Assume that p is any real number, q is any real number and r is any real number. NUMBER OPERATIONS: PROPERTY // ADDITION // MULTIPLICATION CLOSURE p+q is a real number and pq is a real number. COMMUTATIVE p+q = q+p // pq = qp ASSOCIATIVE p+(q+r) = (p+q)+r // p(qr) = (pq)r IDENTITY p+0 = p = 0+p // p1 = p = 1p INVERSE (0 is not equal to 1) p+(-p) = 0 // p(1/p) = 1 DISTRIBUTIVE (FACTORING OF COLLECTING SIMILAR TERMS) p(q+r) = pq+pr ZERO PRODUCT 0p = p0 =0 FACTORS OF ZERO (WHEN THEY EXIST) pq = 0 implies p = 0 or q = 0 NUMBER RELATIONS: PROPERTY // EQUALITY p = q // INEQUALITY p is less than q REFLEXIVE p = p // p is not less than p SYMMETRIC If p = q then q = p. // If p is less than q, then q is not less than p. TRANSITIVE If p = q and q = r then p = r. // If p is less than q and q is less than r, then p is less than r. SUBSTITUTION Any number, letter or algebra combination of numbers or letters may be substituted for p, q, or r in the properties listed above unless stated otherwise. also: If a = b, then b may be substituted for a in any statement. NUMBERS AND GEOMETRY The numbers p and q may locate points on one line so: p and q locate the same point when p = q. p and q locate different points when not equal to each other. If p and q locate points on a horizontal line then the absolute value of ( p - q ) gives the distance between p and q. Absolute value is written | p - q |. Also, if p and q locate points on a horizontal line and p is less than q, then we generally consider p on the left of q. In fact, p is less than 0 is another way to say p is negative. Geometry: distance between p and q corresponds to this absolute value, | p - q |, in algebra.