Algebra 1 Glencoe McGraw-Hill JoAnn Evans Math 8H Properties (Equality, Arithmetic, Identity)

Algebra 1 Glencoe McGraw-Hill JoAnn Evans

Math 8H

Properties

(Equality, Arithmetic, Identity)

Identity Property

Additive Identity: ZERO is the additive identity.

How do you keep the same answer when adding?

Add zero. 5 + 0 = 5 x + 0 = x

Think: The answer must remain identical (the

same) in value.

0

How do you keep the same answer when multiplying? Multiply by one.

-11 • 1 = -11 x • 1 = x

Multiplicative Identity:

ONE is the multiplicative identity.1

Multiplicative Property of Zero

93 • 0 = 0 2 • 0 = 0

x • 0 = 0

Think: The answer MUST be zero if you are

multiplying by zero.

Inverse Property

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.

Think: Opposites Cancel

1x1

x

1

81

8

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

23 + 19 = 42 42 = 23 + 19

a = b b = a

Think: The expressions on the two sides of the equal

sign can change places with each other since they’re equal (symmetrical).

Transitive Property

Think: Logical Reasoning

If a = b and b = c, then a = c.

126

21

,126

63

63

21

thenandIf

Substitution Property

If x = 2, then 5x = 5(2).

If y = 7, then y + 3 = (7) + 3.

Think: A quantity may be substituted for its equal.

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

Commutative Property of Addition: 2 + 3 = 3 + 2 a + b = b + a

Commutative Property of Multiplication: 4 • 7 = 7 • 4

a • b = b • a

Think: It’s okay to Change the Order. (first two letters of the word commutative)

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)

Associative Property of Multiplication:

(1 • 2) • 3 = 1 • (2 • 3)

Remember:

Associative Property means a change of GROUPING.

(a • b) (c) = a • (b • c)

Property of Negative One

Negative one • any number = the opposite of the number.

A negative coefficient is a coefficient of negative one.

Think: A number times negative one equals its opposite.

-1 • 8 = -8 -1 • -3 = 3

-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?

Counterexamples: 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.

21

42 326 No. No.

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.