(M-4) Identities

In some algebraic manipulations entire expressions get multiplied. For instance, one can write

(a + b)c = ac + bc

This is not an equation but an identity, an expression true for any three numbers (a,b,c). For instance if a = 3, b = 7, c = 5, then

(3 + 7)(5) = (3)(5) + (7)(5) = 15 + 35 = 50

If the addition is performed first

(3 + 7)(5) = (10)(5) = 50

Identities do not add any information about the quantities which they contain, because they are true whatevers their values may be. They are however useful in reshuffling equations to new, cleaner forms. The identity written on top is actually one of the basic properties of numbers ("the distributive law"). From it one gets more generally

(a + b)(c + d) = (a + b)c + (a + b)d

which can be further broken up and which holds for any values of (a,b,c,d). In particular

(a + b)² = (a + b)(a + b) = (a + b)a + (a + b)b

= a² + ba + ab + b²

= a² + 2ab + b²

which is quite useful (you can try it out with some specific values for a and b). Similarly

(a - b) ² = (a - b)(a - b) = (a - b)(a) + (a - b)(-b)

= a² - ba - ab + b²

= a² - 2ab + b²

Again, the two last identities

(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²

hold for any values of a and b, and as will be seen, are very useful in proving Pythagoras' Theorem.