How the four operations become two
Awhile back I posted about fraction division; that post made me think of reciprocals, algebra, and even properties of addition and multiplication. You see, when you study abstract algebra, you get to concept of a group , or a ring , or a field - and all of those consist of a set of elements and one (group) or TWO (ring or field) operations. Yet, in school math, we always study the FOUR basic operations. Why are two operations enough in higher-level algebra? Where did they throw subtraction and division? Addition and multiplication Did you ever notice these similarities between addition and multiplication? Both addition and multiplication are commutative: a + b = b + a and ab = ba for all real numbers a, b. Division and subtraction are not. Boht addition and multiplication are associative: (a + b) + c = a + (b + c). Division and subtraction are not. There exists an element so that adding it to a number doesn't change it (ADDITIVE IDENTITY). We call it zero: 0 + x and x + 0 both e...