Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author. Wh

Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author. Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) x (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. No, both of these are true. Your logical answer is correct: 1/4 goes into 1/2 twice because there are two quarters in every half. This is how our ancestors conceived division and fractions, and that's how physically dividing things works in the real world. But division is not an independent operation; it's the inverse of multiplication. As plonh mentions downthread, division is, quite literally, defined on a field (like Q) as multiplying by the reciprocal. (1/2) / (1/4) = 2 because (4/1) = (1/4)^-1 and (1/2) * (4/1) = 2. Any other mathematical explanation is simply justifying this fact in more palatable terms.