Fast multiplication

Distributivity is old. It was used, without a name, by mathematicians in ancient Egypt, Mesopotamia, Greece, China and India: Euclid in geometric form, Āryabhaṭa in algebraic form, both often but without stating it.

The first explicit statement of it is Brahmagupta's, in the Brahmasphuṭasiddhānta (Verse 12.55). He called using it for multiplication khaṇḍa-guṇanam: multiplication by parts (khaṇḍa = part, guṇana = multiplying).

His verse: 'The multiplier is broken up into two or more parts whose sum is equal to it; the multiplicand is then multiplied by each of these and the results added.' In letters, that's exactly (a + b)c = ac + bc.

And in the very next verse (12.56) he shows what it's for: fast multiplication, which he calls ista-gunana. Here's the idea.

Multiply 3874 by 11. Break the 11 into parts: 3874 × 11 = 3874(10 + 1) = 38740 + 3874. Write those two one under the other and look at the columns: each digit gets added to its neighbour.

In general, write a 4-digit number as dcba. Then dcba × (10 + 1) = dcba0 + dcba, whose digits are d, (c + d), (b + c), (a + b), a. So to multiply by 11:

• copy the last digit;
• add the digits in overlapping pairs, from right to left, carrying when a pair is more than 9;
• copy the first digit (plus any carry).

For 3874: last digit 4; then 7 + 4 = 11, write 1 carry 1; then 8 + 7 + 1 = 16, write 6 carry 1; then 3 + 8 + 1 = 12, write 2 carry 1; then 3 + 1 = 4. Reading it off: 42614.

Multiplying by 101 works the same way with a different split: dcba × 101 = dcba × 100 + dcba, giving digits d, c, (b + d), (a + c), b, a: the pairs skip one place instead of being next to each other. Extend the idea to 1001 and 10001, where the pairs skip further still. (And for 99, split the other way: n × 99 = n × 100 − n.)