Working through Serge Lang's Basic Mathematics atm (thx /prog), and I seem to have arrived a different proof for one of the exercises than the answer given. Pretty sure it's solid, but I need to check in case my logic is faulty.
Let a = m/n be a rational number expressed as a quotient of integers m, where neither m nor n = 0. Show that there is a rational number b such that ab = ba = 1.
My answer: If b = 1/a, then ba = ab = a(1/a) = 1
Also, Lang's book rocks.