"Whatever the extent of deficiency."
104² → excess 4, (104+4)×100 + 4² = 10800+16 = 10816
Squaring numbers near a base (10, 100, 1000). Lightning-fast mental squaring!
To square a number near a base, add the excess (or subtract the deficiency) to get one part, then square the excess/deficiency for the other.
This sutra provides instant squaring for numbers near powers of 10. For a number (base + d): the square = base × (number + d) + d². For 104²: base=100, d=4. Left part: (104+4)×100 = 10800. Right part: 4² = 16. Answer: 10816. For numbers below the base, the same applies with deficiency: 96² → base=100, d=−4. Left: (96−4)×100 = 9200. Right: (−4)² = 16. Answer: 9216. This can be applied recursively — to square 12 (near 10): (12+2)×10 + 4 = 144. The method works because (a+d)² = a² + 2ad + d² = a(a+2d) + d².
Squaring techniques have been central to Indian mathematics since the Sulba Sutras, where they were needed for constructing square altars of specific areas. The "near-base squaring" method was implicitly used by Aryabhata for astronomical calculations involving squares of numbers close to standard values. The formal sutra name "Yaavadunam" comes from a verse in the Atharva Veda parishishta according to Bharati Krishna Tirtha. The technique was rediscovered in Europe by mathematicians studying "near-integer" arithmetic in the 18th century, but without the elegant sutra formulation.
The proof is straightforward: (B+d)² = B² + 2Bd + d² = B(B+2d) + d². Since B is a power of 10, multiplying by B is trivial (just shifting digits). This reduces squaring to one simple addition and one small squaring operation. In computational mathematics, this is known as "base shifting" and is used in arbitrary-precision arithmetic libraries (like GMP). In statistics, this technique is used to compute variance: Var(X) = E[(X−μ)²], where μ serves as the "base" near which all values cluster.