Brahmagupta's formula

Drawing a square every time is slow. The pictures point to a rule. Look at 3/4 × 2/5 = 6/20:

• 6 = 3 × 2 (top × top)
• 20 = 4 × 5 (bottom × bottom)

To multiply two fractions: multiply the tops, and multiply the bottoms. Check: 1/2 × 1/4 → 1 × 1 = 1, 2 × 4 = 8 → 1/8, just as the square showed. (Unlike adding, you don't need the bottoms to match first.)

This rule, in its general form, was first written down clearly by Brahmagupta in his book *Brāhmasphuṭasiddhānta* in 628 CE, together with the rules for adding, subtracting and dividing fractions.

It makes areas easy. A rectangle is 3 3/4 ft by 9 3/5 ft. As fractions: 15/4 and 48/5. Tops: 15 × 48 = 720; bottoms: 4 × 5 = 20. Area = 720/20 = 36 square feet.

Does the order matter? 1/2 × 1/4 = 1/8 and 1/4 × 1/2 = 1/8. No. A rectangle has the same area whichever side you call the length. So a/b × c/d = c/d × a/b.