Expressions/Equations Rev 10/8/06 P. W. Thompson Worksheet 3, Page 1 Worksheet 3 Properties of Operations When something is always true about an operation it is called a property of that operation. Addition and multiplication have several properties. One is called commutativity; the other is called associativity. Part I: Commutative Property of Addition Commutative Property of Multiplication When we write (5+4)=(4+5), we are stating that the when we calculate the two expressions we will get the same number. When we write (x+y)=(y+x) we are stating that regardless of the numbers we add, if we change their order, when we add them we will get the same number. The commutative property of addition says that you may add quantities in any order and still get the same result. As below, when you have two lengths a and b, you get the same length whether you put a first or b first. The result, (a + b) is the same either way. abab The commutative property of multiplication says that you may multiply quantities in an order and you will get the same result. This is illustrated by computing the area of a rectangle. It doesn’t matter which side you consider the width, you will get the same area either way. b ab You can change the order in which you add or multiply two numbers and the expression’s value will not change. That is, you can change ♦ + ∆ to ∆ + ♦, or you can change ♦ * ∆ to ∆ * ♦, and the result will not change. Please keep in mind that ♦ and ∆ do not have to be numbers. They can be expressions that evaluate to a number. The next example shows this.Expressions/Equations Rev 10/8/06 P. W. Thompson Worksheet 3, Page 2 Start with (2 * 7) + (8 – 5). The tree diagram for this expression shows that it is the sum of 2*7 and 8-5. +*-(2 * 7) + (8 - 5)2785 We can use the commutative property of addition to change (2 * 7) + (8 - 5), into (8 – 5) + (2 * 7). Commute The result is the expression (8 - 5) + (2 * 7). This means that the expression (8 - 5) + (7 * 2) is equivalent to (8 - 5) + (7 * 2), because they still evaluate to the same number, namely 17. You can also apply the commutative property of multiplication to the sub-expression (2 * 7). +*- (8 - 5) + (2 * 7)2785+*- (8 - 5) + (7 * 2)2785Commute The result is the expression (8 - 5) + (7 * 2). This means that the expression (8 - 5) + (7 * 2) is equivalent to (8 - 5) + (2 * 7), because they still evaluate to the same number, namely 17.Expressions/Equations Rev 10/8/06 P. W. Thompson Worksheet 3, Page 3 Not every operation is commutative. Subtraction is not commutative. 8 – 5 is not the same number as 5 – 8. The expression 8 – 5 is 3. The expression 5 – 8 is -3. That is, subtraction is not commutative, because changing the order of the operands may change the value of the expression. What other operations are not commutative? Your Turn Apply COMMUTE to each of the following expressions. Circle the operation to which you will apply the COMMUTE operation, then write the expression and draw the expression tree as they will appear afterward. a. -*- (8 - 5) - (2 * 7)2785 ________________ b. // (u + 5) / (2 / 7)27u5 ________________+ c. On a separate sheet: Apply COMMUTE to the expressions below that allow it (and write the resulting expression). Explain why the other expressions do not allow COMMUTE. NOTE: It might be useful to draw expression trees before deciding. i) a / b - 4 ii) (x / y) + (u ^ v) iii) 3 + k / (m * 5) iv) 5 * y + u / t + 7Expressions/Equations Rev 10/8/06 P. W. Thompson Worksheet 3, Page 4 Associativity You can change the way that three numbers are grouped (or associated ) when they are all added or when they are all multiplied without changing the expression’s value. Re-associate from left to right: (a + b) + c is the same number as a + (b + c) (a * b) * c is the same number as a * (b * c) Re-associate from right to left: a + (b + c) is the same number as (a + b) + c a * (b * c) is the same number as (a * b) * c Example: ASSOCL(2 * 7) * -42 * (7 * -4) ASSOCL (“assoesh-L”) re-associates expressions from having the grouping on the left to having the grouping on the right, changing (2 * 7) * -4 to the expression 2 * (7 * -4). ASSOCR (“assoesh-R”) re-associates expressions from having the grouping on the right to having the grouping on the left, changing 2 * (7 * -4) to the expression (2 * 7) * -4. Not every set of operations on three numbers is associative. You cannot re-associate (2*7)+8. The expression (2*7)+8 evaluates to 22. The expression 2*(7+8) evaluates to 30. These expressions are not equivalent. (2 * 7) + 82 * (7 + 8) In order to apply associativity, the expression must have two operation signs and they must be both addition or both multiplication. That is, ASSOCL and ASSOCR can be applied only when you have a sum of a sum and something or a product of a product and something. Questions: • Is division associative? That is, is (x / y) / z always equal to x / (y / z)? Try some examples. • What would it mean that subtraction is associative? Is subtraction associative?Expressions/Equations Rev 10/8/06 P. W. Thompson Worksheet 3, Page 5 • What would it mean that exponentiation is associative? Is exponentiation associative? Using Associativity of Addition and Multiplication with Complex Expressions The operands in an expression do not have to be numbers in order to use associativity. The operands can be expressions, too. Example ASSOCR changed 2*7 + (8/4 + 3*-2) to (2*7 + 8/4) + 3*-2. This follows the pattern of changing a + (b + c) to (a + b) + c. In this case 2*7 is a, 8/4 is b, and 3*-2 is c. All that matters is that you have three things, and that they are all added or all multiplied. Your Turn Apply ASSOCL and/or ASSOCR to each of the following expressions. Circle the operation to which you will apply the ASSOCL or ASSOCR operation, then write the expression and draw the expression tree as they will appear afterward. a. ((u + 5) * (2 / 7)) * 827u5**+/8Expressions/Equations Rev 10/8/06 P. W. Thompson Worksheet 3, Page 6 b. g + ((u - 5) + 4)u5-g++4 c. Circle the expressions to which neither ASSOCL nor ASSOCR can be applied. Explain why they cannot be used. It might be useful to draw expression trees before deciding. i) a + b * 4 ii) (x + y) + (u + v) iii) 3 …