Distributivity

Algebra lets you write a general statement in a short form and, as the last chapter showed, prove things instead of just noticing them. This chapter is about one property in particular: distributivity, which links multiplication and addition.

Start with a concrete question. Take 23 × 27. By how much does the product grow if you increase 23 by 1? If you increase 27 by 1? If you increase both?

The second is easiest. Increasing 27 by 1 adds one more 23 to the total, so the product grows by 23. The distributive property says the same thing in symbols.

The distributive property of multiplication over addition: a(b + c) = ab + ac. Picture a rectangle of a rows split into b columns and c columns: the whole is the two parts put together. The a must reach both parts: 23 × (20 + 8) is 23 × 20 + 23 × 8, not 23 × 20 + 8. (Note that a(b + c) means a × (b + c); we usually drop the × next to brackets, as in 5a and xy.)

So with a = 23, b = 27, c = 1: 23(27 + 1) = 23 × 27 + 23. The increase is 23: exactly the other factor. In general, a(b + 1) = ab + a.

There's a matching form for a bracket on the left: (a + b)c = c(a + b) by commutativity, = ca + cb by distributivity, = ac + bc by commutativity again.

Now increase both by 1. Treat (a + 1) as one thing and distribute: (a + 1)(b + 1) = (a + 1)b + (a + 1)1 = ab + b + a + 1. So the product grows by a + b + 1: not just a and b, but one more. (So 98 × 99 becomes 99 × 100 by adding 98 + 99 + 1 = 198, with no multiplying.)

And if one goes up by 1 while the other goes down by 1? (a + 1)(b − 1) = (a + 1)b − (a + 1)1 = ab + b − a − 1. That may be a gain or a loss, depending on whether b is bigger than a + 1.

You don't need to memorise each case: rebuild it by distributing whenever you need it. For example, when does one number going up by 2 and the other down by 4 leave the product unchanged? (a + 2)(b − 4) = ab − 4a + 2b − 8, which equals ab when b = 2a + 4: 1 × 6 = 3 × 2, 2 × 8 = 4 × 4, and so on.