We Distribute, Yet Things Multiply
Algebra is more than a shorthand for arithmetic, it is a tool for proving that patterns hold for every number, not just the ones we happen to test. This chapter takes one humble rule, the distributive property a(b+c) = ab + ac, and shows just how far it can travel. We use it to predict how a product changes when its factors wobble up or down, to invent one-line tricks for multiplying by 11, 101 and 1001, and to unlock the three great algebraic identities, (a+b)^2, (a-b)^2 and (a+b)(a-b). Along the way we hunt for mistakes in other people’s algebra, discover that a single dot pattern can be described four different ways that all turn out to be equal, and trace distributivity back to Brahmagupta’s khaṇḍa-guṇanam, “multiplication by parts.”
Learning Outcomes
By the end of this chapter, you will be able to:
- state and apply the distributive property a(b+c)=ab+ac, including with negative numbers and more than two terms;
- predict how a product changes when its factors are increased or decreased, using the identity (a+m)(b+n)=ab+mb+an+mn;
- use distributivity to perform fast mental multiplication by 11, 101, 1001, 99, 999, \dots;
- expand and use the three key identities (a+b)^2=a^2+2ab+b^2, (a-b)^2=a^2-2ab+b^2 and (a+b)(a-b)=a^2-b^2;
- spot and correct common algebraic mistakes, and prove that two differently-written expressions are actually equal;
- appreciate that one pattern can have many correct descriptions, and trace distributivity through the history of Indian mathematics.