Concepts: We Distribute, Yet Things Multiply (part 1)

1.1 Why distributivity matters

Algebra turns statements about numbers into general rules. Distributivity is the rule that connects multiplication with addition: multiplying a sum by a number gives the same result as multiplying each part separately and adding. Written with letters, if a, b and c are any numbers, then a(b + c) = ab + ac.

The rule looks small, but it is the engine behind almost every expansion in this chapter. It lets us open brackets, collect like terms and prove identities. It also explains quick multiplication methods that people used long before calculators.

1.2 Watching a product grow

Take the product 23 × 27 = 621. Ask what happens when one factor grows by 1.

• If 27 becomes 28, the product becomes 23 × 28 = 644, an increase of 23.
• If 23 becomes 24, the product becomes 24 × 27 = 648, an increase of 27.
• If both become 24 and 28, the product becomes 672, which is 621 + 23 + 27 + 1 = 672.

The pattern comes from distributivity. Increasing b by 1 replaces a × b with a(b + 1) = ab + a, so the product grows by a. Increasing a by 1 adds b. Doing both adds a, adds b, and adds the 1 × 1 corner where the two increases overlap.

1.3 When one factor grows and the other shrinks

Now increase a by 1 and decrease b by 1. Expanding (a + 1)(b − 1) by distributivity gives ab + b − a − 1. So the product increases by b − a − 1 when b is larger, and decreases when a is larger than b by more than 1.

With a = 23 and b = 27 the increase is 27 − 23 − 1 = 3, so 24 × 26 = 624. With a = 10 and b = 5 the change is 5 − 10 − 1 = −6, so 11 × 4 = 44 is smaller than 10 × 5 = 50. A formula that predicts a decrease is not wrong; the negative change is part of the answer.

1.4 The general identity for changing both factors

If one factor grows by m and the other by n, distributivity twice gives four terms: the two original parts, the two new strips and the corner piece. This is Identity 1, and it works for negative values of m and n as well. The four terms match the four rectangles you get when a rectangle of sides a and b grows by m in one direction and n in the other.

Substituting m = 1 and n = −1 gives ab + b − a − 1, the same expression as before. The identity also holds when a and b are negative integers, because distributivity holds for all integers. Check a = −5, b = 8, m = 2, n = −3: the left side is (−3)(5) = −15 and the right side is −40 + 16 + 15 − 6 = −15.

1.5 Brackets with more than two terms

Distributivity does not stop at two terms inside a bracket. The multiplier must meet every term of the second bracket, once each. The textbook's Example 1 expands (3a/2)(a − b + 1/5): multiply 3a/2 by a, then by −b, then by 1/5. The three products are (3/2)a², −(3/2)ab and (3/10)a.

No two of those terms have the same letter part, so they cannot be combined. Terms with the same letter part are called like terms, and only like terms can be added or subtracted into one term.

Example 2 expands (a + b)(a + b) and Example 3 expands (a + b)(a² + 2ab + b²). In Example 3, the products a²b and 2a²b are like terms and add to 3a²b; the products ab² and 2ab² add to 3ab². The result is a³ + 3a²b + 3ab² + b³.

1.6 A pinch of history

Mathematicians in Egypt, Mesopotamia, Greece, China and India used distributivity long before it was written as a formula. Euclid used it in geometric form and Āryabhaṭa used it in algebraic work. Brahmagupta gave the first explicit statement in the Brahmasphuṭasiddhānta, verse 12.55. He called the method khaṇḍa-guṇanam, multiplication by parts: break the multiplier into parts whose sum equals the multiplier, multiply the multiplicand by each part, and add the results. In modern notation, for two parts, that is exactly (a + b)c = ac + bc.

The same idea powers every expansion in the rest of the chapter: open the brackets, multiply each pair, and collect like terms.

1.7 Check an expansion with a substitution

A quick numerical check catches sign slips before they spread. Substitute a = 2 and b = 1 into Example 1: the left side is (3 × 2/2)(2 - 1 + 1/5) = 3 × 1.2 = 3.6. On the right, 3a²/2 = 6, -3ab/2 = -3, and 3a/10 = 0.6, which also total 3.6. The two sides agree. Use small values that keep the arithmetic light, and avoid a value that makes one bracket zero, because zero can conceal a wrong coefficient elsewhere. A substitution cannot prove an identity, but it is a fast check on an expansion you have just written.