"Differences and similarities."
47×53 → avg 50, diff 3 → 50²-3² = 2500-9 = 2491
Multiplying numbers equidistant from a round number using the difference of squares.
Use the identity a² - b² = (a+b)(a-b) to simplify multiplication when numbers are symmetric around a base.
When two numbers are equidistant from a round number, their product equals the square of the average minus the square of the difference. This is the algebraic identity (a+b)(a−b) = a² − b². For 47 × 53: average is 50, distance is 3. So 50² − 3² = 2500 − 9 = 2491. This works spectacularly for mental math because squaring round numbers is easy. More examples: 28 × 32 = 30² − 2² = 896. 196 × 204 = 200² − 4² = 39984. The sutra teaches us to look for the "center" of two numbers and exploit symmetry.
The difference of squares identity was known to Babylonian mathematicians (circa 2000 BCE) and appears in Euclid's Elements (Book II, Proposition 5). In India, this identity was used extensively in the Siddhanta astronomical texts for computing products of large numbers in planetary orbit calculations. Aryabhata and Bhaskara I both used this technique. The Vedic formulation emphasizes the mental process of finding the "center" — a uniquely Indian pedagogical approach. The identity is also central to the factorization methods used in number theory and was key to Fermat's factorization method (1643).
The identity (a+d)(a−d) = a²−d² is a fundamental algebraic truth derivable from the distributive law. It has profound applications: in number theory, it's used for integer factorization (Fermat's method). In physics, it describes the interference pattern of waves — when two frequencies combine, the result involves sum and difference frequencies (heterodyne principle in radio). In optics, the difference of squares appears in the lens-maker's equation. In statistics, the variance formula can be expressed as E[X²] − (E[X])², which is a difference-of-squares structure.