"The product of the sum is equal to the sum of the products."
(8+20)×(16+19) → 28×35 = 980. Sum first, multiply once!
Multiplying two sums together by computing each sum first, then multiplying the results.
Break each factor into a sum, compute the sums separately, then multiply. This simplifies large multiplications into smaller mental additions and one multiplication.
This sutra encodes the distributive property in its most useful form for mental math. Instead of multiplying each pair of terms and adding (FOIL method), you can sometimes simplify by summing first. If you need (a+b)(c+d), compute S1=a+b and S2=c+d, then multiply S1 × S2. This is particularly useful in checking: the digit sum of a product should equal the product of the digit sums of the factors. For example, 23 × 14 = 322. Digit sums: 5 × 5 = 25, digit sum 7. Digit sum of 322 = 7. ✓ Match confirms the answer. This "digit sum check" is one of the most practical applications of this sutra.
The digit-sum checking method (casting out nines) was described in detail in the Vedic Mathematics Teacher's Manual by Kenneth Williams, based on Bharati Krishna Tirtha's original work. However, the technique has much older roots — it appears in Fibonacci's Liber Abaci (1202 CE), which itself drew from Indian and Arabic mathematical traditions. The method was standard practice among medieval Indian merchants and accountants. The formal statement "product of the sum = sum of the products" is a beautiful packaging of what is essentially the distributive law, making it memorable and easy to apply mentally.
The sutra is the distributive property: (a+b)(c+d) can be computed as a single product of the sums. The digit-sum check works because of modular arithmetic: the digit sum of a number is its residue modulo 9. Since (a mod 9)(b mod 9) ≡ ab (mod 9), checking digit sums is equivalent to checking the product modulo 9. This catches roughly 8/9 of all single-digit errors. In abstract algebra, this is a ring homomorphism from the integers to Z/9Z. In software testing, this principle is used for checksums and data integrity verification.