Properties of Operations

Commutative Property

Associative Property

Additive Identity Property

Multiplicative Identity Property

Multiplicative Property of Zero

counterexample

the ORDER in which numbers are ADDED or MULTIPLIED does not change the SUM or PRODUCT

Addition Symbols: x + y = y + x

Addition Example:  3 + 5 = 5 + 3

Multiplication Symbols: a * b = b * a

Multiplication Example:  2 * 4 = 4 * 2

the way in which numbers are GROUPED when they are added or multiplied does not change the SUM or PRODUCT

Addition Symbols: a + (b + c) = (a + b) + c

Addition Example: 2 + (3 + 8) = (2 + 3) + 8

Multiplication Symbols: a * (b * c) = (a * b) * c

Multiplication Example: 3 * (4 * 5) = (3 * 4) * 5

A property is a statement that is true for any number.

When 0 is added to any number, the sum is the number.

Symbols: 0 + x = x

Example: 0 + 9 = 9

When any number is multiplied by 1, the product is the number.

Symbols: x * 1 = x

Example: 1 * 5 = 5

When any number is multiplied by 0, the product is 0.

Symbols: x * 0 = 0

Example: 3 * 0 = 0

You may wonder if any of the properties apply to subtraction or division.

If you can find a counterexample, an example that shows that a conjecture is false, the property does not apply.

Division of whole numbers is commutative.

Write two division expressions using the Commutative Property.

 $15\ \div\ 3\ =\ 3\ \div\ 15$  <-- conjecture 

 $5\ \ne\ \frac{1}{5}$  <-- Divide 

The conjecture is false. We found a counterexample. That is  $15\ \div\ 3\ \ne\ 3\ \div\ 15$ .  So division is NOT commutative.

The difference of two different whole numbers is always less than both of the numbers.

False

Counterexample:

 $\left(7\ +\ g\right)\ +\ 5$  

(7 + g) + 5 = (g + 7) + 5  ---- Commutative Property of Addition 

(7 + g) + 5 = g + (7 + 5) --- Associative Property of Addition

= g + 12 --- Simplify 

(m * 11) * m = (11 * m) * m

(m * 11) * m = (11 * m) * m --- Commutative Property of Multiplication

(m * 11) * m = 11 * (m * m) --- Associative Property of MMultiplication

 $11m^2$  -- Simplify