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Properties of OperatorsThe entities that the familiar artithmetic operators act on are the real numbers. Of course we are quite familiar with the four basic arithmetic operations:
These operators combine two values to produce a third value, and are called binary operations. We typically write a general operation as: r = x op y where x and y are the two initial values, r is the resulting value, and op is the operation that combines x and y. There are several basic properites of operators that are key to performing arithmetic and success in algebra. Commutative PropertyThe Commutative Property involves a single operation and states that the left-to-right order of the two values does not matter. That is:x op y = y op x The familiar operations of addition and multiplication have this property. Indeed, as examples: 3 + 4 = 4 + 3 = 7 and 5·7 = 7·5 = 35 Not all binary arithmetic operations are commutative. The operations of subtraction, division, and raise to a power do not have the commutative property. As examples: 5 − 8 = −3 whereas 8 − 5 = 3 6 ÷ 2 = 3 whereas 2 ÷ 6 = 1/3 23 = 8 whereas 32 = 9 The last example does not "look" like the others. However, if we keep everything on a line and use ^ as the "raise to a power" operator, then we can write it as: 2^3 = 8 whereas 3^2 = 9 In words, the commutative property of addition states that it does not matter if we add x to y or we add y to x. In terms of algebra, we can interchange the two values for addition and multiplication. A slightly different use involves the relational operators such as =, <, ≥, .... If we think of these as a comparison operator that produces a result of True or False, then they behave like other binary arithmetic operators taking two values and producing a single result. Clearly the = operator (as well as the ≠) has the Commutative Property. That is, we can interchange the two sides of an equation: x = 5 is the same as 5 = x. However, the <, ≤, >, and ≥ operators do not have the Commutative Property. If you interchange the two sides of an inequality, you must also change the "sense" of the operator; eg. change < to >. Note, that whereas addition and multiplication (as well as subtraction and division) acting on the Real Numbers are closed; that is, the result is also a Real Number; the relational operators are not closed with respect to the Real Numbers. They produce a result that is not a Real Number but rather a truth value. In an entirely different context, it generally does not matter when we mix two substances which we mix with the other. However, when mixing acid and water, in order to prevent splatter it is better to add the acid to the water rather than the reverse; thus. the chemistry mnemonic: Acid into water, like you had oughter! Associative PropertyThe Associative Property also applies to a single type of operator, but now to a combination of three values and two occurrences of the operation. It is a statement about which of the two operations we perform first. It is normally written as:(x op y) op z = x op (y op z) Here we have used the parentheses to specify which of the two operations we perform first. Note that the left-to-right order of the values does not change. The operations of addition and multiplication have the Associative Property. 4 + 5 + 6 = 9 + 6 = 15 and 4 + 5 + 6 = 4 + 11 = 15 3·2·5 = 6·5 = 30 and 3·2·5 = 3·10 = 30 subtraction, division, and raise to a power do not have the Associative Property as an example will illustrate: 8 − 3 − 2 = 5 − 2 = 3 whereas 8 − 1 = 7 Distributive PropertyThe Distributive Property involves two different operations and three values and gives a rule whereby we can change which operation we do first. Because the operations are different, the property is more complicated than the Associative Property. It is written as:x op1 (y op2 z) = x op1 y op2 x op1 z We have carefully retained the original order of the values. The value x occurs to the left of y and z on both sides of the equality. We can change the order only if the operation, op1 has the Commutative Property. For the ordinary arithmetic operators multiplication and addition have the Distributive Property. In the above definition, op1 is multiplication and op2 is addition or subtraction. As an example: 3·(2 + 4) = 3·2 + 3·4 Both of these evaluate to 18. The left-hand-side becomes 3·6 and the right-hand-side becomes 6 + 12. In algebra, we typically see this as: c(a + b) = ca + cb. For operations, op1, that have the Commutative Property we can write the basic definition as: (y op2 z) op1 x = y op1 x op2 z op1 x For multiplication and addition, an example is: (3 + 4)·2 = 3·2 + 4·2, with both sides evaluating to 14 There are other pairs of the arithmetic operations that also have the Distributive Property: "raise to power" and multiplication: (3·5)2 = 32·52 The left-hand-side is 152 = 225 and the right-hand-side is 9·25 = 225 However, and this is a common error, "raise to power" and addition do not have the Distributive Property. As an example: (3 + 4)2 = 72 = 49 whereas 32 + 42 = 9 + 16 = 25 |
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© 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by Lawrence Turner |
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