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 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 ) = 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.
