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Identity and Equality
Properties
Identity and Equality Properties
• Properties refer to rules that indicate a standard
procedure or method to be followed.
• A proof is a demonstration of the truth of a
statement in mathematics.
• Properties or rules in mathematics are the result
from testing the truth or validity of something by
experiment or trial to establish a proof.
• Therefore, every mathematical problem from the
easiest to the more complex can be solved by
following step by step procedures that are
identified as mathematical properties.
Identity Properties
• Additive Identity Property
• Multiplicative Identity Property
• Multiplicative Identity Property of Zero
• Multiplicative Inverse Property
Additive Identity Property
 For any number a, a + 0 = 0 + a = a.
 The sum of any number and zero is equal to
that number.
 The number zero is called the additive identity.
 Example:
If a = 5 then 5 + 0 = 0 + 5 = 5.
Multiplicative Identity Property
 For any number a, a  1 = 1  a = a.
 The product of any number and one is equal to
that number.
 The number one is called the multiplicative
identity.
 Example:
If a = 6 then 6  1 = 1  6 = 6.
Multiplicative Property of Zero
 For any number a, a  0 = 0  a = 0.
 The product of any number and zero is
equal to zero.
 Example:
If a = 6, then 6  0 = 0  6 = 0.
Multiplicative Inverse Property
 For every non-zero number, a/b,
a b
 1
b a
 Two numbers whose product is 1 are called
multiplicative inverses or reciprocals.
 Zero has no reciprocal because any number times 0 is
0.
3 4
 1
 Example:
4 3
4
3
The fraction
is the reciprocal of
.
3
4
The two fractions are multiplicative inverses of each other.
Equality Properties
• Equality Properties allow you to compute with expressions on both
sides of an equation by performing identical operations on both sides
of the equal sign. The basic rules to solving equations is this:
* Whatever you do to one side of an equation; You must perform the
same operation(s) with the same number or expression on the other
side of the equals sign.
• Substitution Property of Equality
• Addition Property of Equality *
• Multiplication Property of Equality
Substitution Property of Equality
 If a = b, then a may be replaced by b in any expression.
 The substitution property of equality says that a quantity may be
substituted by its equal in any expression.
 Many mathematical statements and algebraic properties are
written in if-then form when describing the rule(s) or giving an
example.
 The hypothesis is the part following if, and the conclusion is the
part following then.
 If 8 + 4 = 7 + 5; since 8 + 4 = 12 or 7 + 5 = 12;
 Then we can substitute either simplification into the original
mathematical statement.
Addition Property of Equality
 If a = b, then a + c = b + c or a + (-c) = b + (-c)
 The addition property of equality says that if you may add or
subtract equal quantities to each side of the equation & still have
equal quantities.
 In if-then form:
 If 6 = 6 ; then 6 + 3 = 6 + 3 or 6 + (-3) = 6 + (-3).
Subtraction Property of Equality
 If a = b, then a -c= b -c or a - (-c) = b - (-c)
 The subtraction property of equality says that if you may
subtract equal quantities to each side of the equation & still have
equal quantities.
 In if-then form:
 If 6 = 6 ; then 6 - 3 = 6-3 or 6 - (-3) = 6 - (-3).
Multiplication Property of Equality
If a = b, then ac = bc
The multiplication property of equality says that if you may
multiply equal quantities to each side of the equation & still
have equal quantities.
In if-then form:
 If 6 = 6 ; then 6 * 3 = 6 * 3.
Division Property of Equality
The
Property
of Equality
TheDivision
Division
Property
of states that if
you divide both sides of an equation by the same
Equality
states
thatremain
if youequal
divide both
nonzero number,
the sides
sides of an equation by the same
nonzero
If
6=6, thennumber,
6/3=6/3 the sides remain
equal
Associative Property
The associative property states that
you can add or multiply regardless of
how the numbers are grouped. By
'grouped' we mean 'how you use
parenthesis'. In other words, if you are
adding or multiplying it does not matter
where you put the parenthesis. Add
some parenthesis any where you like
Commutative Property
Commutative Property is the one
that refers to moving stuff around. For
addition, the rule is "a + b = b + a"; in
numbers, this means 2 + 3 = 3 + 2.
For multiplication, the rule is "ab = ba";
in numbers, this means 2×3 = 3×2