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# 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: XY = YX

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

Multiplicative  Inverse: For any real number X where X0, 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 ) = XY + XZ    (left distributive law) and  (X + Y) Z = XZ + YZ  (right distributive law).  This property also allows like terms to be combined so AX + BX  = (A + B)X.
• Distributive Property Extension: Like Terms May Be Combined
AX + BX  = (A + B)X.
• Distributive Property Extension: The Distributive Property may be used multiple times
(A + B)(C + D)  = AC + AD + BC + BD.
• 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 = kB or B = kA  for 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 =AB 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.

*Note: All properties on this page apply to complex numbers as well.

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