For the order of operations, remember this sentence:
"Please Excuse My Dear Aunt Sally."
Arithmetic Properties of Real Numbers
Let a , b , and c be any real numbers.
Category
Commutative
Associative
Distributive
Identity
Inverse
Multiplication
Equality
Real numbers General
Free World U
Property Name
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Distributive Property
Algebraic Representation
a +b =b +a
ab = ba
a + (b + c ) = (a + b ) + c
a (bc ) = (ab )c
a (b + c ) = ab + ac
P - parentheses
E - exponents
M - multiplication
D - division
A - addition
S - subtraction
What It Says in Words
Order of addition does not matter.
Order of multiplication does not matter.
Grouping does not matter in addition.
Grouping does not matter in multiplication.
A number times a sum equals the sum of the number times
each addend.
Property of Opposites in Products
-(a + b ) = (-a ) + (-b )
a +0=a
a ×1=a
a + (-a ) = 0
a × ¹⁄a = 1
if a × b = 0, then a or b = 0
a (-1) = -a
(-a )b = -ab and (-a )(-b ) = ab
Reflexive Property
Symmetric Property
a =a
if a = b , then b = a
Transitive Property
if a = b and b = c , then a = c
If one number equals a second number, and the second equals
a third number, then the first number equals the third
number.
Addition Property
Subtraction Property
Multiplication Property
Division Property
if a = b , then a + c = b + c
if a = b , then a - c = b - c
if a = b , then ac = bc
if a = b, c ≠ 0, then a /c = b /c
These four versions of the property of equality mean that
whatever arithmetic operations you do to one side of an
equation you are allowed to do to the other side, and the
equation is still true.
Trichotomy Law
for any real a and b ,
either a < b , a = b , or a > b
Density Property
for any real a and b , a < b , there
exists a real c such that a < c < b
Property of the Opposite of a Sum
Identity Property of Addition
Identity Property of Multiplication
Additive Inverse Property
Multiplicative Inverse Property
Zero-Product Property
Multiplicative Property of Negative One
The opposite of a sum is equal to the sum of the opposites.
Zero is the identity element of addition.
One is the identity element of multiplication.
(-a ) is the additive inverse of a .
¹⁄a is the multiplicative inverse (or reciprocal ) of a .
If the product of two numbers is zero, one of the factors is zero
Any number times negative one yields its opposite.
Factors with opposite signs yield a negative product; or two
negative factors yield a positive product.
A number equals itself.
If a number equals another, the other number equals the first.
For any two numbers, they must either equal each other, one
must be less than the other, or one must be greater than the
other.
There is always another real number in between two other
real numbers.
Ninth Grade - Algebra I - Algebra Basics - Using Arithmetic Properties: Operations and Properties