Phase 5 · Indian Mathematics

25.3 Pi = 3.1416 — Aryabhata's Approximation

How Aryabhata computed pi to 4 decimal places in 499 CE with a key insight about irrationality, and how Madhava of Kerala reached 11 decimals 900 years later

~12 min3 pages + knowledge check

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Aryabhata's π — Four Decimals and a Deep Hint

In 499 CE, when European mathematics was still wrestling with Archimedes\' polygon method, Aryabhata gave not just π = 3.1416 in a single verse but added a word ("āsannaḥ") that would remain uniquely perceptive in mathematical history for over 1200 years.

Ganitapada Verse 10 — Text and Interpretation

"chaturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām. ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ"

Translation: "Add 4 to 100, multiply by 8, and add to 62000. This is approximately (āsannaḥ) the circumference of a circle whose diameter is 20000."

Calculation: (100 + 4) × 8 + 62000 = 832 + 62000 = 62832. π = 62832 ÷ 20000 = 3.1416.

Significance of āsannaḥ: "Āsannaḥ" = "approaching" = "not exact." This word signals Aryabhata knew his value was approximate, and likely intuited that no simple fraction could exactly represent π — an implicit recognition of irrationality.

Aryabhata vs. the Contemporary World

Archimedes (~250 BCE): 223/71 to 22/7 — ~2 correct decimals

Aryabhata (499 CE): 3.1416 — 4 correct decimals

China (~480 CE): Zu Chongzhi — 355/113 ≈ 3.14159 — 6 decimals

Madhava (~1375 CE): 3.14159265359 — 11 decimals

Europe (~1600 CE): van Ceulen — 20+ decimals (but no infinite series yet)