Arithmetic of rational numbers

Adding and subtracting: first give both fractions the same denominator (the LCM is simplest), then add or subtract the numerators: a/b + c/b = (a + c)/b. Think of slices of a pizza: you can only add slices once they are the same size.

Multiplying: multiply straight across, top × top and bottom × bottom: (a/b) × (c/d) = ac/bd (b and d not zero).

Dividing: turn the second fraction upside down and multiply: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc (b, c and d not zero).

The usual laws still hold:

• addition and multiplication are commutative: the order does not matter;
• the distributive law: p(q + r) = pq + pr.

Closed again. Add, subtract or multiply two rational numbers and you always get a rational number. Divide, and you do too, as long as you do not divide by zero.

A surprise in Exercise Set 3.3, Q8. (5/6)(x + 3/5) expands to (5/6)x + 1/2, which is exactly the other side. So the equation is true for every x. It is just the distributive law.