Basic Properties of Real Numbers:
Given that X, Y, Z, A, B, C, and D are real numbers, then the following are true:
Commutative Property of Addition: X + Y = Y + X
Commutative Property of Multiplication: X∙Y = Y∙X
Associative Property of Addition: (X+Y) + Z = X + (Y + Z)
Associative Property of Multiplication: (X∙Y) Z = X(Y∙Z)
Additive Identity: For any real number X, X + 0 = X where 0 is the additive identity
Additive Inverse: For any real number X, there exists -X such that X + (-X) = 0
Multiplicative Identity: For any real number X, 1∙X = X where 1 is the multiplicative identity
MultiplicativeInverse: For any real number X where X ≠ 0, there exists 1/X such that X ∙(1/X) = 1
Zero Product Law: If XY = 0, then X = 0 or Y = 0 or both X and Y = 0.
Distributive Properties:
· X(Y + Z) = X∙Y + X∙Z (left distributive law) and (X + Y)Z = X∙Z + Y∙Z (right distributive law). This property also allows like terms to be combined so A∙X + B∙X = (A + B)X.
· Distributive Property Extension: Like Terms May Be Combined
A∙X + B∙X= (A + B)X.
· Distributive Property Extension: The Distributive Property may be used multiple times
(A + B)(C + D) = A∙C + A∙D + B∙C + B∙D. Some people call this “F O I L”.
· NOTE: The Distributive Property Also Justifies Factoring.
Multiplication By A/A = 1: Given that "A" is any algebraic quantity or expression, multiplication of X by the fraction A/A does not change the value of a real or complex quantity X.
Division By A Monomial: (X + Y)/Z = X/Z + Y/Z. Note that this is an application of the Distributive Property since (X + Y)/Z = (1/Z)(X + Y) = (1/Z)∙X + (1/Z)∙Y = X/Z + Y/Z.
Fraction Cancellation Property: (A/B)∙B = (AB)/B = A. This is also known simply as Fraction Cancellation.
Direct Variation Definition: If two quantities A and B vary directly, then A = k∙B or B = k∙Afor some constant k. Direct variation is analogous to being directly proportional and varies directly as.
Inverse Variation Definition: If two quantities A and B vary directly, then A = k/B or B = k/A or k =A∙B for some constant k. Inverse variation is analogous to being inversely proportional and varies inversely as.
Division by Zero Results in an Undefined Result: Any time you divide by zero, the result is an undefined result that is neither equal to a real number nor a complex number.