Transcript Math 8H Properties (Equality, Arithmetic, Identity) Algebra 1 Glencoe McGraw-Hill JoAnn Evans Identity Property Think: The answer must remain identical (the same) in value. Additive Identity: ZERO is the additive.

Math 8H
Properties
(Equality, Arithmetic, Identity)
Algebra 1
Glencoe McGraw-Hill
JoAnn Evans
Identity Property
Think: The answer must
remain identical (the same)
in value.
Additive Identity: ZERO is the additive identity.
How do you keep the same answer when adding?
Add zero.
5 + 0 = 5
x + 0 = x
0
Multiplicative Identity:
ONE is the multiplicative identity.
1
How do you keep the same answer when
multiplying? Multiply by one.
-11 • 1 = -11
x • 1 = x
Multiplicative Property of
Zero
Think: The answer MUST
be zero if you are
multiplying by zero.
93 • 0 = 0
2•0=0
x • 0 = 0
Inverse Property
Think: Opposites Cancel
Additive Inverse: A number plus its additive inverse
(opposite) equals ZERO.
9 + (-9) = 0
x + -x = 0
Multiplicative Inverse: A number times its multiplicative
inverse (RECIPROCAL) equals ONE.
1

8  1
8
1

x  1
x
Reflexive Property
Think: Reflexive = Reflection
(like a mirror)
x=x
3=3
x+2=x+2
This may seem painfully obvious, but it is an
essential property of equality. It clearly
shows the role of the equal sign as stating
thatthe two sides of an equation are equal.
x + 2
x + 2
The Symmetric Property
Think: The expressions on the
two sides of the equal sign can
change places with each other
since they’re equal
(symmetrical).
23 + 19 = 42
a = b
42 = 23 + 19
b = a
Transitive Property
Think: Logical Reasoning
If
1
3

and
2
6
3
6

then
6
12,
1
6

2
12
If a = b and b = c, then a = c.
Substitution Property
Think: A quantity may be
substituted for its equal.
If x = 2, then 5x = 5(2).
If y = 7, then y + 3 = (7) + 3.
Distributive Property
Think: Distribute
(pass out) the
multiplication to each
term.
2(3x + 5y + 4)
= 2(3x + 5y + 4)
= 6x + 10y + 8
a(b + c) = ab + ac
Commutative Property
Think: It’s okay to
Change the Order.
(first two letters of the word
commutative)
Commutative Property of Addition:
2+3=3+2
a + b = b + a
Commutative Property of Multiplication:
4•7=7•4
a • b = b • a
Associative Property
Think: a change of association
(an association is a group)…
Associative Property means a
change of GROUPING.
Associative Property of Addition:
(1 + 2) + 9 = 1 + (2 + 9)
(a + b) + c = a + (b + c)
Remember:
Associative Property means a change of
GROUPING.
Associative Property of Multiplication:
(1 • 2) • 3 = 1 • (2 • 3)
(a • b) (c) = a • (b • c)
Property of Negative One
Think: A number times
negative one equals its
opposite.
Negative one • any number = the opposite of the number.
-1 • 8 = -8
-1 • -3 = 3
A negative coefficient is a coefficient of negative one.
-x = (-1)x
And finally………………The Closure Property
A set of numbers is CLOSED
under an operation if the
result of the operation (the
answer) is in the same number
set as the two numbers used in
the operation.
Example:
Is the set of even integers closed under
the operation of division?
In other words…When you divide an
even integer by an even integer, is the
answer an even integer?
6 2  3
No.
1
24 
2
Counterexamples:
No.
6 divided by 2 results in an odd answer.
2 divided by 4 results in a fractional answer.
The set of even integers is not closed under division.
Example:
Is the set of odd integers closed
under the operation of multiplication?
In other words…When an odd integer
is multiplied times another odd integer,
is the answer an odd integer?
7 • 5 = 35
3 • 11 = 33
5 • 13 = 65
The set of odd integers is closed under the
operation of multiplication.