Arithmetic in the Indian Tradition

Indian arithmetic developed sophisticated techniques for:

• Addition, subtraction, multiplication, division
• Squares, cubes, square roots, cube roots
• Fractions and proportions
• Negative numbers and zero (Brahmagupta, 628 CE)

Brahmagupta's Rules for Zero (Brahmasphuta-siddhanta):

Brahmagupta thus formalised the arithmetic of zero a thousand years before the West.

Geometry — Sulba Sutras

The Sulba Sutras (literally "rules of the cord") describe how to construct fire altars (yajna-vedis) using ropes and pegs. They predate Greek geometry and contain:

Baudhayana's Theorem (~800 BCE):

"The diagonal of a rectangle produces both areas which its length and breadth produce separately."

Written algebraically: c² = a² + b² — exactly Pythagoras' theorem, stated 300 years earlier.

              A
              │\
              │ \
            b │  \  c (diagonal)
              │   \
              │    \
              └─────\
              B  a   C

         c² = a² + b²

Constructions in the Sulba Sutras:

• Squaring a circle (approximate)
• Doubling a square
• Constructing a square whose area equals the sum of two other squares
• Approximation: √2 ≈ 1 + 1/3 + 1/(3·4) − 1/(3·4·34) = 1.4142156 (accurate to 5 decimal places)

Trigonometry — A Distinctly Indian Invention

While Greeks worked with chords, Indians invented the sine function:

            ●
           /│\
          / │ \
         /  │  \
        / R │   \ R
       /    │    \
      /     │     \
     /  jya │      \
    /───────┴───────●
    Centre        Radius

    jya (sine) = half-chord of double the arc
    R · sin θ  in modern notation

Aryabhata's Sine Table (499 CE): He gave sine values for every 3°45' from 0° to 90°, with R = 3438 — accurate to four significant figures. The Sanskrit word jya (chord) was transliterated by Arabs as jiba, mistranscribed as jaib ("bay"), then translated into Latin as sinus — the English sine we use today.

Famous Trigonometric Identity (Madhava, 14th century): The Madhava-Leibniz series for π:

   π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − ...

Madhava derived this 300 years before Leibniz. The Kerala School also derived the Madhava-Newton power series for sine and cosine.

Other Key Geometric Concepts

Key Terms — Lesson 3.2 (Arithmetic, Geometry, Trigonometry)

The Baudhayana theorem statement, Brahmagupta's zero rules and the jya → sine etymology are recurring 3–5 mark questions.

Khachheda — Brahmagupta's term for a quantity divided by zero, left undefined. Sulba Sutras — "Rules of the cord"; altar-geometry manuals containing the earliest Indian geometry. Baudhayana's theorem — The diagonal of a rectangle equals the combined areas of its sides: c² = a² + b², stated about 300 years before Pythagoras. Jya — Half-chord / sine; transliterated by Arabs as jiba → jaib → Latin sinus → English "sine". Madhava–Leibniz series — π/4 = 1 − 1/3 + 1/5 − 1/7 + …, derived by Madhava some 300 years before Leibniz. Kuttaka — Aryabhata's "pulveriser" algorithm for indeterminate (Diophantine) equations. Chakravala — The Brahmagupta–Bhaskara II cyclic method for Pell's equation Nx² + 1 = y².

Worked Example

Use the Sulba Sutra approximation to estimate √2.

The rule gives √2 ≈ 1 + 1/3 + 1/(3×4) − 1/(3×4×34).

• 1/3 = 0.33333
• 1/(3×4) = 1/12 = 0.08333
• 1/(3×4×34) = 1/408 = 0.00245

So √2 ≈ 1 + 0.33333 + 0.08333 − 0.00245 = 1.41421.

The true value is 1.41421356…, so the ancient estimate is accurate to five decimal places — remarkable for roughly 800 BCE.