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

Think: The answer must remain identical (the same) in value. 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.

1 Multiplicative Identity: ONE is the multiplicative identity. How do you keep the same answer when multiplying? Multiply by one. -11 • 1 = -11 x • 1 = x

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 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. Think: Opposites Cancel

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.

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.

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 means a change of GROUPING. 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 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. 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.