 3.1416 (Āryabhaṭa) (ax + by = c) — Kuttaka (Pulverizer) Areacyclic = √[(s−a)(s−b)(s−c)(s−d)] (Brahmagupta) (x² + n y²)(u² + n v²) = (xu − n yv)² + n(xv + yu)² — Bhāvanā sin table ↔ chord/arc relations (Āryabhaṭa)

Cyclic Quadrilateral Area

Enter sides a,b,c,d (cyclic). We compute s and √[(s−a)(s−b)(s−c)(s−d)].

—

Kuttaka (ax + by = c)

Extended Euclid to find one integer solution; parametric family shown.

—

Trig Explorer (Āryabhaṭa)

Angle → sine/versine; equal-arc step intuition for tables.

—

Bhāvanā Identity

Compose (x² + n y²) and (u² + n v²) → (xu − n yv)² + n(xv + yu)²

—

🪷 Aryabhata & Brahmagupta — Guidance (Vedic Gyan × Ganit)

Āryabhaṭa (c. 476–550 CE)

  • Works: Aryabhatiya (Gītikā, Ganit, Kālakriyā, Gol)
  • Themes: Trigonometry (sine/versine table), Kuttaka (pulverizer), Astronomy (earth’s rotation), Place-value
  • Key Ideas: π ≈ 3.1416; eclipse theory by shadow; relative motion viewpoint
  • Legacy: Early systematic trig & indeterminate algorithms

Wikipedia (Hindi) Wikipedia (English)

Brahmagupta (c. 598–668 CE)

  • Works: Brahmasphuṭasiddhānta, Khaṇḍakhādyaka
  • Themes: Zero & negatives arithmetic, Algebra, Cyclic quadrilaterals, Pell-type equations (Bhāvanā)
  • Key Ideas: Formal rules with 0 and negatives; cyclic quadrilateral area; composition identities
  • Legacy: Foundations influencing later Islamic/European math

Wikipedia (Hindi) Wikipedia (English)

Why “Vedic Gyan × Ganit”?

Vedic thought treated number, time, and cosmos as a continuum. Temple architecture, calendar-making, and astronomy all relied on mathematics—hence we present Aryabhata & Brahmagupta together, connecting śāstra with calculational rigor.

Core Contributions (Deep-Dive)

Āryabhaṭa — Highlights

  • Trigonometry: Sine/versine tables; procedures to compute sines; practical celestial calculations.
  • Earth’s Rotation: Day/night explained by Earth’s spin; relative motion viewpoint.
  • π approximation: Verse gives circumference close to 3.1416.
  • Kuttaka (कुट्टक): “Pulverizer” algorithm for linear indeterminate equations (Diophantine type).
  • Place-value: Positional notation in practice.

Brahmagupta — Highlights

  • Arithmetic with Zero & Negatives: Systematic rules (add/subtract/multiply; division caveats).
  • Algebra: Quadratics, Pell-type equations; bhāvanā composition.
  • Geometry: Cyclic quadrilateral area (Brahmagupta formula).
  • Astronomy: Refined parameters; eclipse computations.

सूत्र / Sutras & Key Rules

🧮 Brahmagupta’s Arithmetic Rules (Zero & Negatives)

  1. Signs: धन×धन=धन; ऋण×ऋण=धन; धन×ऋण=ऋण.
  2. Add/Subtract: धन+धन=धन; ऋण+ऋण=ऋण; धन+ऋण ⇒ sign of larger magnitude.
  3. Zero: n+0=n; n×0=0.
  4. Division by 0: text-discussed historically; modern math: undefined.

🔷 Cyclic Quadrilateral Area — Brahmagupta

Area = √[(s−a)(s−b)(s−c)(s−d)], where s=(a+b+c+d)/2.

🧩 Brahmagupta–Fibonacci Identity

(x² + n y²)(u² + n v²) = (xu − n yv)² + n(xv + yu)²

🔁 Bhāvanā for Pell-type

For x² − N y² = 1, composition yields new solutions from known ones.

🧭 Aryabhata’s Kuttaka (Pulverizer)

Linear indeterminate ax + by = c solved via repeated division (extended Euclid style).

  1. Find gcd(a,b) with steps; back-substitute.
  2. Scale to c/g and parametrize all integer solutions.

📐 Trig Table (Sine/Versine)

  • Equal arc-steps → recursive differences.
  • Angle–chord relations for astronomy.

π Approximation

π ≈ 3.1416 (close value via verse).

Mini-Examples (Quick Demos)

Example — Brahmagupta Area

Let a=4, b=5, c=7, d=8 → s=(4+5+7+8)/2=12 → Area=√(8·7·5·4)=√1120≈33.47

Example — Kuttaka

Solve 26x+9y=1. gcd=1; extended Euclid → 26(−1)+9(3)=1 ⇒ x=−1, y=3; general: x=−1+9t, y=3−26t.

Further Reading

  • Aryabhatiya — critical editions & translations
  • Brahmasphuṭasiddhānta — arithmetic & algebra chapters
  • Histories of Indian Mathematics & Astronomy (peer-reviewed sources)