The Śulba Sūtras (800–500 BCE), a set of ancient Indian texts on geometry, contain one of the earliest known references to irrational numbers, particularly 2\sqrt{2}. These texts were written by Vedic scholars such as Baudhāyana, Āpastamba, and Kātyāyana to provide geometric rules for constructing sacrificial altars (yajña-vedis) with precise dimensions.
- The Approximation of 2\sqrt{2}
- The Baudhāyana Śulba Sūtra gives an approximation of 2\sqrt{2} as:
2≈1+13+13×4−13×4×34\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3 \times 4} – \frac{1}{3 \times 4 \times 34}
- In decimal form:
1.4142151.414215
This is accurate up to five decimal places, making it one of the earliest known approximations of an irrational number.
- Geometric Interpretation
The approximation of 2\sqrt{2} was used for constructing squares with areas twice that of a given square.
This was important for altar constructions, where priests needed to double the area while maintaining symmetry.
- Recognition of Irrationality
Though ancient Indian mathematicians did not explicitly call these numbers “irrational,” they understood that certain numbers, like 2\sqrt{2}, could not be expressed exactly as a fraction.
This predates the Greek mathematician Hippasus (c. 5th century BCE), who is often credited with discovering irrational numbers.
Later Indian mathematicians like Aryabhata (5th century CE), Brahmagupta (7th century CE), and Bhaskara II (12th century CE) expanded on these concepts.
The Śulba Sūtras’ treatment of 2\sqrt{2} influenced algebra, trigonometry, and continued fraction methods in later centuries.
| Civilization | Approximation of Irrational Numbers | Approximate Date |
| India (Śulba Sūtras) | 2≈1.414215\sqrt{2} \approx 1.414215 | 800–500 BCE |
| Babylonians | 2≈1.414213\sqrt{2} \approx 1.414213 (on clay tablet YBC 7289) | c. 1800 BCE |
| Greece (Hippasus, Pythagoreans) | Discovery of irrationality of 2\sqrt{2} | c. 5th century BCE |
