"Transpose and adjust."
For 1234 ÷ 12: transpose the 2 to -2, then work digit by digit.
Division by numbers slightly greater than powers of 10.
Transform the divisor to make division simpler by transposing digits and adjusting the calculation.
This sutra simplifies division when the divisor is close to and slightly above a power of 10 (like 12, 13, 102, 103). The key idea is to "transpose" the excess digits by changing their sign. For dividing by 12: the "2" becomes "−2". You then work left to right, bringing down each digit and adjusting with the transposed value. For example, 1234 ÷ 12: bring down 1, multiply by −2 = −2, add to next digit 2 → 0, multiply 0 by −2 = 0, add to 3 → 3, multiply 3 by −2 = −6, add to 4 → −2. The method extends to 3-digit and larger divisors using multiple transposed digits.
Division algorithms have been a cornerstone of Indian mathematics since the Brahmasphutasiddhanta of Brahmagupta (628 CE). The transpose-and-adjust method is a mental simplification of the traditional long division. Bharati Krishna Tirtha highlighted how this sutra makes division by numbers like 12, 13, 102 nearly as easy as single-digit division. The principle of transforming a difficult divisor into an easier one by using its complement has parallels in the European "short division" methods developed centuries later. In Ayurvedic medicine, practitioners used this technique for dosage calculations involving ratios close to powers of 10.
The mathematical foundation is synthetic division. Dividing by (10+a) is equivalent to using −a as the synthetic divisor. This is a special case of the polynomial division algorithm where the divisor is a first-degree polynomial. In numerical analysis, this connects to Horner's method for polynomial evaluation, which is the most efficient algorithm for evaluating polynomials and is still used in modern computer algebra systems. The technique also relates to the theory of continued fractions and convergents used in rational approximation.